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+IEEE ROBOTICS AND AUTOMATION LETTERS, PREPRINT VERSION. ACCEPTED DECEMBER, 2022
+1
+Learning-based Design and Control for
+Quadrupedal Robots with Parallel-Elastic Actuators
+Filip Bjelonic1,2, Joonho Lee1, Philip Arm1, Dhionis Sako1, Davide Tateo2, Jan Peters2, Marco Hutter1
+Abstract—Parallel-elastic joints can improve the efficiency and
+strength of robots by assisting the actuators with additional
+torques. For these benefits to be realized, a spring needs to be
+carefully designed. However, designing robots is an iterative and
+tedious process, often relying on intuition and heuristics. We
+introduce a design optimization framework that allows us to co-
+optimize a parallel elastic knee joint and locomotion controller
+for quadrupedal robots with minimal human intuition. We design
+a parallel elastic joint and optimize its parameters with respect to
+the efficiency in a model-free fashion. In the first step, we train a
+design-conditioned policy using model-free Reinforcement Learn-
+ing, capable of controlling the quadruped in the predefined range
+of design parameters. Afterwards, we use Bayesian Optimization
+to find the best design using the policy. We use this framework to
+optimize the parallel-elastic spring parameters for the knee of our
+quadrupedal robot ANYmal together with the optimal controller.
+We evaluate the optimized design and controller in real-world
+experiments over various terrains. Our results show that the new
+system improves the torque-square efficiency of the robot by 33 %
+compared to the baseline and reduces maximum joint torque by
+30 % without compromising tracking performance. The improved
+design resulted in 11 % longer operation time on flat terrain.
+Index Terms—Legged Robots, Reinforcement Learning, Com-
+pliant Joints and Mechanisms, Mechanism Design
+I. INTRODUCTION
+T
+HE quest of creating a single versatile, efficient and
+strong robotic platform has driven research in legged
+robotics for many years. While controllers are getting more
+robust and intelligent, locomotion performance is limited by
+the available joint speed and joint torque. Better performance
+can be achieved by creating more efficient and powerful actu-
+ators. Adding elastic elements has the promise of supporting
+the actuators with additional torque [1].
+In this letter, we explore the effect of the elastic component
+on energy efficiency during locomotion by attaching a parallel
+spring mechanism on the knee joints of the ANYmal robot
+(Fig. 1). This system is used to experiment and verify the
+benefit of the parallel elasticity.
+A. Robots with elastic actuators
+One of the first approaches in this direction was the Series
+Elastic Actuator (SEA) by Gill Pratt [2] which incorporates
+a series-elastic element between the actuator and the load.
+This design makes the joint positioning error-tolerant, reduces
+Manuscript received: August 27, 2022; Revised November 21, 2022;
+Accepted December 16, 2022.
+This paper was recommended for publication by Editor Abderrahmane A.
+Kheddar upon evaluation of the Associate Editor and Reviewers’ comments.
+This work was supported by a fellowship within the IFI program of the
+German Academic Exchange Service (DAAD).
+1 Authors are with ETH Zurich; Robotic Systems Lab; Leonhardstrasse 21,
+8092 Zurich, Switzerland.
+2 Authors are with TU Darmstadt; Intelligent Autonomous Systems Lab;
+Hochschulstrasse 10, 64289 Darmstadt, Germany
+Digital Object Identifier (DOI): see top of this page.
+Fig. 1.
+The ANYmal robot with parallel-elastically actuated knee joints.
+ANYmal is walking upstairs at the central station of Zurich, which is used
+as one of the experimental sites during this work.
+impact loads, and, most importantly, allows for precise torque
+measurement. The ANYmal quadrupedal robot [3] integrates
+into its ANYdrive actuator a serial elastic spring. More exam-
+ples are ATRIAS [4], a biped that has serial elastic springs
+at the actuator level and Cassie [1] with a 6-bar linkage
+with 2 springs in series. HyQ [5] has a serial elastic spring
+between the knee and the foot of the robot, which reduces foot
+chattering during touch-down.
+Another approach is the Parallel Elastic Actuator (PEA). In
+this setup, the actuator and the spring are in parallel. While
+this approach has been studied in robotic manipulation for
+gravity compensation [6], for pick-and-place [7] and efficient
+oscillation [8], there is no comparative evaluation of walking
+robots with PEA outside of controlled lab environments. One
+example of a legged robot with PEAs is SpaceBok [9]. In a
+lab experiment with simulated moon gravity, PEAs reduced
+the energy required for a jump by a factor of two on this
+robot [10]. Another, more recent example of using PEA is
+BirdBot [11], which has a parallel elastic spring clutching
+mechanism, spanning multiple joints. The avian-inspired leg
+design shows self-stable and robust bipedal locomotion while
+requiring 10 % of the knee-flexing torque compared to a non-
+clutching parallel spring setup. Another example is STEPPR
+[12]. This bipedal robot has a parallel-elastic spring at the hip
+and the ankle. Using only the hip springs during walking, the
+robot consumes 31 % less joint electrical power and reduces
+power consumption overall by 13 %.
+All of the previous works mention the possibility of saving
+energy with the carefully designed springs. Unfortunately,
+most of them are designed based on heuristic and cannot
+exploit the full potential of elastic elements.
+Building upon intuitive design, a common approach starts
+with mimicking nature’s counterparts [13]. Atrias [4] and
+arXiv:2301.03509v1  [cs.RO]  9 Jan 2023
+
+2
+IEEE ROBOTICS AND AUTOMATION LETTERS, PREPRINT VERSION. ACCEPTED DECEMBER, 2022
+BirdBot [11] for instance are inspired by ostriches and the
+emu. The problem with bio-inspired design is the high amount
+of variables that need to be taken into account to fully
+model the targeted animal accurately. Nevertheless, there is
+no systematic way of designing robots in general.
+B. Computational design
+Computational robot design can be divided into gradient-
+based methods that work well with deterministic differentiable
+objective functions, gradient-free algorithms with smooth ob-
+jectives (e.g. trust-region methods), meta-heuristic methods
+that are nature inspired (e.g. simulated annealing, genetic al-
+gorithms), and surrogate methods (e.g. Bayesian Optimization
+(BO)) [14]. Meta-heuristic and surrogate methods have been
+successfully used in black-box optimization problems, where
+the properties of the objective function are not known in
+advance [15] [16].
+A work related to the goal in this work has been done
+by Scalera et al. [7] where the design optimization of elastic
+elements was carried out for a four Degrees of Freedom (DoF)
+parallel robotic arm. Here, the robot achieved an efficiency
+gain of 67 % on a predefined trajectory by defining a non-
+linear optimization problem for finding energy optimal spring
+parameters. This approach is unsuitable for legged locomotion
+since it optimizes over a fixed trajectory that is by no means
+guaranteed to be optimal.
+The approach from De Vincenti et. al. [17] uses a differ-
+entiable trajectory tracking controller such that the overall
+leg design optimization becomes control-aware. Effectively,
+the gradient computation takes the control formulation into
+account in each step. Nevertheless, the trajectory is still fixed
+for all the tasks.
+A co-optimization approach is developed by Dinev et. al.
+[18] for leg lengths, joint positions, trunk shape, and weight
+distribution. Here, motion planning is recomputed in every
+evaluation of the design process. Using finite differences, the
+design optimization increases the energy efficiency of the Solo
+robot by a factor of 3 and shows faster convergence than using
+an evolutionary optimizer (CMA-ES). Using finite differences
+on rough terrain may result in an unstable solver, making this
+approach hard to incorporate into our goals.
+In general, these methods incorporate a design optimization
+that is wrapped around the robot control and planning loop.
+While the approaches incorporate gradient-based or gradient-
+free solvers for the outer loop, the inner loop can be either
+fixed [7] [19], or efficiently re-optimized in every performance
+evaluation [18] [17] [20] [21].
+An interesting simultaneous approach from Chen et al. [22]
+defines a hardware policy besides the control policy, that is
+jointly optimized over the training process with model-free
+Reinforcement Learning (RL). The optimized weights of the
+hardware policy define the hardware parameters and, together
+with the control policy, create the output of the algorithm.
+While this is a fully integrated approach, defining the hardware
+policy as a computational graph is not possible in many cases
+[22].
+Another method by Schaff et al. [23] optimizes an RL policy
+and distribution of design parameters at the same time. The
+agent is able to observe design parameters while the design
+space slowly shrinks toward high-performing designs. This
+approach has been successfully applied on a soft robot crawler
+Shank
+Thigh
+Spring
+Disc
+(a) Design
+Shank
+Thigh
+(b) Elliptic Cam
+Fig. 2. Fig. 2a illustrates a generic two-segment leg with potentially nonlinear
+parallel elastic knee joints. The conceptual implementation of the rotatory
+spring stiffness k in this work is visualized in Fig. 2a. The linear elastic spring-
+wire mechanism connects the thigh with the shank. This creates a spring
+torque τs on the knee. Parts with the same color are physically connected.
+[24] and outperformed a baseline design from an expert with
+the optimal design walking more than 2× as fast.
+Inspired by the co-optimization approach from Dinev et. al.
+[18] and the learning-based approach by Chen et al. [22], the
+following section briefly introduces our design optimization
+framework as well as our main contributions.
+C. Contribution
+We present a systematic approach to designing elastic mech-
+anisms for legged robots by incorporating design-conditioned
+controllers in the optimization. In particular, we present:
+• Co-optimization of the design parameters and the loco-
+motion controller for the PEA-driven legged robot using
+model-free RL and BO.
+• Integration of the optimized design onto the physical
+system and sim-to-real transfer of the learned control
+policy.
+• Real-world experiments to demonstrate the feasibility and
+robustness of our approach followed by the quantitative
+evaluation.
+We would like to emphasize the last contribution because, to
+the authors’ knowledge, this paper provides the first evaluation
+of PEAs on walking robots outside of lab environments.
+II. METHOD
+In this section, we first present our PEA design and then
+present our framework to co-optimize the controller as well as
+design parameters. For any equation, vectors and matrices are
+marked in bold text. Further, we refer to specific legs by their
+position with respect to the base in the anterior and lateral
+direction with the left front (LF), right front (RF), left hind
+(LH), and right hind (RH) leg.
+A. Parallel Elastic Knee
+We design a PEA knee joint for quadrupedal robots seen
+in Fig. 2a. Particularly, a parameterization d ∈ D of the joint
+stiffness k is necessary. We design and implement a spring-
+wire mechanism (Fig. 2a). The wire connects the thigh and
+shank over a generic disc that defines the lever arm for the
+spring force. The disc is attached to the shank of the robot.
+The torque on the knee that is generated by this design can
+be calculated in general by
+
+BJELONIC et al.: LEARNING-BASED DESIGN AND CONTROL FOR QUADRUPEDAL ROBOTS
+3
+Fig. 3. The non-linear trajectory of the spring force’s lever arm ˆlr over on full
+rotation is plotted in pink. The radius of the major and minor axis is 3 and 1
+respectively, while the spring force is assumed to always point upwards. The
+length, as well as the angle of the lever arm, changes dynamically, depending
+on the angle of the knee.
+τs(q) = Fs × ˆlr(θ)
+(1)
+with Fs being the force created by the linear spring, θ ∈
+[0, 2π) defines the boundary of the cam and ˆlr(θ) is the spring
+force’s lever arm. The amplitude of the spring force can be
+calculated by Hooke’s law as fs = ||Fs|| = ks∆ls, with
+ks being the spring stiffness. The spring elongation ∆ls is
+influenced by the length of wire which is wrapped around
+the cam and the position of the lever arm. With this setup,
+the first parameter d1 is the equilibrium position ¯qKFE of
+the linear spring, which is defined as the knee angle where
+Fs = 0. Further parameters are added through the definition
+of the cam. Since the wire is always assumed to be in contact
+with the cam, the lever arm can be calculated by finding the
+point on the cam that is tangent to the spring force. This can
+be formalized by the following equation
+0 = Fs × ∂ˆlr
+∂θ .
+(2)
+We select an elliptic cam as a trade-off between simplicity
+and degrees of freedom of parameterization. In this case, this
+equation has always two solutions depending on the side at
+which the spring force acts. In our case, the left side of the
+lever arm respects the inequality
+�
+Fs, ˆlr(θ)
+�
+≥ 0.
+(3)
+Following, we describe the elliptic cam attached to the
+shank of the robot.
+1) Elliptic Cam: Elliptic cam is defined by
+lr(θ) = Rφ
+�
+a · cos(θ)
+b · sin(θ)
+�
+(4)
+Rφ =
+�
+cos(φ)
+−sin(φ)
+sin(φ)
+cos(φ)
+�
+φ = φ0 + qKFE,
+where θ ∈ [0, 2π) and φ0 being the initial angle of the ellipse
+with respect to the shank’s longitudinal axis at qKFE = 0 rad,
+a and b are the radius of the major and minor axis respectively,
+seen in Fig. 2b. Now, the lever arm ˆlr is not stationary and
+changes during the rotation of the knee. An example trajectory
+of the contact point over one full rotation of 360° for an ellipse
+with a = 3 and b = 1 is illustrated in Fig. 3.
+The contact point ˆlr can be calculated using (2) and (3).
+Fig. 4. The trajectories τ are collected with the trained design-conditioned
+policy in simulation. Afterward, the robot’s performance for a specific design
+choice is measured by a custom objective function f(D) and sent to the BO.
+Using this value, the algorithm builds a surrogate function (blue + cyan color
+in the left plot) and samples new points with respect to its acquisition function
+(green color in the right plot). The blue dots refer to already sampled points
+and the green dots to the next design set to be rolled out.
+Similarly, based on equations (1) - (4), we can compute the
+spring displacement by numerically solving an elliptic integral.
+We skip the derivations for the sake of space. The resulting
+torque is non-linear if a ̸= b
+τs(d) = ψ(qKFE, d)
+(5)
+with the design space d = [¯qKFE, a, b, φ0]T
+∈ R4. An
+animation of the design space is included in the supplementary
+video.
+B. Design Optimization
+Here we present our framework for optimizing the design
+parameters d. The general approach of our design optimization
+strategy is pictured in Fig. 4. We roll out trajectories with the
+design-conditioned policy (explained in section II-C) in the
+environment with each given set of design parameters. We
+define the objective function f for the design optimization
+by the Monte Carlo estimate over a large number of sam-
+ples collected in the simulation. By doing so, we evaluate
+the general performance of a design instance across many
+different scenarios with different initial states, disturbances,
+and commands.
+1) Design Objective: The main objective of our design op-
+timization problem is energy efficiency. Accurately simulating
+the efficiency of a robot is a difficult task due to various
+sources of energy consumption, e.g., mechanical energy at the
+actuators, power used to run computers and sensors, etc. We
+assume that the power loss of the system can be approximated
+by the joule heating of the individual actuators. There are
+other factors like transmission loss and electronics loss that are
+neglected. Joule heating is one of the major terms for energetic
+losses in electric motors and is proportional to the square of the
+actuator torque. Similar to the Cost of Transportation (CoT),
+we define the Cost of Torque (CoTr) as
+CoT ∝ CoTr =
+�
+τ 2dt
+mg∆s
+(6)
+with m being the total mass of the robot, g = 9.81 m/s2 the
+gravitational acceleration and ∆s the traveled distance by the
+robot. By normalizing with m, which depends on the design,
+
+2
+1
+F
+S
+0
+9
+-1
+Ellipse
+-2
+-3
+-2
+-1
+0
+1
+2
+C4
+IEEE ROBOTICS AND AUTOMATION LETTERS, PREPRINT VERSION. ACCEPTED DECEMBER, 2022
+and ∆s, this metric allows for the comparison of different
+designs and walking speeds.
+2) Optimization: The aim of the design optimization step is
+to find optimal design parameters d ∈ D for a specific task t ∈
+T with respect to an objective function f(d|t, π) : T ×D → R,
+given the pre-trained policy π. The objective f evaluates d for
+a fixed task t, which is velocity tracking on rough terrain in
+our setup, and outputs a performance measure.
+A task t defines the specific problem the policy solves.
+These parameters could be for example terrain property (rough
+terrain, stairs, etc.) as well as command amplitude and direc-
+tion. The task parameters are randomly sampled during policy
+training and design optimization.
+The objective f is defined by the physical quantities we
+are optimizing the design, e.g., joint torques or tracking per-
+formance (our setup), which are often not differentiable with
+respect to d. In our setup, we assume f is not differentiable
+because legged locomotion entails many discrete changes in
+dynamics due to foot contact. Thereby we use a black-box
+optimization method.
+The optimization problem can then be mathematically for-
+malized as
+d∗ = arg min Ed∈D,t∈T [f(d|t, π)]
+(7)
+s.t.
+0 ≤ c(d).
+We use the Heteroscedastic Evolutionary Bayesian Optimi-
+sation (HEBO) algorithm [25]. This BO algorithm won the
+NeurIPS2020 black-box optimization challenge [26]. The out-
+come of this challenge is the reason why we chose a surrogate
+method over a meta-heuristic method (compare Sec. I-B).
+C. Design-conditioned Policy
+It is important to have an optimal controller for each
+design instance to evaluate each design instance at its best
+performance. We assume that we can achieve near-optimal
+performance with a neural network policy conditioned on
+design parameters. A recent work by Won et al. [27] showed
+that it is possible to train a shape-conditioned policy for a
+bipedal robot through RL that can maintain a stable gait while
+the shape of its body is dynamically changing.
+Our policy training follows the approach, and we addition-
+ally adapt the privileged learning method by Lee et. al. [28]
+for sim-to-real transfer.
+We train two types of policies:
+• Design-conditioned policy (teacher): This policy directly
+observes design parameters and other environmental pa-
+rameters (e.g., terrain shape and friction coefficient),
+which we call privileged information, from simulation.
+The policy is used in the design optimization loop (see
+Fig. 4).
+• Deployment policy (student): This policy is deployed on
+the robot with noisy measurements as observation. This
+policy does not have access to privileged observations
+and observes the history of noisy proprioceptive measure-
+ments and exteroceptive measurements. The deployment
+policy is explained in section II-D.
+The design-conditioned policy is trained via RL in simulation
+and the student policy is trained via imitation learning with
+simulated sensor noises. Using temporally extended observa-
+tions, e.g., history of proprioceptive measurements [28] or
+noisy exteroception [29], the student policy can estimate the
+Teacher (Reinforcement Learning)
+Learning Environment
+Policy
+Update
+Fig. 5.
+The learning pipeline is adapted from the teacher-student approach
+[28]. The most important change is that the teacher directly observes the
+design parameter in the privileged observation.
+Fig. 6. We performed several tests with AoPS on rough terrain, showing its
+robustness. The policy was extensively tested in the mountains, the forests,
+and the City of Zurich.
+privileged information and adapt to the sim-to-real discrep-
+ancy.
+1) Reinforcement Learning: The design-conditioned policy
+is trained using RL. We model the RL problem as a Markov
+Decision Process (MDP), where the design-conditioned policy
+πθ defines the distribution of at ∈ A conditioned on the ob-
+servation ot ∈ O. The environment updates the robots state in
+each step according to a transition function p(st+1|st, at) and
+gives a reward rt(st, st+1, at). The objective is to maximize
+πθ∗(at|ot) → max E
+�
+�
+∞
+�
+˜t=t
+γ˜t−tr(a˜t, s˜t)
+�
+�
+(8)
+with γ ∈ [0, 1] being the discount factor.
+An MDP is defined by the 4-tuple of O, A, r, p. The state
+transition (p) follows rigid body dynamics in simulation. Each
+other component is explained below.
+We use the Proximal Policy Optimization (PPO) Algorithm
+[30] to update and train our policy.
+ot (∈ R133) contains the base target velocity commands,
+base orientation, base linear and angular velocity, parameters
+for the leg motion primitive (Foot Trajectory Generator (FTG)
+by [28]), a short history of joint positions and joint velocities,
+and the last two joint position targets. The privileged informa-
+tion in R46 includes contact friction, state and force at each
+foot, external forces and torques applied to the base, the design
+parameters, and the robot’s link masses.
+During the policy training, the design parameters are ran-
+domly sampled from D (compare Sec. II-B) per episode. In
+order to avoid tedious design calibration, we provide observa-
+
+contact states
+contact forces
+terrain profile
+contact friction
+disturbancesPolicyaesign params. o
+Encoder BJELONIC et al.: LEARNING-BASED DESIGN AND CONTROL FOR QUADRUPEDAL ROBOTS
+5
+(a) Assembled Spring Setup
+(b) Hip
+(c) Ellipse
+(d) Wire
+(e) Spring
+Fig. 7.
+These images of the robot as well as the individual parts show
+the spring-wire setup developed in this work. The green letters in Fig. 7a
+correspond to the 4 pictures on the right.
+tions of the equilibrium positions of the PEAs separately for
+each leg.
+2) Action: The agent controls the robot through at ∈ R16
+(compare Fig. 5), with the first 4 actions setting the frequency
+of the FTG [28] and 12 additional joint position deltas.
+The FTG outputs vertical foot trajectories with predefined
+clearance that are mapped to desired joint positions using
+inverse kinematics.
+3) Reward: The reward function includes a metric for
+following linear base commands in the x and y directions
+as well as the rotation along the yaw axis. Furthermore,
+we punish undesired movement in the base (z velocity, roll,
+and pitch angular velocity). For smooth and realistic torque
+commands, we penalize the acceleration with which the joint
+position targets change over time. For the agent to find optimal
+and efficient behavior, we penalize the L2 norm of the actuator
+torques. Lastly, we penalize joint velocities that exceed the
+actuator limits and foot slippage, which reduces foot strain
+due to sliding.
+4) Architecture: The design-conditioned policy is modeled
+as a Multi Layer Perceptron (MLP) and an auto-encoder
+network. The encoder network takes the privileged information
+and outputs an embedding vector ¯lt. Finally, the proprioceptive
+observations and this vector ¯lt are used as the input to the
+policy network (compare Fig. 5).
+D. Deployment Policy
+The deployed policy does not have access to privileged
+information. Instead, it uses a sequence of past observations
+to infer the unobserved state of the environment [29]. The
+student policy is constructed by a Recurrent Neural Network
+(RNN) [31] to effectively handle the sequential data. Similarly
+to Lee et al. [28], the training is done by imitation learning
+with an additional reconstruction loss for the embedding of
+the privileged information (¯lt).
+The observation of the deployed policy consists of pro-
+prioceptive measurements from the IMU and joint encoders
+and exteroceptive measurements from depth sensors. Both
+modalities are simulated with noise during the training, which
+is not added to the design-conditioned policy’s observation.
+The action space of the deployment policy is the same as the
+design-conditioned policy.
+An important factor for the sim-to-real transfer is to account
+for the model mismatch of the springs. During the student
+policy training, the design parameters are perturbed by 10 %
+from the optimized parameter to emulate limited manufactur-
+ing precision (see Sec. III-A). The design-conditioned policy
+observes the exact values as privileged information while
+the student policy does not have direct access to the design
+parameter.
+III. EXPERIMENTS
+We report the results of five different experiments to
+quantify the effectiveness of our approach as well as the
+performance gained by our new parallel elastic knee. The first
+experiment in Sec. III-B shows that our design optimization
+framework can find optimal parameters with respect to our
+design-conditioned policy in various tasks and with high
+repeatability. Experiments 2 and 3, in Sec. III-C and Sec. III-D
+respectively, are hardware experiments on flat terrain, showing
+that the parallel-elastic robot is more efficient than the baseline
+and requires less torque in forward walking as well as tracking
+random commands. The fourth experiment in Sec. III-E shows
+that the novel design can traverse difficult terrain. Lastly, Sec.
+III-F reports the last experiment, using the robot on a running
+track, which shows that the newly designed robot can operate
+longer with the same battery charge.
+A. Setup
+The task t for which the robot is optimized is forward
+walking at 1 m s−1 in an environment with stepping stones,
+flat terrain, and rough terrain with base perturbations of up to
+50 N force and 50 N m torque. The contact friction that the
+robot experiences is in the range µ = [0.5, 2]. The objective
+function f is chosen as the average reward
+f = 1
+N
+N
+�
+i=0
+r(ati, sti).
+(9)
+We use 1000 different episodes to estimate the expectation of
+the objective.
+We optimize the design parameters (II-A1) and build the
+elliptic cam in Fig. 2b for the hardware experiments. The
+physical parts that we created are illustrated in Fig. 7. Our
+final design consists of a linear spring with stiffness ks =
+4154 N m−1 and the four optimal design parameters, namely
+the radius of the major axis a = 8.1 cm and minor axis
+b = 6.0 cm, initial angle φ0 = 0.0 rad and the equilibrium
+position of ¯qKFE = 0.36 rad. The wires in Fig. 7d define
+the equilibrium position ¯qKFE of each leg and are due to
+manufacturing constraints not equally long. We randomize
+these values separately for each leg during the student training
+to account for unsymmetrical spring parameters. The policies
+use the spring exclusively in the pulling direction. Thus, we
+can implement the design with one tension spring per knee.
+After training the design-conditioned agent, we create two
+student policies. For the distillation, we fix our design pa-
+rameters in the demonstrations from the design-conditioned
+policy to the optimal design (parallel-elastic knee joint) and
+to a = 0 cm and b = 0 cm (rigid baseline). This allows us to
+create a comparative evaluation of having parallel elastically
+
+AAwmal6
+IEEE ROBOTICS AND AUTOMATION LETTERS, PREPRINT VERSION. ACCEPTED DECEMBER, 2022
+Fig. 8.
+This figure illustrates a contour plot of the design space in the
+case of a linear characteristic (using a circular shape). The objective is the
+Average Learning Reward. Additionally, we report 25 iterations of our design
+optimization framework progressing from blue dots to pink dots. The green
+line shows the first-principles design which is derived from a conventional
+design approach. The yellow star indicates the optimal design and the green
+star is the optimal first-principles design.
+actuated knee joints with respect to the baseline. The baseline
+is referred to as ANYmal and the optimal design as AoPS with
+a total mass of 51.3 kg and 52.5 kg respectively.
+B. Simulation-based Results
+This simulation-based experiment shows that our design
+optimization framework can find optimal design parameters
+within a given interval for PEAs. In order to visualize the
+result, we optimize the elliptic cam from Fig. 2b and set
+a = b = r. Therefore, since the design is point symmetric with
+the origin, this design has only 2 parameters d = [¯q, r]T ∈ R2.
+The plot in Fig. 8 shows a contour plot of the average learning
+reward in the design space D. The contour is obtained by
+sampling 40 points for each design parameter and 200 robots
+per design (320.000 simulated trajectories). Additionally, we
+report 25 iterations of our design optimization framework in
+Fig. 8. From the contour of the objective, it is observable that
+the optimal value lies around ¯q ≈ 0.0 rad and r ≈ 6.0 cm
+(yellow star). Within the first iterations, the framework is
+already close to the optimal value and still explores the design
+space for other optimal parameters.
+The green first-principles design curve in Fig. 8 is defined by
+a conventional design approach. This design compensates the
+gravity of the robot at the average joint configuration while
+walking with normal ANYmal, which is 1.3 rad. We would
+like to minimize the torque in the flight phase (q > 1.3) which
+results in ¯q being as small as possible. The optimal design
+(green star) is ¯q = 0.0 rad and r = 9.02 cm.
+The x-axis, where r = 0 cm, corresponds to our baseline
+since the torque is zero due to a zero lever arm. While the
+average reward differs in about 1 %, the optimal parameter
+reduces the CoTr by 33 % in comparison to the baseline.
+In contrast, the first-principles design reduces the CoTr by
+only 8 %. This shows that our design optimization effectively
+finds the best design parameters given the conditioned control
+policy. In this case, the highest average reward results in the
+lowest CoTr. Please note that the CoTr is a subset of the
+average reward CoTr ⊂ AverageReward (compare Sec. II-C).
+Using the full design space, we trained policies with 5
+different random seeds and optimized the parameters for
+forward walking at 1 m s−1 on flat terrain (see Fig. 9d). The
+standard deviation is below 3 ◦ for the angles (¯qKFE, φ0)
+and below 1 mm for the radii (a, b). This shows that our
+(a) Standing
+(b) Payload
+(c) Stairs
+(d) Flat
+(e) Rough
+Fig. 9. On the top row, 5 different environments are shown for which each
+design is trained and optimized. From left to right, the task is standing on
+flat terrain, carrying 20 kg payload on flat terrain, walking on stairs, flat-
+and rough terrain. The bottom row shows each optimal design found by
+our framework in the equilibrium position of the spring. For the hardware
+experiments, we built the design in 9e
+Fig. 10.
+This bar plot illustrates the efficiency gain by adding springs on
+AoPS (purple bars) compared to ANYmal (red bars). The former can reduce
+the needed torques to travel 15m by 32.8 % compared to the latter.
+design optimization method is repeatable and does not produce
+random designs over multiple runs.
+Finally, we optimize the design of the robot for 5 different
+tasks shown in Fig. 9 The walking experiments are optimized
+for 1 m s−1. The resulting designs in the bottom row show the
+knee configurations at the equilibrium positions and optimized
+cam shapes. This result shows the effectiveness of our method
+for finding different optimal designs depending on different
+scenarios.
+C. Forward Walking
+In this first hardware experiment, we compare the per-
+formance difference of ANYmal and AoPS on flat terrain,
+walking 16 m forwards and backward in a straight line (see
+supplementary video). Each robot walks 3× forward and 3×
+backward.
+The CoTr is visualized as a bar plot in Fig. 10. Both
+robots have only little variance in each test, with ANYmal
+experiencing a CoTr ≈ 12 N m s while AoPS drives the cost
+down to 8 N m s. On average, AoPS is 33 % more efficient
+with respect to CoTr than the baseline ANYmal.
+As shown by Fig. 10, our optimized design does not
+sacrifice the tracking performance for efficiency. Both AoPS
+and ANYmal could track the target velocity with an error
+less than 0.25m s−1. The figure shows slightly better tracking
+for AoPS, but the difference is negligible considering the
+confidence intervals (error bars).
+D. Random Command Tracking
+In our second hardware experiment, we test how the per-
+formance translates to a more versatile task. We send 10
+
+25
+10
+0.92
+Iteration in BO
+cm
+Radius r in 
+6
+0.91
+2
+First-principles design
+0.90
+0.0
+0.5
+1.0
+Equilibrium a in rad12
+ANYmal
+AoPS
+ms
+8
+9
+4
+2
+0 -
+Forwards
+BackwardsBJELONIC et al.: LEARNING-BASED DESIGN AND CONTROL FOR QUADRUPEDAL ROBOTS
+7
+Fig. 11. This figure compares the command tracking performance of ANYmal
+(red) and AoPS (purple) for the forward walking experiment. The dashed lines
+show the desired velocity in the x and y direction and around the yaw axis
+respectively.
+(a) ANYmal
+(b) AoPS
+Fig. 12.
+These two graphs show box plots of the torques needed for each
+joint separately in one experiment where the robots are tracking the 10 random
+commands. For readability reasons, only the LF leg is presented. Furthermore,
+the distribution for each joint torque is indicated by colored violin plots. This
+plot transfers similarly to the other legs as well.
+random commands for 3s each to the robots while the com-
+mands change dynamically (see supplementary video). The
+commands are randomly sampled between [−1.2, 1.2]m s−1
+in x direction, [−0.6, 0.6]m s−1 in the y direction, and
+[−1.2, 1.2]rad s−1 around the yaw axis and the same for
+both robots. The efficiency gain for the execution of all the
+commands is again around 30 % for AoPS while the tracking
+performance was similar to ANYmal.
+Additionally, Fig. 12 reports the joint torques for the left
+front leg of the robots as a boxplot with an overlaying violin
+plot. Regarding the KFE joint (knee), the average torque is
+around 26 N m for ANYmal in Fig. 12a while AoPS is around
+7 N m. Basically, the whole distribution shifts down thanks to
+the parallel elastic spring, which reduces the CoTr notably. As
+a result, the maximum absolute torque that AoPS needs for the
+same task is 52 N m, which is only 71 % of ANYmal (73 N m).
+Furthermore, the HFE joint average torque for AoPS is closer
+to 0 N m than ANYmal, while at the same time requiring less
+variance. This also drives down the CoTr. Expectedly, the
+HAA joint is unaffected by the parallel elastic spring, and
+for both systems mostly the same.
+E. Rough Terrain
+For the fourth and fifth tests, we adapted the perceptive
+learning from Miki et. al. [29] and included exteroceptive
+observations during the student distillation. Using this adapted
+policy, we performed several outdoor experiments with our
+parallel-elastic robot. We climbed several inclinations, tra-
+versed different types of stairs, went through confined spaces,
+walked over forest ground, inclined gravel paths, etc. A few
+snapshots are presented in Fig. 6 and videos in the supplemen-
+tary material. The robot did not fall once during the tests and
+reports the robustness of the controller and the novel design.
+Fig. 13.
+The state of charge for ANYmal (Red) and AoPS (Purple) over
+time during the experiment in Sec. III-E shows that our optimized design can
+achieve higher operating times with the same battery.
+TABLE I
+RUNNING TRACK PERFORMANCE
+AoPS
+ANYmal
+Number of Rounds
+7.5
+6.6
+Traveled distance [m]
+3000
+2640
+Initial Charge [%]
+92
+89
+Final Charge [%]
+11
+10
+Operation Time [min]
+68
+59
+Average Velocity [m/s]
+0.735
+0.740
+Efficiency [%]
+111
+100
+Outside Temperature [°C]
+31
+26
+This shows that adding parallel elastic springs does not affect
+the robustness negatively.
+F. Battery Life
+Finally, we used both robots sequentially on a running
+track of 400 m length and let the robots walk with the same
+battery until the battery was fully depleted. The battery was
+as much as possible fully charged before and after the first
+run with AoPS to ensure a fair evaluation. Both robots were
+commanded 1 m s−1 and carefully steered to stay in the inner
+path of the track. The performance of each robot is reported
+in Tab. I. This experiment shows that the overall traveled
+distance of our quadrupedal robot can be increased by at least
+11 % from 2640 m to 3000 m. We introduce the following
+efficiency metric as the quotient in covered distance scaled
+by the mismatch in battery charge (2 %).
+Efficiency = 3000 m
+2640 m ∗ 0.89 − 0.10
+0.92 − 0.11 = 1.11.
+(10)
+We also report the state of the charge over time in Fig. 13.
+Besides the faster drop for ANYmal, this shows that the battery
+that we used is internally calibrated and the linear scaling in
+(10) can compensate for the 2 % difference in charge.
+IV. CONCLUSION
+This paper shows that, with the co-optimization of the de-
+sign and controller, parallel springs on the knee of quadrupedal
+robots can increase locomotion efficiency without compromis-
+ing the command tracking performance and robustness. While
+it is well studied that gravity compensation with PEAs is
+energetically beneficial for static tasks [6], the PEA’s contri-
+bution during the dynamic locomotion is relatively unstudied.
+The effect of PEA is nontrivial during the locomotion since
+the actuators have to repeatedly work against the spring. A
+key takeaway of our work is that PEAs can also increase the
+performance during dynamic locomotion.
+We co-optimized design parameters and locomotion con-
+trollers that act optimally for a given set of design parameters
+
+- Command
+1.00
+1.00
+S
+ANYmal
+S
+Linear Velocity in m/
+AoPS
+0.75
+0.75
+0.50
+0.50
+0.25
+0.25
+0.00
+0.00
+-0.25
+-0.25
+Velocity  Velocity y Velocity :ANYmal
+80
+AoPS
+%
+.≤
+Charge
+60
+JO
+40
+State
+20
+0
+10
+20
+30
+40
+50
+60
+Time in min8
+IEEE ROBOTICS AND AUTOMATION LETTERS, PREPRINT VERSION. ACCEPTED DECEMBER, 2022
+and task. With a parallel elastic knee actuator designed by
+our approach, we could reduce the required joint torques,
+which yields a higher operation time for our quadrupedal robot
+ANYmal during locomotion.
+An important thing to note from our hardware experiments
+is the robustness of our controller to the model uncertainty,
+which shows the practical benefit of the RL-based control
+method. Trained by the privileged learning method [28] with
+randomized spring parameters, our controller tolerates possible
+model mismatches on the physical system without accurate
+spring calibration procedures, thus, removing the need to run
+any complex system identification routine.
+As we showed the potential of PEAs in legged robotics,
+further investigations in this direction have to follow. Firstly,
+the physical system’s energy consumption must be better
+modeled. This work assumes that the CoTr measurement is
+proportional to the battery life of the robot. Nevertheless,
+during the experiments in Sec. III-C and Sec. III-F, we found
+a discrepancy. There are unmodeled factors such as electrical
+and mechanical losses which we did not identify in this work.
+Secondly, the design-conditioned policy cannot be guaranteed
+to be as performant as a policy trained for each design
+parameter. The discrepancy was negligible in the setup covered
+in this paper. A previous study on this topic was conducted by
+us [32]. Lastly, a more elaborate design should be introduced.
+Our current design limits the workspace of the knee joint
+and the implementation of the cable-spring mechanism can be
+inaccurate. Additionally, research will be devoted to including
+other parameters in the design process like link masses or leg
+lengths.
+ACKNOWLEDGMENT
+The authors would like to thank the RSL Design Team for
+their insightful discussions and Marko Bjelonic for his great
+support on the Cluster and for helping with the state estimation
+on AoPS.
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+

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+Are high-energy photoemission final states
+free-electron-like?
+V.N. Strocov,1 L.L. Lev,1,2 F. Alarab,1 P. Constantinou,1 T. Schmitt,1
+T. J. Z. Stock,3 L. Nicolaï,4 J. Očenášek4 & J. Minár4
+1Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen-PSI, Switzerland
+2Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141701, Russia
+3London Centre for Nanotechnology, University College London, London WC1H 0AH, UK
+4University of West Bohemia, New Technologies Research Centre, 301 00 Plzeň, Czech Republic
+Abstract
+Three-dimensional (3D) electronic band structure is fundamental for understanding a vast diversity of
+physical phenomena in solid-state systems, including topological phases, interlayer interactions in van
+der Waals materials, dimensionality-driven phase transitions, etc. Interpretation of ARPES data in terms
+of 3D electron dispersions is commonly based on the free-electron approximation for the photoemission
+final states. Our soft-X-ray ARPES data on Ag metal reveals, however, that even at high excitation
+energies the final states can be a way more complex, incorporating several Bloch waves with different
+out-of-plane momenta. Such multiband final states manifest themselves as a complex structure and
+excessive broadening of the spectral peaks from 3D electron states. We analyse the origins of this
+phenomenon, and trace it to other materials such as Si and GaN. Our findings are essential for accurate
+determination of the 3D band structure over a wide range of materials and excitation energies in the
+ARPES experiment.
+
+Introduction
+Knowledge of electronic band structure resolved in three-dimensional (3D) electron momentum (k) is
+fundamental for understanding a vast diversity of physical phenomena in crystalline solid-state systems.
+Recently, the interest in 3D band structure has been boosted due to its essential role in topological
+phases such as Weyl semimetals characterised by 3D cones of linear electron dispersion (see, for
+example, refs. 1,2) as well as their generalisation to high-fold chiral fermions3,4 and high-dimensional
+degeneracies such as the Hopf links and nodal lines, chains and knots in 3D k-space (see the reviews5–8
+and the references therein). Less straightforward but equally important implications of the 3D band
+structure include, for example, interlayer interaction and 3D charge-density waves in van der Waals
+materials9–11, formation of quantum-well states at interfaces and heterostructures12–16 as well as
+minibands in semiconductor superlattices17, k-dependent electron-phonon interactions18,
+dimensionality-driven phase transitions19,20, 3D quantum Hall effect21, and many more properties of
+solid-state systems.
+High-energy angle-resolved photoelectron spectroscopy (ARPES), operating in the soft- and hard-X-ray
+photon energy (hv) regions, has pushed the k-resolving spectroscopic abilities of this technique from the
+conventional surface science to the intrinsic electronic structure deep in the bulk, buried interfaces and
+heterostructures, and diluted impurity systems (see the recent reviews22–26 and the references therein).
+The main advantage of high photoelectron energies is an increase of the photoelectron mean free path
+(λPE) to a few nanometres and more27. Crucial for the experimental determination of 3D band structure,
+the increase of λPE translates, via the Heisenberg uncertainty principle, to sharpening of the intrinsic
+resolution of the ARPES experiment in the out-of-plane momentum (kz) which is defined as Δkz = λPE
+-1 28.
+The sharp resolution in kz underlies the applications of high-energy ARPES for accurate determination of
+the electronic band structure resolved in 3D k-space as illustrated by many of the works cited above.
+In contrast to the in-plane momentum k// = (kx,ky), conserved in the photoemission process because of
+the in-plane periodicity of the system, the kz component is distorted upon the photoelectron escape from
+the crystal to vacuum. It can however be reconstructed based on its conservation in the photoexcitation
+process in the bulk (corrected for the photon momentum phv) if the final-state kz is known. Conventionally,
+the final-sate dispersion is modelled within the free-electron (FE) approximation, where kz is found as
+, with Ek and K// being the photoelectron kinetic energy and in-plane
+𝑘𝑧 =
+2𝑚 
+ħ
+𝐸𝑘 −
+ħ
+2
+2𝑚 𝐾//
+2 − 𝑉0 
+momentum, respectively, m the free-electron mass, and V0 the inner potential. Somewhat stretching this
+formula, an energy dependence of the dynamic exchange-correlation29,30 can be accommodated via an
+energy-dependent V0. Importantly, the FE approximation implies that the final-state wavefunction is a
+plane wave, where the finite λPE is described by an imaginary part of kz. It has since long been realised
+that at low excitation energies used in the conventional VUV-ARPES the FE approximation may in many
+cases fail even for metals31–34 and all the more for semiconductors35 and more complex materials, for
+example, transition metal dichalcogenides36–38. For high-energy ARPES, however, the relevance of this
+approximation is commonly taken for granted. Being quintessential for 3D band mapping with
+high-energy ARPES, this assumption is based on a physically appealing argument that at high excitation
+energies Ek of photoelectrons much exceeds modulations of the crystal potential V(r), and they can be
+considered as free particles.
+Here, we analyse soft-X-ray ARPES data on Ag the metal and demonstrate that even at high excitation
+energies the complexity of the final states can go far beyond the FE picture. In particular, they can be
+composed of multiple Bloch waves having different kzs which manifest themselves as complex structure
+of the spectral peaks or their excessive broadening. This analysis extends to GaN and Si the
+semiconductors. We theoretically demonstrate the origin of these non-trivial effects as resulting from
+
+hybridization of plane waves on the crystal potential, and elucidate how they should be taken into
+account for accurate determination of 3D valence-band dispersions in the high-energy ARPES
+experiment.
+Results
+Fig. 1 presents the Brillouin zone (BZ) of the fcc Ag (a) and the experimental out-of-plane cross-section
+of the Fermi surface (FS) in the ГXW symmetry plane measured under variation of hv (b). The indicated
+kzs, running through a sequence of the Г and X points, were rendered from the hv values assuming FE
+final states with V0 = 10 eV. In-plane cross-sections measured at two hv values, bringing kz to the Г and
+X points, are presented in the two panels (c). In general, the experimental out-of-plane FS follows a
+pattern of repeating rounded contours characteristic of the states near the Fermi level (EF) formed by the
+sp-band of Ag. This pattern is reproduced by our one-step ARPES calculations (e) where FE-like final
+states were used. Surprisingly, however, a closer look at the experimental FS reveals significant
+deviations: (1) Multiple FS contours, offset in kz, can be resolved in some (E,k) regions such as those
+marked by magenta arrows. The corresponding multiple dispersions coming from the sp-band are
+apparent, for example, in the ARPES image measured at hv = 997 eV (d, top) and the corresponding
+momentum-distribution curve as a function of kx at EF (kx-MDC, yellow line). This multiple-dispersion
+pattern contrasts to the clean dispersions at hv = 894 eV (d, bottom). As we discuss in more detail below,
+such replica spectral structures demonstrate that the final states incorporate multiple bands with different
+kzs – hereinafter called multiband final states (MBFSs) – which is a phenomenon beyond the
+conventional picture of FE-like final states implying one single band with one kz. In our case the
+separation of the kzs in these MBFSs is larger than the intrinsic Δkz (according to the λPE values from the
+TPP-2M formula, varying from ~0.15 Å-1 at 300 eV to 0.056 Å-1 at 1300 eV); (2) The second type of
+deviations from the FE final states, seen in the out-of-plane FS (b), is a notable spectral intensity
+spreading into the X points where the sp-band is unoccupied. Furthermore, broadening of the FS
+contours in kz irregularly varies through k-space, and in some (E,k) regions (such as those marked by
+yellow arrows) can be excessively large. These two effects are also caused by the MBFSs, but in this
+case the kzs are separated less than Δkz. We note that in the extremes of the E(kz) dispersion (dkx/dkz=0
+in the out-of-plane FS) the MBFSs have only a second-order effect on the ARPES structure; however,
+even in this situation a large enough kz separation within the MBFSs can cause multiple FS contours, as
+seen in the in-plane FS map measured at hv = 712 eV (c, magenta arrow). Obviously, the MBFS effects
+are not reproduced by the ARPES calculations (e) employing FE final states. Although presently on a
+qualitative level, these effects are reproduced by our one-step ARPES calculations (Supplemental
+Material) where the final states are treated within the multiple-scattering formalism, naturally
+incorporating the non-FE effects including the MBFSs.
+
+Fig. 1. FS cross-sections for Ag(100): Theoretical FS (a), its experimental out-of-plane cross-section (b),
+and two in-plane cross-sections (c) measured at the indicated hv values, bringing kz to the Г and X points
+(lower and upper panels, respectively). Replicas and broadening of the FS contours in certain (E,k)
+regions (such as those marked by magenta and yellow arrows, respectively) manifest MBFSs. These
+effects are particularly clear in the ARPES image and kx-MDC at hv = 997 eV (d, top) in contrast to those
+at hv = 894 eV (bottom). These effects are beyond the one-step ARPES calculations with FE-like final
+states (e).
+In Fig. 2, the theoretical E(k) along the ГX direction (a) is compared with the experimental out-of-plane
+band dispersions E(kz) at kx=0 (b) and the in-plane E(k//) images (c) measured at kz running through the
+successive Г points (energies as binding energies Eb relative to EF). Again, the gross structures of the
+experimental E(kz) follow the expected periodic pattern with the sp-band crossing EF as reproduced by
+our one-step ARPES calculations in (e) with the FE-like final states. We see, however, replicas and
+anomalous broadening of the sp-band (such as marked by magenta arrows) as well as significant
+spectral intensity around the X point. These anomalies appear most clearly in the zoom-in of the sp-band
+and the kz-MDC at EF (d, yellow line) where we observe a complex multi-peak structure of the spectral
+intensity around the X point. Again, these effects are manifestations of the MBFSs, with the ARPES
+dispersions originating from the individual final-state bands marked by the magenta arrows. Again, they
+are absent in the ARPES calculations employing FE final states (e) but are qualitatively reproduced upon
+inclusion of multiple-scattering final states (Supplemental Material). The MBFS effects could not be
+observed in the first soft-X-ray study on Ag(100) focused on the 3d states39 because the smaller kz
+dispersion of these states compared to the sp ones could not provide sufficient separation of the spectral
+peaks from the different bands in the MBFS. We note in passing that the experimental 3d states appear
+in ~1 eV below the LDA-DFT energies; such an energy shift, already noticed for Cu, is a pronounced
+self-energy effect due to non-local exchange interaction of the 3d electrons strongly localized in the core
+region40.
+
+b
+(d)
+a
+1200
+997 eV
+800
+894eV
+600
+572 eV
+400Fig. 2. Band dispersions along the ГX direction for Ag(100): Theoretical E(k) (a) compared with the
+experimental out-of-plane ARPES dispersions at kx=0 (b, the spectral intensity represented in
+logarithmic scale) and (c) in-plane dispersions for the indicated hv values, bringing kz to the successive
+Г point. A zoom-in of the sp-band (d) shows its replicas and excessive broadening (such as marked by
+magenta arrows) most evident in the kz-MDC at EF (yellow line) as multiple and broadened spectral
+peaks, manifesting the MBFSs. These effects are beyond the one-step calculations of the ARPES
+intensity and kz-MDC with FE-like final states (e).
+Discussion
+Origin of the MBFSs
+By definition, a FE-like final state in the crystal is one single plane wave ei(k+G)r which matches the
+outgoing photoelectron plane wave. In the whole multitude of bands, formally available under Ek and K//
+conservation, this plane wave corresponds to one single band that we will refer to as primary, relaying
+Mahan's primary photoemission cones 41. All other bands in the multitude give strictly zero contribution to
+the photocurrent. We will be calling them secondary, relaying Mahan's secondary cones. The MBFS
+effects, observed in our ARPES data, indicate that the corresponding final states may include, for given
+Ek and K//, several bands with different kzs giving comparable contributions to the ARPES intensity.
+These effects obviously fall beyond the FE-like picture. As the first-principles calculations can not yet
+exhaustively describe our experimental results, we will analyse the MBFS effects based on insightful
+model calculations.
+The non-FE effects in the final states, in particular their multiband composition, is certainly a
+phenomenon not new for low-energy ARPES. They have been studied experimentally and theoretically
+for 3D bulk band dispersions in various materials including Cu31,32, Mg34 and even Al the paradigm FE
+metal14,42, semiconductors35, various transition metal dichalcogenides36–38 as well as surface states, in
+particular for the Al(100) and (111) surfaces33. However, it is intriguing to observe such effects in our
+soft-X-ray energy range. Why do they appear in spite of the fact that the photoelectron Ek is
+overwhelmingly large compared to the V(r) modulations?
+
+(b)
+(d)
+(e)
+894eV
+1268 eV
+310eV
+572eVWe will now build a physically appealing picture of the non-FE effects in the photoemission final states
+using their standard treatment as the time-reversed LEED states43. They are superpositions of damped
+Bloch waves фk(r) with complex kz, whose imaginary part Imkz represents the (1) inelastic electron
+scattering, described by a constant optical potential Vi (imaginary part of the self-energy), and (2) elastic
+scattering off the crystal potential44–47. The amplitudes Ak of these фk(r), determining their contribution to
+the total ARPES signal, were determined within the matching approach of the dynamic theory of
+LEED17,38,44,45,48,49 where the electron wavefunction in the vacuum half-space (superposition of the
+incident plane wave eiK0r and all diffracted ones ei(K+g)r, g being the surface reciprocal vectors) is matched,
+at the crystal surface, to that in the crystal half-space (superposition of фk(r) satisfying the
+surface-parallel momentum conservation k//=K//+g). The underlying complex bandstructure calculations
+utilised the empirical-pseudopotential scheme, where фk(r) are formed by hybridization of plane waves
+ei(k+G)r, G being 3D reciprocal-lattice vectors. The Fourier components V��K = <ei(k+G)r|V(r)|ei(k+G')r> of the
+local pseudopotential V(r) were adjustable parameters.
+We start from the ideal FE case, where V(r) is constant and equal to V0 (so-called empty lattice). The
+corresponding calculations are plotted in Fig. 3 (a) as the E(Rekz) bands (the corresponding E(Imkz)
+bands are not shown here for brevity). Due to the absence of hybridization between the plane waves in
+the empty-lattice case, each фk(r) contains one single plane wave corresponding to a certain G vector.
+Typical of high energies, we observe a dense multitude of bands brought in by an immense number of all
+G vectors falling into our energy region. Starting from the ultimate V0 = 0 case, when the vacuum
+half-space is identical to the crystal one, it is obvious that only one band will couple to the photoelectron
+plane wave in vacuum eiKr and thus be effective in the ARPES final state, specifically, only the primary
+band whose plane wave – in the context of LEED often called conducting plane wave – has k+G equal to
+the photoelectron K. The whole multitude of the secondary bands, whose plane wave's k+G is different
+from K, will give no contribution to the photocurrent. In our more general case V(r) = V0, the kz
+component of the photoelectron distorts upon its escape to vacuum, and the above momentum-equality
+condition to identify the conducting plane wave should be cast in terms of the in-plane components as k//
++ G// = K//. In a formal language, these intuitive considerations can be expressed through the partial
+contributions of each фk(r) into the total current absorbed in the sample in the LEED process, which are
+the so-called partial absorbed currents Tk ∝ Vi⋅
+, with the integration extending from the
+0
+∞
+∫ 𝐴𝑘ϕ𝑘(𝑧)
+|
+|
+2𝑑𝑧
+crystal surface into its depth31,32,37. Importantly in the ARPES context, the Tk values multiplied by the
+photoemission matrix elements define the partial photocurrents emanating from the individual фk(r) in the
+MBFS31. In Fig. 3(a) the calculated Tk are marked in blue colorscale. As expected for the empty-lattice
+case, Tk is equal to 1 for the primary (in the LEED context often called conducting) band and strictly zero
+for all other ones, realising the ideal FE final state containing one single plane wave. In Mahan's
+language, only the primary-cone photoemission is active in our ideal FE case.
+We will now introduce spatial modulations of V(r) as expressed by VΔK for non-zero ΔK. The plane waves
+start to hybridise through the VΔK matrix elements, and each фk(r) becomes a superposition of a few
+plane waves as фk(r) = ΣGCGei(k+G)r. In this case not only one but several фk(r) can acquire a certain
+admixture of the k// + G// = K// conducting plane wave – in the formal language, their Tk becomes
+non-zero – and give a certain contribution to the total photocurrent. Our model calculations for this case
+are sketched in Fig. 3 (b). The ARPES final state appears multiband in a sense that it consists of several
+фk(r) with different kzs (typically alongside the primary band) which give comparable contributions to the
+total ARPES signal as quantified by the corresponding Tk. In Mahan's language, the qualitative
+distinction between the primary- and secondary-cone photoemission dissolves. Correspondingly, the
+ARPES spectra will show up several peaks corresponding to different kz or, if the separation of these kzs
+is smaller than the intrinsic Δkz, excessive broadening of the spectral peaks. This is exactly what we
+have just seen in our ARPES data on Ag(100). We note in passing that on the qualitative level the bands
+
+contributing to the photocurrent can be easily identified based on the Fourier expansion of their фk(r)
+which should have a substantial weight of the k// + G// = K// conducting plane wave50.
+Whereas for the sake of physical insight we have intentionally simplified the above picture, the exact
+treatment of the MBFSs based on the matching approach of LEED has been developed in a series of
+previous works albeit limited to relatively low final-state energies31,34,37,38. Finally, we note that the MBFS
+phenomenon can also be understood within the simplified three-step model of photoemission, where the
+whole quantum-mechanical photoemission process is splitted into the photoexcitation of a photoelectron,
+its transport out of the crystal, and escape to vacuum. In this framework, the MBFSs can be viewed as
+resulting from multiple scattering of photoelectrons on their way out of the crystal that creates multiple
+Bloch-wave modes of the scattered wavefield.
+Fig. 3. Band structure of the final-state Bloch waves E(Rekz) in a model fcc crystal along the ГX
+direction (a) in the empty-lattice case V(r) = V0 and (b) with a more realistic spatially modulated
+pseudopotential, sketched in the insert. The dense multitude of bands is formed by an immense
+number of G vectors falling into our high-energy region.The contributions of each band into the total
+photocurrent are quantified by Tk (blue colorscale). Whereas in the first case the photocurrent
+emanates from one single FE band (marked with the corresponding G vectors), in the second case it
+may distribute over a few bands alongside the FE dispersion, which form a MBFS incorporating a few
+kzs.
+Whereas the effects of MBFSs have already been established at low excitation energies, their survival in
+high-energy ARPES might seem puzzling. In a naive way of thinking, photoelectrons with energies much
+higher than the modulations of V(r) should not feel them, recovering the FE case with one single фk(r).
+However, VΔK as the strength of hybridization between two plane waves depends, somewhat
+counter-intuitively, not on energy but rather on ΔK between them. As sketched in the insert in Fig. 3 (b),
+VΔK typically has its maximal negative value at ΔK = 0 (which is the V0), and with increase of ΔK sharply
+rises and then asymptotically vanishes. Importantly, however high the energy is, the multitude of the
+plane waves always contains pairs of those whose ΔK is small. The corresponding bands can be
+identified by close dispersions. For such pairs VΔK is large, giving rise to their strong hybridization.
+Importantly, all bands hybridising with the k// + G// = K// plane wave will receive non-zero Tk and thus
+
+1200
+(a)
+G00-6
+(b)
+VAK
+1100
+△K
+1000
+G005
+900
+800
+k
+0.5
+E
+700
+G004
+600
+500
+400
+Rekz
+Rekzcontribute to the total photocurrent, as shown in Fig. 3 (b). This forms the MBFSs that should survive
+even at high energies.
+Effect of MBFSs on the spectral structure
+We will now follow in more detail how the MBFSs affect the ARPES spectra. As an example, we will
+analyse the experimental kz-MDC from Fig. 2(d) in the region of the X point at hv ~ 1100 eV, reproduced
+in Fig. 4 (with the linear background subtracted). Within the FE approximation, we might expect to
+observe here two Lorentzian peaks, placed symmetrically around the X point and broadened by the
+same intrinsic Δkz. However, the kz-MDC shows three distinct peaks A-C, with the peak B coming from a
+final-state band falling beyond the FE approximation. Moreover, Lorentzian fitting of the peaks finds that
+whereas the peak C has a relatively small width of 0.11 Å-1, the widths of the peaks A and B are more
+than twice larger, 0.30 and 0.32 Å-1, respectively. The picture of MBFSs neatly explains this observation,
+suggesting that whereas the peak C is formed by a final state having one dominant kz contribution, and
+the peaks A and B by final states incorporating a multitude of kzs separated less than Δkz. Whereas it is
+generally believed that the intrinsic broadening of the ARPES peaks in kz is determined exclusively by
+finite λPE the photoelectron mean free path, our example demonstrates that the multiband final-state
+composition may not only create additional spectral peaks but also be an important factor of their
+broadening additional to λPE.
+Fig. 4. kz-MDC at EF from Fig. 2(d) in the hv region around 1100 eV (vicinity of the X point) decomposed in three
+Lorentzians. The presence of the peak B and the larger broadening of the peaks A and B compared to C are
+caused by MBFSs.
+Intriguingly, however, we note that even the narrowest peak C is almost twice broader than Δkz ~ 0.065
+Å-1 expected from λPE ~ 15.5 Å suggested by the TPP-2M formula 51 well-established in XPS and Auger
+electron spectroscopy. One explanation might be that already the peak C would incorporate multiple
+final-state bands with smaller kz separation compared to other two peaks. Another explanation would
+trace back to quasielastic electron-electron or electron-phonon scattering, which would increase with
+energy owing to the increase of the phase-space volume available for such scattering. Altering k of
+photoelectrons, it should destroy the coherence of photoelectrons and thus reduce λPE as reflected in the
+observed Δkz. At the same time, the quasielastic scattering should have only a little effect on attenuation
+of the k-integrated signal of the core-level or intrinsically incoherent Auger electrons. In other words, the
+effective λPE in ARPES should be smaller than that in XPS/Auger spectroscopy, described by the
+TPP-2M and related formalism. Such intriguing fundamental physics certainly deserves further
+investigation.
+
+15
+15.5
+16
+16.5
+17
+17.5
+18
+18.5MBFS phenomena through various materials
+The phenomenon of MBFSs surviving at high excitation energies is certainly not restricted to Ag only
+and, strengthening with the strength of V(r) modulations, should be fairly general over various materials.
+Even for Al the paradigm FE metal, astonishingly, such MBFSs can be detected at least up to excitation
+energies of a few hundreds of eV14,42. Quite commonly the MBFS effects at high energies are observed
+in van-der-Waals materials such as MoTe2
+52, which should be connected with a large modulation of V(r)
+across the van-der-Waals gap.
+Another vivid example of the MBFS effects is the soft-X-ray ARPES data for GaN presented in Fig. 5,
+compiled from the previously published results on AlN/GaN(1000) heterostructures13. The panel (a)
+shows the ARPES spectral structure plot expected from the DFT valence bands and FE final states with
+V0 = 5 eV. With the non-symmorphic space group of bulk GaN, the ARPES dispersions allowed by the
+dipole selection rules (though in our case somewhat relaxed due to the band bending in GaN) are
+marked bold. The panels (b,c) present the experimental out-of-plane ARPES dispersions measured at kx
+in two formally equivalent
+points of the surface BZ,
+0 in the first and
+1 in the second zone. As
+Г
+Г
+Г
+expected because of weaker electron screening of the atomic potential and thus sharper modulations of
+V(r) in the covalent GaN compared to the metallic Ag, the deviations of experimental dispersions from
+the predictions of the FE approximation are much stronger than for Ag. One can clearly see the MBFSs
+where the individual bands (marked by arrows at their top) are separated in kz more than the intrinsic Δkz
+broadening. In the multitude of the experimental ARPES dispersions, one can identify the one which can
+be associated with the primary-cone photoemission (bold arrows) although in the
+0 data this band
+Г
+cannot be traced below 1000 eV. Remarkably, for the same initial-state E(k) the ARPES dispersions
+measured at the
+0 and
+1 points appear completely different, identifying different final-state bands
+Г
+Г
+selected from the continuum of all unoccupied states available for given final-state energy and K//. These
+bands are identified by their leading plane-wave component to have k// + G// = K//, where K// of the
+photoelectron changes between the surface BZs32.
+Fig. 5. Out-of-plane ARPES dispersions for GaN(1000): (a) Expected from the DFT valence bands and
+FE final states with V0 = 5 eV. With the non-symmorphic space group of bulk GaN, the dispersions
+allowed by the dipole selection rules are shown bold; (b,c) Measured at kx = 0 projecting onto the Г0
+and Г1 points over two BZs. The experiment clearly resolves individual final-state bands (marked by
+arrows) whose separation in kz is larger than the intrinsic Δkz broadening.
+
+(b)
+(a
+(c)
+A
+A
+-10The high-energy final states in Si are a counter-example though. Fig. 6 presents soft-X-ray ARPES data
+on a few-nm thick layer of Si(100) n-doped with As53 as the out-of-plane band dispersions (b) and iso-EB
+contours (c), respectively. The panel (a) shows the ARPES spectral structure plot expected from the
+DFT-GGA calculated valence bands and FE final states with V0 = 10 eV, with the bold lines indicating the
+dispersions allowed by the selection rules (for in-depth discussion see Ref. 54). Because of the covalent
+character of Si, one might again expect that the non-FE effects here would be comparable to those for
+GaN and in any case stronger than for the metallic Ag. Contrary to such expectations, however, the
+experimental data in (b,c) does not show any clear signatures of the MBFSs in Si in the shown (Ek,k)
+region, although at low excitation energies they are profound35. At the moment we can not decipher any
+simple arguments that would relate the strength of the non-FE effects in the high-energy electron states
+to any obvious electronic-structure parameters of various materials.
+Fig. 6. Out-of-plane ARPES data for Si(100): (a) ARPES dispersions expected from the DFT valence
+bands and FE final states with V0 = 10 eV, with dispersions allowed by the selection rules shown bold;
+(b) Experimental band dispersions and (c) iso-EB contours in 2 eV below the valence-band maximum.
+No clear signatures of the MBFSs can be identified in these data.
+Non-FE effects beyond ARPES
+The non-FE effects in high-energy electron states such as MBFS manifest themselves not only in the
+ARPES dispersions. Another manifestation will be the circular dichroism in the angular distribution of
+photoelectrons (CDAD) that necessitates that the final-state wavefunctions deviate from the free-electron
+plane waves55,56. The CDAD has indeed been observed already in the early soft-X-ray ARPES study on
+Ag(100)39. Another example is the orbital tomography of adsorbed molecules (see, for example, Refs.
+57–59) which takes advantage of the Fourier relation between the angle distribution of photoelectrons
+and electron density of the valence electron orbitals. The non-FE effects introduce additional plane-wave
+components in the final states, calling for refinement of the straightforward Fourier-transform processing
+of the experimental data59. Beyond ARPES, the very fact of electron diffraction at crystalline surfaces
+identifies non-FE effects in the electron states in the crystal, because otherwise the incident electrons
+would upon entering the crystal follow the same FE wavefunction and thus would not reflect. The
+Reflection High-Energy Electron Diffraction (RHEED) evidences that the non-FE effects survive even in
+the energy range of a few tens of keV, when ΔK between the incident and diffracted plane waves is small
+
+(a)
+(b)and thus the corresponding VΔK large. These considerations suggest that the MBFSs should survive
+even in hard-X-ray ARPES, waiting for a direct experimental observation.
+Finally, we should point out that the coherent photoemission process underlying the ARPES experiment
+discussed above (as well as the orbital tomography) is fundamentally different to the essentially
+incoherent process of X-ray photoelectron diffraction (XPD) (see, for example, the reviews60–62. In the first
+case, all photoelectron emitters (atoms) throughout the crystal surface region within the depth λPE are
+coherent – or entangled, in the modern quantum mechanics discourse – and emit a coherent
+photoelectron wavefield characterised by a well-defined k. The resulting ARPES intensity as a function of
+Ek and θ bears sharp structures reflecting, through the momentum conservation, the k-resolved band
+structure of the valence states. In the XPD, other way around, the coherence between the emitters
+throughout the surface region is lost. This takes place, for example, for isolated impurity atoms or
+adsorbed molecules, localised core levels, where the initial-state wavefunctions at different atoms are
+decoupled from each other, or when the coherence of photoelectrons is broken by thermal or defect
+scattering, or when the signal from certain valence-band states, like d-states, is integrated in energy63,64.
+The result is that each photoelectron emitter creates scattered waves within a sphere of the radius λPE,
+which interfere with each other incoherently with the waves emanating from another emitter. Typical of
+diffraction with a few interfering rays, the resulting XPD intensity distribution as a function of Ek and θ is
+fairly smooth, and reflects the local atomic structure. With Ek increasing into the hard-X-ray energy
+range, λPE and thereby the number of coherently scattered waves increases. This forms sharp
+Kikuchi-like structures in the XPD angular distribution, reflecting the long-range atomic structure62. In any
+case, the XPD stays incoherent between the emitters. This fundamental difference between the coherent
+photoemission and incoherent XPD processes is stressed, for example, by the fact that in the first case
+the photoelectron angular distribution follows pph, shifting with hv, and in the second case it is insensitive
+to pph
+19.
+Conclusion
+Our analysis of extensive soft-X-ray ARPES data on the Ag metal has demonstrated that even at high
+excitation energies the photoemission final states may, intriguingly, in some energy and k-space regions
+feature pronounced multiband composition beyond the conventional FE approximation. The
+corresponding Bloch waves have different kz momenta, typically alongside the FE dispersion, and give
+comparable contribution to the ARPES spectra. Using empirical-pseudopotential simulation of the final
+states, where these contributions were quantified as proportional to the partial current in each Bloch
+wave determined within the wavefunction-matching formalism of LEED, we have demonstrated that the
+MBFSs appear due to hybridization of plane waves through low-K components of the crystal potential.
+Depending on the kz separation of the individual Bloch waves, the MBFSs give rise to multiple ARPES
+peaks from 3D valence-band dispersions or become an important factor of their broadening in addition to
+the intrinsic Δkz broadening due to the finite λPE. From the first principles, these effects can be described
+by one-step ARPES calculations with the final states treated within the multiple-scattering or Bloch-wave
+approaches. Although our KKR-based calculations on Ag were able to qualitatively describe the
+experimental results, further theoretical effort is required to achieve a quantitative agreement at high
+excitation energies. Besides Ag, the MBFS phenomena are observed, for example, in previous soft-X-ray
+data on the covalent GaN and even Al, the paradigm FE metal. They are surprisingly weak, however, for
+the covalent Si. The MBFS phenomenon, typically strengthening with the sharpness of the
+crystal-potential modulations, should be fairly general over a wide range of materials and excitation
+energies even into the hard-X-ray range.
+
+Methods
+Experiment
+The experiments were performed at the soft-X-ray ARPES facility65 installed at the high-resolution
+ADRESS beamline66 of the Swiss Light Source, Paul Scherrer Institute, Switzerland. X-rays irradiated the
+sample with a flux of ~1013 photons/s at a grazing-incidence angle of 20o. A single crystal of Ag(100)
+(MaTecK) was cleaned by a few cycles of Ar ion sputtering/annealing. The sample was cooled down to
+~12K in order to quench relaxation of k-conservation due to thermal motion of the atoms67, with the
+coherent spectral fraction enhanced by subtracting the angle-integrated spectrum scaled under the
+condition of non-negativity of the remaining spectral weight. The measurements were performed with
+p-polarised X-rays at a combined energy resolution varying from ~50 to 180 meV when going from hv =
+300 to 1300 eV, which is about twice better than in the first soft-X-ray ARPES study on Ag(100) 39. The
+FS maps were integrated over an EB window from -75 to 25 meV relative to EF. Angular resolution of the
+analyzer PHOIBOS-150 was ~0.1o. Other relevant experimental details, including the conversion of Ek
+and emission angle θ to k, corrected for pph, can be found elsewhere65. The data on GaN and Si from the
+previous ARPES works, discussed below, were taken under the same experimental conditions, but with
+the energy resolution relaxed to ~80 to 250 meV in the same hv range.
+Calculations
+In our simulations of the photoemission final states, the use of an empirical local pseudopotential has
+allowed reduction of the secular equation on complex kz to an eigenvalue problem for a complex
+non-Hermitian matrix17,45. For the energy range of our simulation extending to 1200 eV, the basis set
+included all plane waves below an energy cutoff of 1800 eV. The inner potential V0 was set to 10 eV, all
+VΔK to 5 eV for ΔK2 < 48 and to zero for larger ΔK2, and Vi to 5 eV. The accuracy of the calculations was
+controlled via the current conservation generalised for non-zero Vi on the crystal side. For our qualitative
+analysis of the final states, no attempt has been made to fit these parameters to our particular case.
+Details of the calculations can be found elsewhere32.
+The first-principles ARPES calculations were performed using the SPR-KKR package68 relying on the
+multiple scattering theory using the Korringa-Kohn-Rostoker (KKR) method. The ground-state properties
+of the Ag(001) surface were derived from density-functional-theory (DFT) calculations within the
+local-density approximation (LDA) carried out with full potential. The ARPES spectra were calculated
+within the one-step model of photoemission in the spin-density-matrix formulation69 taking into account all
+aspects of the photoemission process for the actual experiment including pph, matrix elements and final
+states constructed as the time-reversed LEED states. Taking into advantage the predominance of
+forward scattering at Ek above ~400 eV70 the calculations used the single-site scattering approximation.
+The final-state damping was described via constant Vi = 3 eV set to reproduce λPE = 10.2 Å at Ek = 600
+eV given by the TPP-2M formula51. For further computational details see Supplemental Material. The
+main paper presents the results obtained with FE final states, and the effects of multiple-scattering final
+states and various computational approximations are discussed in Supplemental Material.
+Data availability
+The raw and derived data presented are available from the corresponding authors upon a reasonable
+request.
+
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+Acknowledgements
+V.N.S. thanks E.E. Krasovskii for illuminating discussions and critical reading of the manuscript, and J. H.
+Dil for valuable exchange on physics of XPD. J.M. is grateful to D. Sébilleau, S. Tricot and A. Koide for
+sharing their scattering-amplitude calculations. The authors thank N.J. Curson and S.R. Schofield for
+giving access to Si samples prepared at University College London. J.M. and L.N. acknowledge the
+support of the Czech Ministry of Education, Youth and Sports via the grant CEDAMNF
+CZ.02.1.01/0.0/0.0/15_003/0000358 and the support from GACR Project No. 2018725S. L.L.L.
+acknowledges the financial support from the Ministry of Science and Higher Education of the Russian
+Federation, grant #075-11-2021-086. T.J.Z.S. acknowledges the financial support of the Engineering and
+
+Physical Sciences Research Council (grants nos. EP/R034540/1, EP/W000520/1), and Innovate UK
+(grant no. 75574).
+Author contributions
+V.N.S. and J.M. conceived the SX-ARPES experiment at the Swiss Light Source. V.N.S., L.L.L., F.A. and
+L.N. performed the experiment supported by T.S. T.J.Z.S. fabricated the thin-film Si samples. V.N.S.
+processed and interpreted the data, and performed computational simulation of the final states supported
+by P.C. J.M. performed the first-principles ARPES calculations. V.N.S. wrote the manuscript with
+contributions from J.M., L.L.L., P.C., T.J.Z.S. and J.O. All authors discussed the results,
+interpretations, and scientific concepts.
+Competing interests
+The authors declare no competing interests.
+
+Supplemental Material: KKR calculations with
+multiple-scattering final states
+Computational scheme
+In the first step of our theoretical investigations, we performed self-consistent electronic structure
+calculations within the ab-initio framework of the spin-density functional theory in order to generate the
+self-consistent-field (SCF) potential for further photoemission calculations. The LDA potential of Vosko et
+al. was used1. The electronic structure of semi-infinite crystal was calculated within the relativistic
+multiple scattering approach using the Green's function Korringa-Kohn-Rostoker (KKR) formalism in the
+tight binding mode2. The experimental lattice constant (a = 4.09 Å) was used. In order to achieve precise
+description of the most subtle details of the SCF potential, important for photoemission at high excitation
+energies, the multipole expansion of the Green’s function employed an unusually large
+angular-momentum cutoff lmax of 5. In addition, a large number of k-points (36x36x36) in the first surface
+BZ was used. The self-consistent calculations have been performed in two modi, within the so-called
+atomic sphere approximation (ASA) and in the full potential (FP) mode.
+The obtained SCF potential was used for the photoemission calculations within the one-step model. The
+final states (the time-reversed LEED state) were treated using the so-called layer KKR technique3,
+allowing accurate description of these states in a wide hv range starting from 6 eV up to several keV. To
+ensure the convergence of the multiple scattering between the layers, our calculations used a
+plane-wave basis where the number of the surface reciprocal lattice vectors g was increased to 147
+instead of the default value 372. Another important ingredient of the multiple-scattering calculations is an
+accurate description of the kinematic and dynamic effects in both initial and final states. For the latter, the
+dynamic effects are taken into account via the X-matrix4 which represents the energy-dependent multiple
+scattering within a single layer. Whereas in VUV-ARPES the kinematic and dynamic effects are
+comparable, in the soft- and hard-X-ray regime the dynamic effects weaken, whereby the X-matrix
+approaches zero, leading to the so-called single-site scattering approximation. Another important
+parameter in the description of multiple scattering is connected with the expansion of all physical
+quantities in terms of angular momentum l, i.e. using the Bauer’s identity to represent plane waves
+(scattering between the layers) and spherical waves (inside the layer). These expansions involve a
+summation over l that must be truncated at a certain value lmax. In this context, the increase of lmax should
+be viewed rather as an extension of the basis set for accurate description of the multiple scattering than
+physically meaningful l-channels in the scattering process. A simple assessment of lmax can be obtained
+from the radial Schrödinger equation where, in order to scatter on the spherical potential, the electron
+must first overcome the centrifugal barrier l(l+1)/a2 (a is the atomic radius). This implies only the partial
+waves, whose l satisfies the inequality k2 > l(l+1)/a2, should be included into the l-expansion. The higher
+Ek, the larger lmax needs to be used (for the detailed explanation see Ref. 5). For Ek in the range
+300-1300eV, considered here, lmax falls between 4 and 5. The calculations have been performed for a
+finite temperature of 20K leading to an additional final-state k-broadening, increasing with hv6.
+
+Effect of various approximations for the multiple-scattering process
+We have made an effort to elaborate our ARPES calculations towards their quantitative agreement with
+the experiment in a few successive steps:
+– Fig. S1 shows the results obtained with multiple-scattering final states, as opposed to the FE final
+states used for the calculations in Figs. 1 and 2 in the main text, under successive refinements of the
+computational approximations:
+– The results in Fig. S1(a) were obtained within the ASA and lmax = 3. Due to the non-FE effects
+described by the multiple-scattering final states, they already show spectral structures due to the MBFSs
+(such as where marked by magenta arrows) although mostly on the low-energy end of our hv range and
+not exactly in the same k-space regions compared to the experiment;
+– The inclusion of warping of the potential in the interstitial and surface regions within the FP scheme7,
+Fig. S1(b), does not result in any significant improvement in our case of Ag. Nevertheless, we anticipate
+that the accurate FP will be crucial for more open crystal structures, covalent materials, van-der-Waals
+materials, etc. where the potential modulations are sharper7. As we have seen in the present work, their
+accurate description should be particularly important at high Ek where the final states are highly sensitive
+to the high-frequency modulations of V(r) and thus to the accurate representation of its real-space
+variations;
+– Another step, the inclusion of the full X-matrix compared to the single-site approximation, presented in
+Fig. S1(c), considerably improves the description of the relative intensity variations in the hv interval
+between 400 and 500 eV (magenta arrow, for example) but does not notably affect the spectral intensity
+at higher energies. This observation can be understood from analysis of scattering amplitude fk(θ), giving
+more insight into the scattering process. The calculations of fk(θ) for Ag by Sébilleau et al.8, reproduced
+in Fig. S2, demonstrate that for Ek above ~400 eV it is strongly dominated by forward scattering. In
+practice, this means that for these energies the electrons scatter essentially along the rows of atoms,
+justifying the single-site approximation for the multiple scattering;
+– Finally, at the last step of our computational refinement presented in the Fig. S1(d), we increased lmax
+from 3 to 5. As expected, not only has this returned a vivid pattern of the MBFS-induced replica bands
+and excessive spectral broadening at low Ek (such as where marked by magenta and yellow arrows,
+respectively) but also pushed these effects to yet higher hv up to 700 eV (magenta arrow). Further
+increase of lmax would inflate the computational time beyond presently realistic.
+Although these successive refinements of the computational scheme do move towards a better
+description of the experiment, the achieved agreement with the experimental results can only be
+considered as qualitative. We conjecture that the remnant deviation may trace back to quite small
+sensitivity of the total energy to high-frequency components of the crystal potential. Therefore, the
+total-energy minimization used to generate the self-consistent potential in the DFT calculations may not
+ensure sufficient accuracy of its high-frequency components which critically affect the hybridization and
+thus non-FE effects in the final states at high energies. The accuracy of the final states used in the
+ARPES calculations can in principle be verified independently from the initial states by calculating the
+LEED spectra and their fitting to the experiment using the methodology previously developed for very low
+energies (see Refs. 9–11 and the references therein). In any case, including the subtle details of V(r)
+within the FP approach and the use of sufficiently large lmax give the best possible single-particle
+description of the photoemission final state.
+
+Fig. S1. One-step ARPES calculations as in Fig. 1(d) but using multiple-scattering final states under
+successive refinements of their treatment: (a) standard spherical-wave expansion and single-site
+scattering approximation; (b) adding full potential; (c) the full X-matrix beyond the single-site scattering;
+(d) increasing the angular momentum expansion to lmax = 5. The calculations reproduce the multiple
+spectral peaks (magenta arrows) and the excessive spectral broadening (yellow) induced by the
+MBFSs.
+Fig. S2. Scattering amplitude fk(θ) for Ag as a function of Ek (left panels) and the total forward and
+backward scattering contributions (right panel).
+
+hv (eV)
+(a)
+(b)
+(c)
+(d)
+1200
+1000
+800
+600
+400
+0Ag Scattering Factor @ E = 50.00 eV
+Ag Scattering Factor @ E = 500.00 eV
+90*
+90°
+[0)]
+135°
+(9(e)
++SET
+((e)
+3([6))
+(e))C
+180°
+180°
+Ag Forward and Backward scattering amplitudes
+225*
+225*
+315
+Forward
+270*
+270*
+Backward
+Ag Scattering Factor @ E = 100.00 eV
+Ag Scattering Factor @ E = 700.00 eV
+90°
+135*
+[0]]
+g((e)
+135*
+s((e)
+([(0)
+3(6))
+180°
+180*
+2
+225*
+225*
+315
+270*
+270*
+200
+400
+600
+800
+1000
+Ag Scattering Factor @ E = 300.00 eV
+Ag Scattering Factor @ E = 900.00 eV
+90°
+Kinetic Energy [eV]
+135*
+[0]]
+[6]]
+135*
+3(R0)
+3((6))
+180*
+225
+225*
+/315
+270*
+270*References
+1. S. H. Vosko, L. Wilk, and M. Nusair, Accurate Spin-Dependent Electron Liquid Correlation Energies for
+Local Spin Density Calculations: A Critical Analysis, Canadian J. Phys. 58 (1980) 1200
+2. H. Ebert, D. Ködderitzsch, and J. Minár, Calculating Condensed Matter Properties Using the
+KKR-Green’s Function Method – Recent Developments and Applications, Rep. Prog. Phys. 74 (2011)
+096501.
+3. J. M. MacLaren, S. Crampin, D. D. Vvedensky, and J. B. Pendry, Layer Korringa-Kohn-Rostoker
+Technique for Surface and Interface Electronic Properties, Phys. Rev. B 40 (1989) 12164
+4. J. Braun, J. Minár, and H. Ebert, Correlation, Temperature and Disorder: Recent Developments in the
+One-Step Description of Angle-Resolved Photoemission, Phys. Rep. 740 (2018) 1.
+5. Multiple Scattering Theory for Spectroscopies, eds. D. Sébilleau, K. Hatada and H. Ebert. Springer
+Proc. Phys. 204 (2018).
+6. J. Braun, The theory of angle-resolved ultraviolet photoemission and its applications to ordered
+materials. Rep. Prog. Phys. 59 (1996) 1267.
+7. J. Braun, J. Minár, S. Mankovsky, V. N. Strocov, N. B. Brookes, L. Plucinski, C. M. Schneider, C. S.
+Fadley, and H. Ebert, Exploring the XPS Limit in Soft and Hard X-Ray Angle-Resolved Photoemission
+Using a Temperature-Dependent One-Step Theory, Phys. Rev. B 88 (2013) 205409
+8. D. Sébilleau, S. Tricot, and A. Koide, unpublished (2022).
+9. V. N. Strocov, H. Starnberg & P. O. Nilsson. Excited-state bands of Cu determined by VLEED band
+fitting & their implications for photoemission, Phys. Rev. B 56 (1997) 1717.
+10. V. N. Strocov, R. Claessen, G. Nicolay, S Hüfner, A Kimura, A. Harasawa, S. Shin, A. Kakizaki, P.O.
+Nilsson, H.I. Starnberg & P. Blaha. Three-dimensional band mapping by angle-dependent
+very-low-energy electron diffraction and photoemission: Methodology and application to Cu. Phys.
+Rev. B 63 (2001) 20510.
+11. V. N. Strocov, E.E. Krasovskii, W. Schattke, N. Barrett, H. Berger, D. Schrupp & R. Claessen.
+Three-dimensional band structure of layered TiTe2: Photoemission final-state effects. Phys. Rev. B 74
+(2006) 195125.
+

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+Research in Astronomy and Astrophysics manuscript no.
+(LATEX: ms2022-0349.tex; printed on January 10, 2023; 1:43)
+Limiting Magnitudes of the Wide Field Survey Telescope (WFST)
+Lei Lei (雷磊)1,2, Qing-Feng Zhu (朱青峰)1,3, Xu Kong (孔旭)1,3, Ting-Gui Wang (王挺贵)1,3,
+Xian-Zhong Zheng (郑宪忠)1,2, Dong-Dong Shi (师冬冬)2, Lu-Lu Fan (范璐璐)1,3 and Wei Liu
+(刘伟)2
+1 School of Astronomy and Space Science, University of Science and Technology of China, Hefei
+230026, China; [email protected]
+2 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China;
+3 Deep Space Exploration Laboratory / Department of Astronomy, University of Science and Technology
+of China, Hefei 230026, China;
+Received 20xx month day; accepted 20xx month day
+Abstract Expected to be of the highest survey power telescope in the northern hemisphere,
+the Wide Field Survey Telescope (WFST) will begin its routine observations of the northern
+sky since 2023. WFST will produce a lot of scientific data to support the researches of time-
+domain astronomy, asteroids and the solar system, galaxy formation and cosmology and so
+on. We estimated that the 5 σ limiting magnitudes of WFST with 30 second exposure are
+u = 22.31 mag, g = 23.42 mag, r = 22.95 mag, i = 22.43 mag, z = 21.50 mag, w = 23.61
+mag. The above values are calculated for the conditions of airmass = 1.2, seeing = 0.75
+arcsec, precipitable water vapour (PWV) = 2.5 mm and Moon-object separation = 45◦ at
+the darkest New Moon night of the Lenghu site (V=22.30 mag, Moon phase θ = 0◦). The
+limiting magnitudes in different Moon phase conditions are also calculated. The calculations
+are based on the empirical transmittance data of WFST optics, the vendor provided CCD
+quantum efficiency, the atmospherical model transmittance and spectrum of the site. In the
+absence of measurement data such as sky transmittance and spectrum, we use model data.
+Key words: techniques: photometric — surveys — telescopes
+1 INTRODUCTION
+The Wide Field Survey Telescope (WFST; Lou et al. 2016; Shi et al. 2018; Lou et al. 2020; Lin et al. 2022) is
+an optical telescope to be installed at the Lenghu site, located on Saishiteng mountain near Lenghu Town in
+Qinghai Province on the Tibetan Plateau, China in 2023. The WFST has a 2.5-m diameter primary mirror
+and a 5-lens corrector to form a prime-focus optics (Wang et al. 2016;Lou et al. 2016; Lou et al. 2020).
+The detector of WFST consists of nine 9K×9K CCD chips and has 0.9 Giga pixels. The entire system is
+arXiv:2301.03068v1  [astro-ph.IM]  8 Jan 2023
+
+2
+Lei et al.
+optimized for the wavelength range from 3200 ˚A to 9600 ˚A (Lou et al. 2016; Chen et al. 2019). With the aid
+of an active optics system and an ADC (atmospheric dispersion compensator), WFST can achieve an image
+quality of 0.4 arcsec 80% energy enclosed across a field of view of 3 degree diameter and ∼7 deg2 area.
+WFST is able to survey ∼ 2 × 104 deg2 northern sky in ugrizw six bands. As a powerful survey telescope,
+its scientific data will greatly support researches of time-domain astronomy, asteroids and the solar system,
+the Milky Way and its satellite dwarf galaxies, galaxy formation and cosmology and so on.
+In recent years, many large ground-based optical survey telescopes have been built or planned all over
+the world. SDSS (Kent 1994; Fukugita et al. 1996), Pan-STARRS (Jedicke & Pan-STARRS 2007; Chambers
+& Pan-STARRS Team 2016), SkyMapper (Schmidt et al. 2005; Rakich et al. 2006), ZTF (Bellm 2014;
+Bellm et al. 2019; Graham et al. 2019) and other built telescopes have produced a large amount of observa-
+tion data, which has greatly promoted astronomical researches and solved many scientific problems. Soon
+new, survey telescopes such as LSST (Hlozek et al. 2019), Mephisto (Liu 2019; Lei et al. 2021; Yuan et al.
+2020) and WFST will join their peers and conduct deeper multi-band surveys to provide crucial data to
+astrophysical researches. Combined with China Space Station Telescope (CSST; Zhao et al. 2016; Yuan
+et al. 2021) and other space telescopes, WFST will greatly improve human understanding of the universe
+and promote more important scientific discoveries.
+The parameters of a survey telescope, such as the diameter of the primary mirror, quantum efficiency
+(QE) of CCD, band transmittance, etc., determine the throughput of the telescope. The site conditions of an
+astronomical observatory, such as atmospheric transmittance, altitude, seeing and sky background bright-
+ness, affect the depth of a survey program.. The limiting magnitude of a survey telescope is an important
+guide for planning research objectives and project scopes. It is also the key for designing exposure time
+plans and survey strategies. Therefore, an accurate estimation of the limiting magnitudes are needed for the
+successful commission of a new telescope.
+In this work, we introduce the estimation of the limiting magnitudes of WFST. In Sect. 2 we describe
+the method we adopt. In Sect. 3 we show our results of limiting magnitudes of the WFST.
+2 LIMITING MAGNITUDES
+2.1 Throughput of WFST
+The throughput of an astronomical observation (Ttot) is limited by the atmospheric transmittance (Tatmo),
+the transmittance of optics (Topt), the transmittance of the filters Tband) and the quantum efficiency of CCD
+(QECCD).
+Ttot = Tatmo · Topt · Tband · QECCD
+(1)
+The optical system of WFST consists of a 2.5 meter diameter primary mirror with a 760 mm diameter
+central hole, five corrector lenses, a ADC made with two glass wedges and ugrizw six switchable filters
+(Lou et al., 2016). Among five correcting lenses, only one is made of the N-BK7HT glass. The others and
+ADCs are made of the fused quartz. Since the transmittance of a fused quartz blank can be neglected, we
+simulate the total transmittance of the five-lens corrector and the ADC with the product of the transmittance
+of a 35 mm thick N-BK7 glass blank and the transmittance of 14 layers of anti-reflection (AR) coatings.
+
+Limiting Magnitudes of WFST
+3
+The transmittance of the N-BK7 glass is obtained from SCHOTT1 and the transmittance of the AR coating
+is from Institute of Optics and Electronics (IOE) ’s measurements. Because of the oversized Primary Focus
+Assembly (PFA), the actual aperture obscuration is 1 m diameter.
+Because we don’t have atmospheric transmitance and spectrum measurements at the site, we adopt
+SkyCalc2 (Version 2.0.9) to obtain model curves. SkyCalc is developed by astronomers in ESO based on
+the Cerro Paranal Advanced Sky Model (Noll et al. 2012; Jones et al. 2013; Moehler et al. 2014). The
+atmospheric transmittance is affected by altitude, humidity, dust, precipitable water vapour (PWV), among
+which the altitude is the most important factor. SkyCalc only provides the atmospheric transmittance at
+three astronomical sites: Paranal (2400 m), La Silla (2600 m) and Extremely Large Telescope (ELT) site
+(3060 m). Figure 1 shows the three transmittance curves of the sites. We can see that three curves have
+same features that they are scaled according to different altitudes of three sites. This is reasonable because
+the geographic features of the three sites are very similar. The Paranal Observatory is on the Cerro Paranal
+mountain, which is in the Atacama Desert of northern Chile. The La Silla Observatory is located on the
+outskirts of the Chilean Atacama Desert. The 40-metre-class ELT is on the Cerro Armazones mountain in
+the central part of the Atacama Desert. WFST is on the Saishiteng mountain in the Gobi desert area on
+the Tibetan Plateau. We consider that the geographic features of the area are more similar to those of sites
+in the high-altitude Atacama desert than those of oceanic mountain areas, such as Mauna Kea in Hawaii.
+It is a reasonable choice to obtain the atmospheric transmittance of the WFST site by using the spectra
+from SkyCalc. So we get the atmospheric transmittance curve of the WFST site at an altitude of 4200 m
+by scaling the atmospheric transmittance curves of paranal, lasilla and ELT sites. In our simulations, we
+assume that airmass = 1.0 and precipitable water vapour (PWV) = 2.5 mm. Figure 1 also shows the result
+of scaling.
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+wavelength(Å)
+0.1
+1.0
+Transmission
+10550
+10560
+10570
+10580
+10590
+10600
+10610
+0.974
+0.976
+0.978
+0.980
+3060 m
+2640 m
+2400 m
+4200 m
+Fig. 1: The atmospheric transmittance curves of Paranal Observatory (2400 m), La Silla Observatory (2600
+m) and the Extremely Large Telescope site (3060 m), the scaled transmittance curve of the WFST site (4200
+m). We assume the WFST site has the same geographic features (i.e. high altitude Mountain and dry air) as
+the three ESO sites.
+1 https://refractiveindex.info/?shelf=glass&book=BK7&page=SCHOTT
+2 https://www.eso.org/observing/etc/bin/gen/form?INS.MODE=swspectr+INS.NAME=SKYCALC
+
+4
+Lei et al.
+As shown in Figure 2(a), combined system throughput, individual transmittance curves of the atmo-
+sphere and the corrector, the reflectivity of the primary mirror and the quantum efficiency of the CCD are
+plotted respectively. We also plot the original estimate of the system throughput for WFST by Shi et al.
+(2018). We can see that the updated system throughput is higher than the early expectation in most wave-
+lengths (Shi et al., 2018). The major reason is that the transmittances of ADC and optical lenses are higher
+than the early estimate. In order to obtain high efficiency in short wavelengths, WFST selects e2v standard
+Si back-illuminated CCD detectors with the astro multi-2 coating. The QE of the CCDs is also increased.
+Figure 2(b) shows the transmittance of the filters and the total throughput of WFST in six bands, which
+is calculated by using Equation 1.
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+wavelength (Å)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Efficiency
+a)
+Atmosphere
+Primary Mirror
+Lenses 
+CCD
+Atmosphere + Primary Mirror + Lenses + CCD. Shi et al. 2018 
+Atmosphere + Primary Mirror + Lenses + CCD. This work
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+wavelength (Å)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+b)
+u
+g
+r
+i
+z
+w
+Fig. 2: (a) The transmittance curves of the atmosphere (blue dot-dashed line), the optics including lenses
+and the ADC (green line), the reflectivity of the primary mirror (yellow line) and the quantum efficiency of
+the CCDs (cyan dot-dashed line). The combined efficiency of the atmosphere, the optics and the CCDs is
+in purple. The red line shows the total efficiency of Shi et al.(2018); (b) The transmittance of each filter and
+the total efficiency in each WFST filter transmittance.
+2.2 Noise of WFST
+Noise in astronomical CCD images mainly consists of the contributions from the artificial light on the
+ground, astrophysical sources, sky background, CCD dark current, and CCD readout noise. The site of
+WFST is on Saishiteng mountain on the Tibetan Plateau, where the nearest residential area is Lenghu town
+which is ∼ 50 km from the observatory site and has a population of 200. There is no industrial activity and
+the ground light pollution. The Haixi Mongolian and Tibetan Autonomous Prefecture of Qinghai Province
+has announced the 17800 square kilometres area of Lenghu as a dark night protecting region in the local law.
+It protects the good observational conditions of the Lenghu astronomical site. Deng et al. (2021) studied
+long-term astronomical conditions of the Lenghu site and pointed out that the sky background of a New
+Moon night can reach 22.3 mag arcsec−2 in the V band and the average night-sky brightness is around 22.0
+mag arcsec−2 when the Moon is below the horizon. We adopt AB magnitude system in this work.
+The sky background spectrum of the Lenghu site is also calculated by the software SkyCalc (Noll et al.
+2012; Jones et al. 2013; Moehler et al. 2014). The monthly averaged solar radio flux is equal to 130.00
+sfu, that is the solar 10.7 cm radio flux in the Sun median active level (Sparavigna 2008; Petrova et al.
+2021). Because the solar activities will affect the sky background brightness, it is necessary to take the
+
+Limiting Magnitudes of WFST
+5
+solar activities into account when we estimate the sky background. We adopt the median values obtained
+from long-term solar monitoring programs as the baseline solar brightness. The spectral flux of one sky
+region is related to the Moon-Target separation and the Moon phase. We designate the Moon phase with the
+Moon phase angle (θ) 0◦ (New Moon), 45◦ (Waxing), 90◦ (Half Moon Waxing), 135◦ (Waxing), 180◦ (Full
+Moon) respectively. We assume the separation of the Moon and a target is always 45◦ in our calculation.
+So the spectral flux of one region is only dependent on the altitude and the Moon phase. We get the sky
+background spectra towards the Zenith under different Moon phase conditions at the Lenghu Observatory
+site by scaling the spectra of three ESO sites provided by SkyCalc. Figure 3(a) shows the sky background
+spectrum at the altitude of 4200 m (θ = 180◦, here we just plot the spectra at a full Moon night because
+it is easier to see their difference.), and the spectra of the three ESO sites at Full Moon night. Figure 3(b)
+shows the sky spectra at Lenghu site under six different Moon phase conditions. As shown in the detail part
+of Figure 3(b), the sky background spectrum at a New Moon night (θ = 0◦) and the sky spectrum at a Dark
+night have almost the same flux.
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+wavelength(Å)
+0.01
+0.10
+1.00
+10.00
+flux (photons s
+1 Å
+1 arcsec
+2 m
+2)
+a)
+= 180
+5175
+5200
+5225
+5250
+5275
+5300
+5325
+0.450
+0.475
+0.500
+0.525
+0.550
+0.575
+0.600
+4200 m
+3060 m
+2640 m
+2400 m
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+wavelength(Å)
+0.00
+0.01
+0.10
+1.00
+10.00
+flux (photons s
+1 Å
+1 arcsec
+2 m
+2)
+b)
+4500
+4600
+4700
+4800
+4900
+0.012
+0.014
+Dark night
+= 0
+= 45
+= 90
+= 135
+= 180
+Fig. 3: (a) The sky background spectrum (purple) at Zenith at the altitude of 4200 m in the Full Moon
+condition, and the spectra of three ESO sites when the Moon phase is θ = 180◦. (b) The Zenith sky spectra
+of the 4200 m site in different Moon phase conditions. The sky background spectrum of Moon phase
+θ = 180◦ and the spectrum of a dark night (when the Moon is under the horizon) have almost the same
+flux.
+We can get the magnitude mV by integrating the sky background spectrum multiplied by the V band
+filter transmission curve:
+mV = −2.5 ×
+�
+log10
+� ∞
+0
+fλTband,λdλ
+� ∞
+0
+Tband,λdλ
+�
+− 21.1
+(2)
+where fλ is sky background spectral flux, Tband is the Johnson V band transmission curve (Bessell, 1990),
+ZP = −21.1 is the zero point (Bessell & Murphy 2012). The modeled sky emission radiance flux from
+SkyCalc is in units of photon/s/m2/micron/arcsec2. The Johnson V band sky background magnitude
+of a New Moon night with the SkyCalc model spectrum at an altitude of 4200 m is 21.74 mag arcsec−2.
+We scale the 4200 m sky background spectrum so that the resulting spectrum has a V-band magnitude
+of 22.3 or 22.0 mag arcsec−2, corresponding to the best and the average sky brightness conditions at the
+Lenghu site. As shown in Figure 3(b), there are differences among the sky spectra under different Moon
+
+6
+Lei et al.
+phase conditions. We scale these sky spectra at different Moon phases use the the same scaling factor
+in the new moon case, where we scale the spectrum from V θ=0◦ = 21.74 to 22.30 mag arcsec−2, so
+that differences among spectra at different Moon phases are not changed. The estimated V band Zenith
+sky background magnitudes at the Lenghu site with different Moon phases are: V θ=0◦, V θ=45◦, V θ=90◦,
+V θ=135◦, V θ=180◦=22.30, 22.10, 21.29, 20.28, 18.90 mag arcsec−2.
+The Lenghu sky background spectrum is calculated for airmass = 1.0. It can be scaled to another
+airmass by multiplying a factor a (Krisciunas & Schaefer 1991).
+a = 10−0.172 (X−1)X
+2.5
+(3)
+when airmass = 1.2, X is
+X =
+1
+�
+(1 − 0.96 × sin (arccos (
+1
+airmass))2)
+≈ 1.18958
+(4)
+Based on the sky background spectrum of the Lenghu site, we estimate the magnitudes of the sky
+background in each band mAB
+band:
+mAB
+band = −2.5 × log10
+�Skyband
+ZPband
+�
+(5)
+where
+ZPband =
+� ∞
+0
+fluxABTband,λdλ
+(6)
+where fluxAB = 3631Jy for all frequencies, and Tband,λ is the transmittance curve of a particular band.
+Skyband =
+� ∞
+0
+fλTband,λdλ
+(7)
+where fλ is sky background spectral flux. The Table 1 shows the sky background magnitudes mAB in
+WFST six bands.
+Table 1: The sky background brightness mAB of WFST six bands in units of mag arcsec−2.
+Moon Phases
+u
+g
+r
+i
+z
+w
+0◦
+23.27
+22.82
+21.80
+20.99
+20.05
+21.78
+45◦
+23.02
+22.49
+21.66
+20.93
+20.03
+21.64
+90◦
+22.00
+21.37
+20.99
+20.61
+19.90
+21.01
+135◦
+20.86
+20.21
+20.08
+20.01
+19.61
+20.12
+180◦
+19.30
+18.73
+18.78
+18.97
+18.92
+18.80
+Note: The sky spectra is calculated by SkyCalc when airmass = 1.0, PWV = 2.5 mm. We calculated the sky
+background brightness mAB when airmass = 1.2.
+2.3 Limiting Magnitudes of WFST
+Assuming the signal to noise ratio of WFST in all bands for a point source is S/N, we can write the formula
+of the S/N as :
+S
+N =
+S · A · τ
+�
+S · A · τ + 2 · npix · [(Sky · A · αpix + D) · τ + R2]
+(8)
+where S is the source signal with a constant spectral flux, τ is the standard exposure time (30 s), A is
+the effective area of the primary mirror (∼ 4.12 × 104 cm2), αpix = 0.111 arcsec2 is the area of one
+
+Limiting Magnitudes of WFST
+7
+pixel, D is the dark current of the CCD (D = 0.005 e−/pixel/s, @−100◦C), R2 is the readout noise of
+the CCD (R = 8 e− rms), npix is the total pixel number in the point spread function (PSF), the usage
+of a factor 2 is because we assume the calculation is performed on sky subtracted images. An optimal
+PSF aperture of 1.18 times of the full width at half maximum (FWHM) is adopted for a non-Adaptive
+Optics case according to the Integration Time Calculator (ITC) of Gemini3. And the FWHM of the seeing
+degrades with the airmass and the wavelength as (airmass)0.6 × λ−0.2
+eff . Here λeff takes the value of
+356.17, 476.34, 620.57, 753.07, 870.45, 612.15 nm in the six bands ugrizw given by the Equation 9.
+λeff =
+� ∞
+0
+λTband dλ
+� ∞
+0
+Tband dλ
+(9)
+With the seeing = 0.75 arcsec measured by Deng et al. (2021) at 500 nm (Tokovinin et al., 2003), we
+estimated the seeing values in different bands and airmass conditions.
+The sky signal actually lands on the detector is:
+Sky =
+� ∞
+0
+fλToptTbandQECCD dλ
+(10)
+where Topt is the throughput of the optics (including the primary mirror, ADC and the 5 corrector lenses),
+QECCD is the quantum efficiency of the CCD.
+We can solve the Equation 8 to obtain the signal of an astronomical object required at the detection limit
+of S/N = 5 and find the corresponding limiting magnitude mlim:
+mlim = −2.5 × log10
+�
+S
+0.61 · ZPlim
+�
+(11)
+A factor 0.61 is used because according to the description of ITC, the 1.18 FWHM sized aperture will
+contain 61% energy of a point source. The ZPlim is the system zero point flux:
+ZPlim =
+� ∞
+0
+fluxABTatmoToptTbandqeCCD dλ
+(12)
+Table 2 lists the calculated limiting magnitudes of ugrizw six bands. We calculated the limiting mag-
+nitudes of WFST at different Moon phases when the sky background brightness is V=22.0 mag and 22.3
+mag, respectively. The results of a single exposure of 30 s and of coadded 100 frames with a total integration
+time of 100 × 30 s are listed. It shows that WFST can reach 23.42 (25.95) mag in the g band with a 30 s
+(100 × 30 s) exposure under the conditions with the sky background brightness V=22.3 mag, seeing =0.75
+arcseconds, airmass = 1.2 and PWV=2.5 mm. If the sky background is V=22.0 mag, the above values are
+23.32 (25.85) mag for 30 s (100 × 30 s).
+3 DISCUSSION AND CONCLUSIONS
+In the current work, by considering the observational conditions of WFST, including throughput, quantum
+efficiency, the noise, the area of the primary mirror and the sky background brightness, we compute the
+limiting magnitudes of WFST. We get the sky background magnitudes in AB magnitude system in the
+Lenghu site at the New Moon night when airmass = 1.2: u, g, r, i, z, w=23.27, 22.82, 21.80, 20.99, 20.05,
+21.78 mag arcsec−2. For the Lenghu darkest night condition (V=22.3 mag arcsec−2) and a exposure time
+3 https://www.gemini.edu/observing/resources/itc/itc-help
+
+8
+Lei et al.
+Table 2: 5σ limiting magnitudes of WFST when airmass=1.2, seeing = 0.75 arcsec, precipitable water
+vapour (PWV) = 2.5 mm and Moon-object separation is 45◦.
+Exposure time
+Moon Phase
+V band sky
+u
+g
+r
+i
+z
+w
+30 s
+0◦
+22.30
+22.31
+23.42
+22.95
+22.43
+21.50
+23.61
+30 s
+45◦
+22.10
+22.27
+23.30
+22.89
+22.40
+21.49
+23.54
+30 s
+90◦
+21.29
+22.04
+22.86
+22.62
+22.26
+21.43
+23.23
+30 s
+135◦
+20.28
+21.64
+22.34
+22.21
+21.99
+21.31
+22.79
+30 s
+180◦
+18.90
+20.97
+21.62
+21.58
+21.49
+21.00
+22.13
+100 × 30 s
+0◦
+22.30
+24.86
+25.95
+25.48
+24.96
+24.03
+26.13
+100 × 30 s
+45◦
+22.10
+24.82
+25.84
+25.42
+24.93
+24.02
+26.06
+30 × 100 s
+90◦
+21.29
+24.58
+25.38
+25.14
+24.78
+23.96
+25.74
+100 × 30 s
+135◦
+20.28
+24.17
+24.85
+24.72
+24.51
+23.83
+25.30
+100 × 30 s
+180◦
+18.90
+23.48
+24.12
+24.09
+24.01
+23.51
+24.64
+30 s
+0◦
+22.00
+22.26
+23.32
+22.83
+22.30
+21.37
+23.47
+30 s
+45◦
+21.80
+22.21
+23.19
+22.77
+22.28
+21.37
+23.40
+30 s
+90◦
+20.99
+21.95
+22.74
+22.48
+22.12
+21.29
+23.09
+30 s
+135◦
+19.98
+21.52
+22.19
+22.07
+21.85
+21.18
+22.64
+30 s
+180◦
+18.60
+20.83
+21.47
+21.44
+21.35
+20.86
+21.99
+100 × 30 s
+0◦
+22.00
+24.81
+25.85
+25.36
+24.83
+23.90
+25.99
+100 × 30 s
+45◦
+21.80
+24.76
+25.72
+25.30
+24.80
+23.89
+25.92
+100 × 30 s
+90◦
+20.99
+24.48
+25.25
+25.01
+24.65
+23.83
+25.60
+100 × 30 s
+135◦
+19.98
+24.05
+24.71
+24.58
+24.37
+23.70
+25.15
+100 × 30 s
+180◦
+18.60
+23.34
+23.98
+23.95
+23.86
+23.38
+24.49
+Note: The V-band sky brightness is the Zenith sky background magnitudes.
+of 30 s, the 5σ limiting magnitudes of WFST are: ulim, glim, rlim, ilim, zlim, wlim = 22.31, 23.42, 22.95,
+22.43, 21.50, 23.61 mag. The current estimates of limiting magnitudes are deeper than those in Shi et al.
+(2018). This is because the current total throughput of WFST is higher than previous value, especially
+the throughput increases by ∼ 50% from ∼ 0.4 to ∼ 0.6 in gri bands (see Figure 2(a)), and the current
+Dark night sky background is lower than the previous estimation. Figure 4 compares the sky spectrum of
+New Moon night of the Lenghu site and the atmospheric transmittance curve between this work and Shi
+et al. (2018). We used SkyCalc to estimate the sky background spectrum and atmospheric transmittance,
+while Shi et al. (2018) used the software MODTRAN4 for estimating the atmospheric transmittance at the
+5130 m Ali area and used a Hawaii sky background spectrum as a sky background spectral template. The
+Hawaii sky brightness in ugz bands is brighter than the current model when we scaled both of them into
+the same conditions of mV = 22.3 mag arcsec−2 and airmass = 1.2 (see Figure 4 (a)), Shi et al. (2018)
+assumed the V band sky brightness is mV = 21.50 mag arcsec−2. There is little difference between the
+current atmospheric transmittance model and the spectrum in Shi et al. (2018) (see Figure 4(b)). Our scaled
+atmospheric transmittance is close to the model of MODTRAN.
+We also obtain the limiting magnitudes of WFST under various conditions (Figure 5). In Figure 5, the
+panel (a) shows the WFST limiting magnitudes of different signal-to-noise ratio when the exposure time
+equals to 30 s and 100 × 30 s respectively, the panel (b) shows the limiting magnitudes of different seeing
+4 http://modtran.spectral.com/
+
+Limiting Magnitudes of WFST
+9
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+wavelength (Å)
+10
+3
+10
+2
+10
+1
+10
+0
+10
+1
+flux (photons s
+1 Å
+1 arcsec
+2 m
+2)
+a)
+Shi et al.(2018)
+This work
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+11000
+wavelength (Å)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Atmospheric Transmission
+b)
+Shi et al.(2018)
+This work
+Fig. 4: (a) The red line shows the sky background spectrum of Lenghu site at New Moon night (Moon
+phase θ = 0◦), mV = 22.3 mag, airmass = 1.2. The black dashed line shows the Hawaii sky background
+spectrum scaled into mV = 22.3 mag and airmass = 1.2 in Shi et al. (2018); (b) The red line shows the
+atmospheric transmittance curve of Lenghu site estimated by SkyCalc in this work. The black dashed line
+shows the atmospheric transmittance curve of Shiquanhe astronomical site at an altitude of 5130 m at the
+Ali Area on the Tibetan Plateau estimated by the software MODTRAN.
+conditions when signal-to-noise ratio = 5 and the exposure time = 30 s, 100 × 30 s respectively, and the
+panel (c) shows the 5σ limiting magnitudes of different exposure times. These results are calculated with
+the sky spectrum scaled into airmass = 1.2 condition at a New Moon night (Moon phase θ = 0◦).
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+S/N
+19
+20
+21
+22
+23
+24
+25
+26
+27
+mag
+a)
+exposure time = 30 s
+exposure time = 100 × 30 s
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+seeing (arcsec)
+20
+21
+22
+23
+24
+25
+26
+mag
+b)
+exposure time = 30 s
+exposure time = 100 × 30 s
+50
+100
+150
+200
+exposure time (s)
+20.0
+20.5
+21.0
+21.5
+22.0
+22.5
+23.0
+mag
+c)
+u g r
+i z w
+Fig. 5: (a) The limiting magnitudes of different S/N values when the exposure time is 30 s (dot-dashed
+line) and 100 × 30 s (solid line); (b) The 5σ limiting magnitudes of different seeing conditions when the
+exposure time is 30 s (dot-dashed line) and 100 × 30 s (solid line); (c) The 5σ limiting magnitudes of
+different exposure times when the seeing is 0.75 arcsec. The conditions for a New Moon night (θ = 0◦) and
+airmass = 1.2. Note: The limiting magnitude curves of the g band (blue) and the w band (red) are so close
+that we can not distinguish the two curves easily.
+The WFST survey data will cover the entire northern sky. Its stacked scientific image data can be used
+to study asteroids, solar system, galaxies and cosmology. Its light curves can be used to discover variable
+objects. The estimated WFST limiting redshift of Type Ia supernovae (SNe Ia) can reach z∼0.64 (luminosity
+distance ∼ 6.3 × 103 Mpc) and z∼1.67 (∼ 1.2 × 104 Mpc) when the exposure time is 30 s and 100 × 30 s.
+SNe Ia can be used to constrain the dark energy in the universe (Riess et al., 1998) and directly measure the
+Hubble constant (Riess et al., 2022). By simulating observations of the SNe Ia with the WFST at the Lenghu
+site, Hu et al. (2022) estimate that above 104 pre-maximum SNe Ia will be discovered in one-year during the
+
+10
+Lei et al.
+wide or deep observations, which suggests that WFST will be a powerful facility in revealing the physics of
+SNe Ia. Lin et al. (2022) computed the prospects of finding Tidal Disruption Events (TDEs) with the WFST.
+Their mock observations on 440 deg2 field (CosmoDC2 catalogue) show that ∼ 30 TDEs can be found per
+year if observed at ugrizw bands with 30 s exposures every 10 days. According to Gao et al. (2022), the
+event rate for galaxy-lensed orphan afterglows of γ-ray bursts (GRBs) is to be less than 0.7 yr−1 for the
+whole sky survey of the WFST. Yu et al. (2021) estimated the multi-messenger detection rate of Binary
+Neutron Star Mergers is about 300-3500 yr−1 with a GECAM-like detector for γ-ray emissions and an
+LSST/WFST detector for optical afterglows. Zhu et al. (2021) and Zhu et al. (2022) showed that the optimal
+detection rates of the KN-dominated and AG-dominated GRB afterglows events are ∼0.2/0.5/0.8/20 yr−1
+and ∼500/300/600/3000 yr−1 for ZTF/Mephisto/WFST/LSST, respectively. There are also some studies
+looking forward to detecting Active galactic nucleus (AGN) and researching AGN physics using WFST
+survey data (Xu-Fan Hu et al. in preparation; Su et al. in preparation).
+There are large sky survey telescopes that have been built around the world, and a number of large sky
+survey telescopes are being built. These projects have produced or will generate a large amount survey data
+and have an important impact in all fields of astronomy. Among them, the WFST will be completed in 2023.
+In the future, WFST (Lin et al. 2022; Shi et al. 2018), together with Mephisto (Lei et al. 2021; Lei et al.
+2022; Chen et al. in preparation), Pan-STARRS (Jedicke & Pan-STARRS 2007; Chambers & Pan-STARRS
+Team 2016), SkyMapper (Schmidt et al. 2005; Rakich et al. 2006), ZTF (Bellm et al. 2019; Graham et al.
+2019) and other telescopes will be able to carry out relay observations of the entire sky with large percentage
+time coverage, which will greatly enhance the development of the time-domain astronomy.
+Acknowledgements This work is supported by the Strategic Priority Research Program of Chinese
+Academy of Sciences (Grant No. XDB 41000000, XDB 41010105), the National Science Foundation of
+China (NSFC, Grant No. 12233008, 12173037, 11973038), the China Manned Space Project (No. CMS-
+CSST-2021-A07) and the Cyrus Chun Ying Tang Foundations. We thank Fredrik T Rantakyrand Rodolfo
+Angeloni from Gemini Observatory for their patient elaboration on the Hawaii sky spectrum model and sky
+brightness measurements.
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+

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@@ -0,0 +1,1259 @@
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL
+REGRESSION
+TORSTEN REUTER AND RAINER SCHWABE
+Abstract. Improvements in technology lead to increasing availability of large
+data sets which makes the need for data reduction and informative subsamples
+ever more important.
+In this paper we construct D-optimal subsampling
+designs for polynomial regression in one covariate for invariant distributions
+of the covariate.
+We study quadratic regression more closely for specific
+distributions. In particular we make statements on the shape of the resulting
+optimal subsampling designs and the effect of the subsample size on the design.
+To illustrate the advantage of the optimal subsampling designs we examine the
+efficiency of uniform random subsampling.
+1. Introduction
+Data Reduction is a major challenge as technological advances have lead to a
+massive increase in data collection to a point where traditional statistical methods
+fail or computing power can not keep up. In this case we speak of big data. We
+typically differentiate between the case where the number of covariates is much
+larger than the number of observations and the case where the massive amount of
+observations is the problem. The first case is well studied, most notably by Tibshirani
+(1996) introducing LASSO, which utilizes ℓ1 penalization to find sparse parameter
+vectors, thus fusing subset selection and ridge regression. The second case, often
+referred to as massive data, can be tackled in two ways. Firstly in a probabilistic
+fashion, creating random subsamples in a nonuniform manner. Prominent studies
+include Drineas et al. (2006), Mahoney (2011) and Ma et al. (2014). They present
+subsampling methods for linear regression models called algorithmic leveraging
+that sample according to probabilities based on the normalized statistical leverage
+scores of the covariate matrix. More recently Derezi´nski and Warmuth (2018) study
+volume sampling, where subdata is chosen proportional to the squared volume of
+the parallelepiped spanned by its observations. Conversely to these probabilistic
+methods one can select subdata by applying deterministic rules. Shi and Tang
+(2021) present such a method, that maximizes the minimal distance between two
+observations in the subdata. Wang et al. (2021) propose orthogonal subsampling
+inspired by orthogonal arrays. Most prominently, Wang et al. (2019) introduce
+the information-based optimal subdata selection (IBOSS) to tackle big data linear
+regression in a deterministic fashion based on D-optimality.
+In this paper we study D-optimal subsampling designs for polynomial regression
+in one covariate, where the goal is to select a percentage α of the full data that
+maximizes the determinant of the information matrix. For the conventional study of
+2020 Mathematics Subject Classification. Primary: 62K05. Secondary: 62R07.
+Key words and phrases. Subdata, D-optimality, Massive Data, Polyonmial Regression.
+Corresponding author: Torsten Reuter. E-mail address: [email protected].
+1
+arXiv:2301.03295v1  [math.ST]  9 Jan 2023
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+2
+approximate designs in this setting we refer to Gaffke and Heiligers (1996). Heiligers
+and Schneider (1992) consider specifically cubic regression on a ball. We consider
+D-optimal designs with measure α that are bounded from above by the distribution
+of the known covariate. Such directly bounded designs were first studied by Wynn
+(1977) and Fedorov (1989). Pronzato (2004) considers this setting using a form
+of the subsampling design standardized to one and bounded by α−1 times the
+distribution of the covariates. More recently, Pronzato and Wang (2021) studies
+the same in the context of sequential subsampling. For the characterization of the
+optimal subsampling designs we make use of an equivalence theorem by Sahm and
+Schwabe (2001). This equivalence theorem enables us to construct such designs for
+various settings of the distributional assumptions on the covariates. Here we will
+only look at distributions of the covariate that are invariant to a sign change, i.e.
+symmetric about the vertical axis. We discuss the shape of D-optimal subsampling
+designs for polynomial regression of degree q first. We conclude that the D-optimal
+design is equal to the bounding distribution in its support and the support of the
+optimal design will be the union of at most q + 1 intervals that are symmetrically
+placed around zero. We then study quadratic regression under several distributional
+assumptions more closely. In particular we take a look at the percentage of mass of
+the optimal design on the outer intervals compared to the inner one, which changes
+drastically given the distribution of the covariate. In addition we examine the
+efficiency of uniform random subsampling to illustrate the advantage of the optimal
+designs. All numerical results are obtained by the Newton method implemented in
+the R package nleqslv by Hasselman (2018).
+The rest of this paper is organized as follows.
+In Section 2 we specify the
+polynomial model. In Section 3 we introduce the concept of continuous subsampling
+designs and give characterizations for optimization. In Sections 4 and 5 we present
+optimal designs in the case of linear and quadratic regression, respectively, for
+various classes of distributions of the covariates. Section 6 contains some efficiency
+considerations showing the strength of improvement of the performance of the
+optimal design compared to random subsampling. The paper concludes with a
+discussion in Section 7. Proofs are deferred to an Appendix.
+2. Model Specification
+We consider the situation of pairs (xi, yi) of data, where yi is the value of the
+response variable Yi and xi is the value of a single covariate Xi for unit i = 1, . . . , n,
+for very large numbers of units n. We assume that the dependence of the response
+on the covariate is given by a polynomial regression model
+Yi = β0 + β1Xi + · · · + βqXq
+i + εi
+with independent, homoscedastic random errors εi having zero mean (E(εi) = 0,
+Var(εi) = σ2
+ε > 0). The largest exponent q denotes the degree of the polynomial
+regression, and p = q + 1 is the number of regression parameters β0, . . . , βq to be
+estimated, where, for each k = 1, . . . , q, the parameter βk is the coefficient for the kth
+monomial xk, and β0 denotes the intercept. For example, for q = 1, we have ordinary
+linear regression, Yi = β0 + β1Xi + εi, with p = 2 parameters β0 (intercept) and β1
+(slope) and, for q = 2, we have quadratic regression, Yi = β0 + β1Xi + β2X2
+i + εi,
+with p = 3 and an additional curvature parameter β2. Further, we assume that
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+3
+the covariates Xi are identically distributed and that all covariates Xi and random
+errors εi′ are independent.
+For notational convenience, we write the polynomial regression as a general linear
+model
+Yi = f(Xi)⊤β + εi ,
+where f(x) = (1, x, . . . , xq)⊤ is the p-dimensional vector of regression functions and
+β = (β0, β1, . . . , βq)⊤ is the p-dimensional vector of regression parameters.
+3. Subsampling Design
+We are faced with the problem that the responses Yi are expensive or difficult
+to observe while the values xi of all covariates Xi are available. To overcome this
+problem, we consider the situation that the responses Yi will be observed only for a
+certain percentage α of the units (0 < α < 1) and that these units will be selected
+on the basis of the knowledge of the values xi of the covariate for all units. As
+an alternative motivation, we can consider a situation where all pairs (xi, yi) are
+available but parameter estimation is computationally feasible only on a percentage
+α of the data. In either case we want to find the subsample of pairs (xi, yi) that
+yields the most precise estimation of the parameter vector β.
+To obtain analytical results, the covariates Xi are supposed to have a continuous
+distribution with density fX(x), and we assume that the distribution of the covariates
+is known. The aim is to find a subsample of this distribution that covers a percentage
+α of the distribution and that contains the most information. For this, we will
+consider continuous designs ξ as measures of mass α on R with density fξ(x)
+bounded by the density fX(x) of the covariates Xi such that
+�
+fξ(x) dx = α and
+fξ(x) ≤ fX(x) for all x ∈ R. A subsample can then be generated according to such
+a continuous design by accepting units i with probability fξ(xi)/fX(xi).
+For a continuous design ξ, the information matrix M(ξ) is defined as M(ξ) =
+�
+f(x)f(x)⊤fξ(x) dx. In the present polynomial setup, M(ξ) = (mj+j′(ξ))j′=0,...,q
+j=0,...,q ,
+where mk =
+�
+xkfξ(x) dx is the kth moment associated with the design ξ. Thus, it
+has to be required that the distribution of Xi has a finite moment E(X2q
+i ) of order
+2q in order to guarantee that all entries in the information matrix M(ξ) exist for all
+continuous designs ξ for which the density fξ(x) is bounded by fX(x).
+The information matrix M(ξ) measures the performance of the design ξ in the
+sense that the asymptotic covariance of the least squares estimator ˆβ based on a
+subsample according to the design ξ is proportional to the inverse M(ξ)−1 of the
+information matrix M(ξ) or, more precisely, nα( ˆβ−β) is asymptotically normal with
+mean zero and covariance matrix σ2
+εM(ξ)−1. Note that for continuous designs ξ the
+information matrix M(ξ) is of full rank and, hence, the inverse M(ξ)−1 exists. Based
+on the relation to the asymptotic covariance matrix, it is desirable to maximize
+the information matrix M(ξ). However, as well-known in design optimization,
+maximization of the information matrix cannot be achieved uniformly with respect
+to the Loewner ordering of positive-definiteness. Thus, commonly, a design criterion
+which is a real valued functional of the information matrix M(ξ) will be maximized,
+instead. We will focus here on the most popular design criterion in applications, the
+D-criterion, in its common form log(det(M(ξ))) to be maximized. Maximization
+of the D-criterion can be interpreted in terms of the asymptotic covariance matrix
+to be the same as minimizing the volume of the confidence ellipsoid for the whole
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+4
+parameter vector β based on the least squares estimator or, equivalently, minimizing
+the volume of the acceptance region for a Wald test on the whole model. The
+subsampling design ξ∗ that maximizes the D-criterion log(det(M(ξ))) will be called
+D-optimal, and its density is denoted by fξ∗(x).
+To obtain D-optimal designs, we will make use of standard techniques coming
+from constrained convex optimization and symmetrization. For convex optimization
+we employ the directional derivative
+FD(ξ, η) = lim
+ϵ→0+
+1
+ϵ (log(det(M((1 − ϵ)ξ + ϵη))) − log(det(M(ξ))))
+of the D-criterion at a design ξ with non-singular information matrix M(ξ) in
+the direction of a design η, where we allow here η to be a general design of
+mass α that has not necessarily a density bounded by fX(x). Evaluating of the
+directional derivative yields FD(ξ, η) = p − trace(M(ξ)−1M(η)) (compare Silvey,
+1980, Example 3.8) which reduces to FD(ξ, ξx) = p − αf(x)⊤M(ξ)−1f(x) for a
+one-point design η = ξx which assigns all mass α to a single setting x in the
+design region. Equivalently, for one-point designs η = ξx, we may consider the
+sensitivity function ψ(x, ξ) = αf(x)⊤M(ξ)−1f(x) which covers the essential part
+of the directional derivative (ψ(x, ξ) = p − FD(ξ, ξx)). For the characterization
+of the D-optimal continuous design, the constrained equivalence theorem under
+Kuhn-Tucker conditions (see Sahm and Schwabe, 2001, Corollary 1 (c)) can be
+reformulated in terms of the sensitivity function.
+Theorem 3.1. The design ξ∗ is D-optimal if and only if there exist a threshold s∗
+and settings a1 > · · · > a2r for some r (1 ≤ r ≤ q) such that
+(i) the D-optimal design ξ∗ is given by
+fξ∗(x) =
+�
+fX(x)
+if x ∈ X ∗
+0
+otherwise
+(ii) ψ(x, ξ∗) ≥ s∗ for x ∈ X ∗, and
+(iii) ψ(x, ξ∗) < s∗ for x ̸∈ X ∗,
+where X ∗ = �r
+k=0 Ik and I0 = [a1, ∞), Ir = (−∞, a2r], and Ik = [a2k+1, a2k], for
+k = 1, . . . , r − 1, are mutually disjoint intervals.
+The density fξ∗(x) = fX(x)1X ∗(x) = �r
+k=0 fX(x)1Ik(x) of the D-optimal design
+ξ∗ is concentrated on, at most, q + 1 intervals Ik. Here, 1A(x) denotes an indicator
+function on the set A, i. e. 1A(x) = 1 for x ∈ A, and 1A(x) = 0 otherwise. The
+density fξ∗(x) has a 0−1-property such that it is either equal to the density fX(x) of
+the covariates (on X ∗) or equal to 0 (on the complement of X ∗). Then the generation
+of a subsample according to the optimal continuous design ξ∗ can be implemented
+easily by accepting all units i for which the value xi of the covariate is in X ∗ and
+rejecting all other units with xi ̸∈ X ∗. The threshold s∗ can be interpreted as the
+(1 − α)-quantile of the distribution of the sensitivity function ψ(Xi, ξ∗) as a function
+of the random variable Xi (see Pronzato and Wang, 2021).
+A further general concept to be used is equivariance. This can be employed
+to transform the D-optimal design simultaneously with a transformation of the
+distribution of the covariates. More precisely, the location scale transformation
+Zi = σXi + µ of the covariates and their distribution is conformable with the
+regression function f(x) in polynomial regression, and the D-criterion is equivariant
+with respect to such transformations.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+5
+Theorem 3.2. Let fξ∗(x) be the density for a D-optimal design ξ∗ for covariates
+Xi with density fX(x). Then fζ∗(z) = 1
+σfξ∗( z−µ
+σ ) is the density for a D-optimal
+design ζ∗ for covariates Zi = σXi + µ with density fZ(z) = 1
+σfX( z−µ
+σ ).
+In particular, also the optimal design ζ∗ is concentrated on, at most, p = q + 1
+intervals, and its density fζ∗(z) is either equal to the density fZ(z) of the covariates
+Zi (on Z∗ = σX ∗ + µ) or it is equal to 0 (elsewhere) such that the optimal
+subsampling can be implemented quite easily.
+A further reduction of the optimization problem can be achieved by utilizing
+symmetry properties. Therefore, we consider the transformation of sign change,
+g(x) = −x, and assume that the distribution of the covariates is symmetric,
+fX(−x) = fX(x) for all x. For a continuous design ξ, the design ξg transformed by
+sign change has density fξg(x) = fξ(−x) and, thus, satisfies the boundedness condi-
+tion fξg(x) ≤ fX(x), when the distribution of Xi is symmetric, and has the same
+value for the D-criterion as ξ, log(det(M(ξg))) = log(det(M(ξ))). By the concavity
+of the D-criterion, standard invariance arguments can be used as in Pukelsheim
+(1993, Chapter 13) and Heiligers and Schneider (1992). In particular, any con-
+tinuous design ξ is dominated by its symmetrization ¯ξ = (ξ + ξg)/2 with density
+f¯ξ(x) = (fξ(x) + fξ(−x))/2 ≤ fX(x) such that log(det(M(¯ξ))) ≥ log(det(M(ξ)))
+(Pukelsheim, 1993, Chapter 13.4). Hence, we can restrict the search for a D-optimal
+design to symmetric designs ¯ξ with density f¯ξ(−x) = f¯ξ(x) which are invariant with
+respect to sign change (¯ξg = ¯ξ). For these symmetric designs ¯ξ, the moments mk(¯ξ)
+are zero for odd k and positive when k is even. Hence, the information matrix M(¯ξ)
+is an even checkerboard matrix (see Jones and Willms, 2018) with positive entries
+mj+j′(¯ξ) for even index sums and entries equal to zero when the index sum is odd.
+The inverse M(¯ξ)−1 of the information matrix M(¯ξ) shares the structure of an even
+checkerboard matrix. Thus, the sensitivity function ψ(x, ¯ξ) is a polynomial with
+only terms of even order and is, hence, a symmetric function of x. This leads to a
+simplification of the representation of the optimal design in Theorem 3.1 because
+the support X ∗ of the optimal design ξ∗ will be symmetric.
+Corollary 3.3. Let the distribution of Xi be symmetric. Then, for the D-optimal
+design ξ∗ with density fξ∗(x) = �r
+k=0 fX(x)1Ik(x) the boundaries a1, . . . , a2r of
+the intervals Ik = [a2k+1, a2k] are symmetric, i. e. a2r+1−k = −ak and, similarly,
+Ir+2−k = −Ik for the intervals.
+This characterization of the optimal design ξ∗ will be illustrated in the next two
+sections for ordinary linear regression (q = 1) and for quadratic regression (q = 2).
+4. Optimal Subsampling for Linear Regression
+In the case of ordinary linear regression Yi = β0 + β1Xi + εi we have
+M(ξ∗) =
+�
+α
+m1(ξ∗)
+m1(ξ∗)
+m2(ξ∗)
+�
+,
+for the information matrix of the optimal design ξ∗.
+The sensitivity function
+is a polynomial of degree 2. To obtain the D-optimal continuous design ξ∗ by
+Theorem 3.1, the boundary points a1 and a2 have to be determined to solve the
+two nonlinear equations
+P(Xi ≤ a2) + P(Xi ≥ a1) = α
+(1)
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+6
+and
+ψ(a1, ξ∗) = ψ(a2, ξ∗) .
+(2)
+The D-optimal continuous design ξ∗ has density fξ(x) = fX(x) for x ≤ a2 and for
+x ≥ a1 while fξ(x) = 0 for a2 < x < a1. The corresponding subsampling design
+then accepts those units i for which xi ≤ a2 or xi ≥ a1, and rejects all units i for
+which a2 < xi < a1.
+When the distribution of Xi is symmetric, Corollary 3.3 provides symmetry
+a2 = −a1 of the boundary points. Hence, by the symmetry of the distribution,
+P(Xi ≤ a2) = P(Xi ≥ a1) = α/2, and a1 has to be chosen as the (1 − α/2)-quantile
+of the distribution of Xi to obtain the D-optimal continuous design.
+Example 4.1 (normal distribution). If the covariates Xi come from a standard
+normal distribution, then the optimal boundaries are the (α/2)- and the (1 − α/2)-
+quantile ±z1−α/2, and unit i is accepted when |xi| ≥ z1−α/2.
+For Xi having a general normal distribution with mean µ and variance σ2, the
+optimal boundaries remain to be the (α/2)- and (1−α/2)-quantile a2 = µ−σz1−α/2
+and a1 = µ + σz1−α/2, respectively, by Theorem 3.2.
+This approach applies accordingly to all distributions which are obtained by a
+shift transformation of a symmetric distribution: Units will be accepted if their
+values of the covariate lie in the lower or upper (α/2)-tail of the distribution. This
+procedure can be interpreted as a theoretical counterpart in one dimension of the
+IBOSS method proposed by Wang et al. (2019).
+However, for asymmetric distributions, the optimal proportions for sampling from
+the upper and lower tail will differ.
+Example 4.2 (exponential distribution). If the covariates Xi come from a standard
+exponential distribution with density fX(x) = e−x, x ≥ 0, we conclude from
+Theorem 3.1 that fξ∗(x) = fX(x)1(−∞,a]∪[b,∞)(x). We can calculate the entries of
+M(ξ∗) as functions of a1 = a and a2 = b as
+m1(ξ∗) = 1 + (a + 1)e−a − (b + 1)e−b
+m2(ξ∗) = 2 + (a2 + 2a + 2)e−a − (b2 + 2b + 2)e−b .
+To obtain the optimal solutions for a and b, the two nonlinear equations (1) and (2)
+come here to be
+1 − e−b + e−a = α
+and
+f(a)⊤M(ξ∗)−1f(a) = f(b)⊤M(ξ∗)−1f(b) .
+The results for selected values of α can be seen in Table 1. Additionally to the optimal
+values for a and b, also the proportions P(Xi ≤ b) and P(Xi ≥ a) are presented in
+Table 1 together with the percentage of mass allocated to the left interval [0, b]. In
+Figure 1, the density fξ∗ of the optimal design ξ∗ and the corresponding sensitivity
+function ψ(x, ξ∗) are exhibited for α = 0.5 and α = 0.3. Vertical lines indicate the
+positions of the boundary points a and b, and the dotted horizontal line displays the
+threshold s∗.
+As could have been expected, less mass is assigned to the right tail
+of the right-skewed distribution because observations from the right tail are more
+influential and, thus, more observations seem to be required on the lighter left tail
+for compensation.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+7
+Table 1. Numeric values for a and b for selected values of α in
+the case of standard exponential Xi
+α
+b
+P(Xi ≤ b)
+a
+P(Xi ≥ a)
+% of mass on [0, b]
+0.5
+0.39572
+0.32681
+1.75335
+0.17319
+65.36
+0.3
+0.21398
+0.19264
+2.23153
+0.10736
+64.21
+0.1
+0.06343
+0.06146
+3.25596
+0.03854
+61.46
+0.01
+0.00579
+0.00577
+5.46588
+0.00423
+57.71
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+1
+2
+3
+4
+5
+x
+Density
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+x
+Sensitivity function
+(a) α = 0.5
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+1
+2
+3
+4
+5
+x
+Density
+1
+2
+3
+4
+0
+1
+2
+3
+4
+5
+x
+Sensitivity function
+(b) α = 0.3.
+Figure 1. Density of the optimal design (solid line) and the stan-
+dard exponential distribution (dashed line, upper panels), and
+sensitivity functions (lower panels) for α = 0.5 (left) and α = 0.3
+(right)
+For Xi having an exponential distribution with general intensity λ > 0 (scale 1/λ),
+the optimal boundary points remain to be the same quantiles as in the standard
+exponential case, a1 = a/λ and a2 = b/λ, by Theorem 3.2.
+5. Optimal Subsampling for Quadratic Regression
+In the case of quadratic regression Yi = β0 + β1Xi + β2X2
+i + εi we have
+M(¯ξ) =
+�
+�
+α
+0
+m2(¯ξ)
+0
+m2(¯ξ)
+0
+m2(¯ξ)
+0
+m4(¯ξ)
+�
+� ,
+for the information matrix of a symmetric design ¯ξ. The corresponding sensitivity
+function ψ(x, ¯ξ) is a polynomial of degree 4 and is symmetric in x.
+According to Corollary 3.3, the density fξ∗(x) of the D-optimal continuous design
+ξ∗ has, at most, three intervals that are symmetrically placed around zero, where
+the density is equal to the bounding density fX(x), and fξ∗(x) is equal to zero
+elsewhere. Thus the density fξ∗(x) of the D-optimal design has the following shape.
+fξ∗(x) = fX(x)1(−∞,−a]∪[−b,b]∪[a,∞)(x) ,
+where a > b ≥ 0 and where we formally allow b = 0 which means that ψ(0, ξ∗) ≤
+0 and that the density fξ∗(x) is concentrated on only two intervals, fξ∗(x) =
+fX(x)1{x∈(−∞,−a]∪[a,∞)}. Although the information matrix will be non-singular
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+8
+even in the case of two intervals (b = 0), the optimal design will typically include a
+non-degenerate interior interval [−b, b], b > 0, as illustrated below in Theorem 5.2.
+To obtain the D-optimal continuous design ξ∗ by Corollary 3.3, the boundary
+points a = a1 and b = a2 (resp. b = 0) have to be determined to solve the two
+nonlinear equations
+P(|Xi| ≤ b) + P(|Xi| ≥ a) = α
+(3)
+and
+ψ(a, ξ∗) = ψ(b, ξ∗) .
+(4)
+For finding the optimal solutions, we use the Newton method implemented in the R
+package nleqslv by Hasselman (2018) to calculate numeric values for a and b based
+on equations (3) and (4) for various symmetric distributions.
+Example 5.1 (normal distribution). For the case that the covariates Xi come from
+a standard normal distribution, results are given in Table 2 for some values of α.
+Additionally to the optimal values for a and b, also the proportions P(Xi ≥ a) =
+Table 2. Numeric values for a and b for selected values of α in
+the case of standard normal Xi
+α
+a
+1 − Φ(a)
+b
+2Φ(b) − 1
+% of mass on [−b, b]
+0.5
+1.02800
+0.15198
+0.24824
+0.19605
+39.21
+0.3
+1.34789
+0.08885
+0.15389
+0.12231
+40.77
+0.1
+1.88422
+0.02977
+0.05073
+0.04046
+40.46
+0.01
+2.73996
+0.00307
+0.00483
+0.00386
+38.55
+P(Xi ≤ −a) = 1 − Φ(a) and P(−b ≤ Xi ≤ b) = 2Φ(b) − 1 are presented in Table 2
+together with the percentage of mass (2Φ(b) − 1)/α allocated to the interior interval
+[−b, b]. In Figure 2, the density fξ∗ of the optimal design ξ∗ and the corresponding
+sensitivity function ψ(x, ξ∗) are exhibited for α = 0.5 and α = 0.1. Vertical lines
+0.0
+0.1
+0.2
+0.3
+0.4
+−2
+0
+2
+x
+Density
+1.7
+1.8
+1.9
+2.0
+−2
+0
+2
+x
+Sensitivity function
+(a) α = 0.5
+0.0
+0.1
+0.2
+0.3
+0.4
+−2
+0
+2
+x
+Density
+1.75
+2.00
+2.25
+2.50
+2.75
+−2
+0
+2
+x
+Sensitivity function
+(b) α = 0.1
+Figure 2. Density of the optimal design (solid line) and the stan-
+dard normal distribution (dashed line, upper panels), and sensitivity
+functions (lower panels) for α = 0.5 (left) and α = 0.1 (right)
+indicate the positions of the boundary points −a, −b, b, and a, respectively. In the
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+9
+subplots of the sensitivity function, the dotted horizontal line displays the threshold
+s∗. For other values of α, the plots are looking similar.
+The numerical results in Table 2 suggest that the interior interval [−b, b] does
+not vanish for any α (0 < α < 1). This will be established in the following theorem.
+Theorem 5.2. Let the distribution of Xi be standard normal.
+For all α ∈
+(0, 1), there exists b > 0 such that the D-optimal design ξ∗ has density fξ∗(x) =
+fX(x)1{x∈(−∞,−a]∪[−b,b]∪[a,∞)}.
+For Xi having a general normal distribution with mean µ and variance σ2, the
+optimal boundary points remain to be the same quantiles as in the standard normal
+case, a1, a4 = µ ± σa and a2, a3 = µ ± σb, by Theorem 3.2.
+Example 5.3 (uniform distribution). If the covariates Xi are uniformly distributed
+on [−1, 1] with density fX(x) = 1
+21[−1,1](x), we can obtain analytical results for the
+dependence of the subsampling design on the proportion α to be selected.
+The distribution of Xi is symmetric. By Corollary 3.3, the density of the D-
+optimal continuous design ξ∗ has the shape
+fξ∗(x) = 1
+21[−1,−a]∪[−b,b]∪[a,1](x)
+where a and b are the solution of the following two equations
+1 − a + b = α
+and
+f(a)⊤M(ξ∗)−1f(a) = f(b)⊤M(ξ∗)−1f(b) ,
+where the entries in the even checkerboard matrix M(ξ∗) are m0(ξ∗) = α, m2(ξ∗) =
+1
+3(1 − a3 + b3), and m4(ξ∗) = 1
+5(1 − a5 + b5). Solving these equations results in
+a(α) = 1
+2
+�
+1 − α +
+�
+45 − 15α + 15α2 − 45α3 + 20α4
+45(1 − α)
+− 4α
+√
+5
+√
+45 − 90α + 90α2 − 75α3 + 57α4 − 27α5 + 5α6
+45(1 − α)
+�1/2 �
+(5)
+and
+b(α) = a − (1 − α)
+(6)
+for the dependence of a and b on α. The values of a and b are plotted in Figure 3.
+There it can be seen that a and b both tend to 1/
+√
+5 as α tends to 1. Similar to the
+normal distribution, the resulting values and illustrations are given in Table 3 and
+Figure 4.
+Also here, vertical lines indicate the positions of the boundary points −a,
+−b, b, and a, and the dotted horizontal line displays the threshold s∗. Moreover,
+the percentage of mass at the different intervals is displayed in Figure 5.
+The results in Table 3 and Figure 5 suggest that the percentage of mass on all
+three intervals [−1, −a], [−b, b], and [a, 1] tend to 1/3 as α tends to 0. We establish
+this in the following theorem.
+Theorem 5.4. Let Xi be uniformly distributed on [−1, 1] and ξ∗
+α be the optimal
+subsampling design for α, 0 < α < 1, defined in equations (5) and (6). Then
+limα→0 ξ∗
+α([−b, b])/α = 1/3.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+10
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+α
+a,b
+Figure 3. Boundary points a and b in dependence on α
+Table 3. Numeric values for a and b for selected values of α in
+the case of uniform Xi on [−1, 1]
+α
+a
+1 − P(Xi ≥ a)
+b = P(−b ≤ Xi ≤ b)
+% of mass on [−b, b]
+0.5
+0.70983
+0.14508
+0.20983
+41.97
+0.3
+0.81737
+0.09132
+0.11737
+39.12
+0.1
+0.93546
+0.03227
+0.03546
+35.46
+0.01
+0.99336
+0.00332
+0.00336
+33.55
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+x
+Density
+2.00
+2.25
+2.50
+2.75
+3.00
+−1.0
+−0.5
+0.0
+0.5
+1.0
+x
+Sensitivity function
+(a) α = 0.5
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+x
+Density
+2.0
+2.5
+3.0
+3.5
+4.0
+−1.0
+−0.5
+0.0
+0.5
+1.0
+x
+Sensitivity function
+(b) α = 0.1
+Figure 4. Density of the optimal design (solid line) and the uni-
+form distribution on [−1, 1] (dashed line, upper panels), and sen-
+sitivity functions (lower panels) for α = 0.5 (left) and α = 0.1
+(right)
+It is worth-while mentioning that the percentages of mass displayed in Figure 5
+are not monotonic over the whole range of α ∈ (0, 1), as, for example the mass at
+the interior interval [−b, b] is increasing from 0.419666 at b = 0.50 to 0.448549 at
+b = 0.92 and then slightly decreasing again to 0.447553 at b = 0.99.
+In the two preceding examples it could be noticed that the mass of observations
+is of comparable size for the three supporting intervals in the case of a normal and
+of a uniform distribution with light tails. This will be different in the case of a
+heavy-tailed distribution for the covariates Xi.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+11
+0.28
+0.30
+0.32
+0.25
+0.50
+0.75
+α
+(1−a)/2α
+0.33
+0.36
+0.39
+0.42
+0.45
+0.25
+0.50
+0.75
+α
+b/α
+Figure 5. Percentage of mass on [a, 1] (left) and [−b, b] (right) as
+a function of α
+6. Efficiency considerations
+To exhibit the gain in using a D-optimal design compared to random subsampling,
+we consider the performance of the uniform random subsampling design ξα of size
+α, which has density fξα(x) = αfX(x), compared to the D-optimal subsampling
+design ξ∗
+α with mass α. More precisely, the D-efficiency of any design ξ with mass
+α is defined as
+effD,α(ξ) =
+� det(M(ξ))
+det(M(ξ∗α))
+�1/p
+,
+where p is the dimension of the parameter vector β.
+For this definition the
+homogeneous version (det(M(ξ)))1/p of the D-criterion is used which satisfies
+(det(λM(ξ)))1/p = λ(det(M(ξ)))1/p (see Pukelsheim, 1993, Chapter 6.2).
+For uniform random subsampling, the information matrix is given by M(ξα) =
+αM(ξ1), where M(ξ1) is the information matrix for the full sample with raw moments
+mk(ξ1) = E[Xk
+i ] as entries in the (j, j′)th position, j +j′ = k. Thus, the D-efficiency
+effD,α(ξα) can be nicely interpreted: When uniform random subsampling is used, the
+inverse of the efficiency effD,α(ξα)−1 times α gives the sample size (mass) required
+to obtain the same precision in terms of the D-criterion as when the D-optimal
+design ξ∗
+α of mass α is used. For example, if the efficiency effD,α(ξα) is equal to 0.5,
+then twice as many observations would be needed under uniform random sampling
+than for a D-optimal subsampling design of size α. Of course, the full sample has
+higher information than any proper subsample such that, obviously, effD,α(ξα) ≥ α
+holds for uniform random subsampling.
+For the examples of Sections 4 and 5, the efficiency of uniform random subsampling
+is given in Table 4 for selected values of α and exhibited in Figure 6 for the full
+range of α between 0 and 1 (solid lines). Here the determinant of the information
+matrix is determined as in the examples of Sections 4 and 5 for the optimal designs
+ξα∗ either numerically or by explicit formulas where available.
+Both Table 4 and Figure 6 indicate that the efficiency of uniform random subsam-
+pling is decreasing in all cases when the proportion α of subsampling gets smaller.
+In the case of uniformly distributed covariates, the decrease is more or less linear
+with a minimum value of approximately 0.58 for quadratic regression when α is
+small. In the other cases, where the distribution of the covariates is unbounded, the
+efficiency apparently decreases faster, when the proportion α is smaller than 10%,
+and tends to 0 for α → 0.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+12
+Table 4. Efficiency for selected values of α
+α
+0.5
+0.3
+0.1
+0.01
+linear regression
+normal
+0.73376
+0.61886
+0.47712
+0.34403
+exponential
+0.73552
+0.61907
+0.46559
+0.30690
+quadratic regression
+normal
+0.73047
+0.59839
+0.41991
+0.24837
+uniform
+0.78803
+0.70475
+0.62411
+0.58871
+0.4
+0.6
+0.8
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+α
+Efficiency
+(a) Linear regression, normal covariates
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+α
+Efficiency
+(b) Linear regression, exponential covari-
+ates
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+α
+Efficiency
+(c) Quadratic regression, normal covariates
+0.6
+0.7
+0.8
+0.9
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+α
+Efficiency
+(d) Quadratic regression, uniform covari-
+ates
+Figure 6. Efficiency of uniform random subsampling (solid line)
+and of an IBOSS-type design (dashed line)
+The latter property can be easily seen for linear regression and symmetric
+distribution: There, the efficiency effD,α(ξα) of uniform random sampling is bounded
+from above by c/q1−α/2, where c = E(X2
+i )1/2 is a constant and q1−α/2 is the (1−α/2)-
+quantile of the distribution of the covariates. When the distribution is unbounded
+like the normal distribution, then these quantiles tend to infinity for α → 0 and,
+hence, the efficiency tends to 0. Similar results hold for quadratic regression and
+asymmetric distributions.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+13
+In any case, as can be seen from Table 4, the efficiency of uniform random
+subsampling is quite low for reasonable proportions α ≤ 0.1 and, hence, the gain in
+using the D-optimal subsampling design is substantial.
+By equivariance arguments as indicated above in the examples of Sections 4 and
+5, the present efficiency considerations carry over directly to covariates having a
+general normal, exponential, or uniform distribution, respectively.
+In the IBOSS approach by Wang et al. (2019), half of the proportion α is taken
+from both tails of the data. The corresponding continuous subsampling design ξ′
+α
+would be to choose the boundary points a1 and a2 to be the (1 − α/2)- and (α/2)-
+quantile of the distribution of the covariates, respectively. For linear regression,
+it can been seen from Corollary 3.3 that the design ξ′
+α is D-optimal when the
+distribution of the covariates is symmetric. As the IBOSS procedure does not
+use prior knowledge of the distribution, it would be tempting to investigate the
+efficiency of the corresponding continuous subsampling design ξ′
+α under asymmetric
+distributions. For the exponential distribution, this efficiency effD,α(ξ′
+α) is added to
+the upper right panel in Figure 6 by a dashed line. There the design ξ′
+α shows a
+remarkably high efficiency over the whole range of α with a minimum value 0.976
+at α = 0.332.
+As an extension of IBOSS for quadratic regression, we may propose a procedure
+which takes proportions α/3 from both tails of the data as well as from the center
+of the data. This procedure can be performed without any prior knowledge of the
+distribution of the covariates. The choice of the proportions α/3 is motivated by
+the standard case D-optimal design on an interval where one third of the weight is
+allocated to each of the endpoints and to the midpoint of the region. For a symmetric
+distribution, the corresponding continuous subsampling design ξ′′
+α can be defined by
+the boundary points a and b to be the (1 − α/3)- and (1/2 + α/6)-quantile of the
+distribution of the covariates, respectively. In the case of the uniform distribution,
+the design ξ′′
+α is the limiting D-optimal design for α → 0 by Theorem 5.4. For the
+whole range of α and for the normal distribution, the efficiency effD,α(ξ′′
+α) is shown
+in the lower panels of Figure 6 by dashed lines. In both cases, the design ξ′′
+α is
+highly efficient over the whole range of α with minimum values 0.994 at α = 0.079
+for the normal distribution and 0.989 at α = 0.565 for the uniform distribution,
+respectively.
+7. Concluding Remarks
+In this paper we have considered a theoretical approach to evaluate subsampling
+designs under distributional assumptions on the covariates in the case of polynomial
+regression on a single explanatory variable. Main emphasis was on D-optimal
+designs. But many of the results may be extended to other optimality criteria like A-
+and E-optimality from the Kiefer’s Φq-class of optimality criteria, IMSE-optimality
+for predicting the mean response, or optimality criteria based on subsets or linear
+functionals of parameters.
+The D-optimal designs show a high performance compared to uniform random
+subsampling. In particular, for small proportions, the efficiency of uniform random
+subsampling tends to zero. This property is in accordance with the observation that
+estimation based on subsampling according to IBOSS is “consistent” in the sense
+that the mean squared error goes to zero with increasing population size even when
+the size of the subsample is fixed.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+14
+We propose a generalization of the IBOSS method to quadratic regression which
+does not require prior knowledge of the distribution of the covariates and which
+performs remarkably well compared to the optimal design. However, an extension
+to higher order polynomials, does not seem to be obvious.
+Appendix A. Proofs
+Before proving Theorem 3.1, we establish two preparatory lemmas on properties
+of the sensitivity function ψ(x, ξ) for a continuous design ξ with density fξ(x) and
+reformulate an equivalence theorem on constraint design optimality by Sahm and
+Schwabe (2001) for the present setting. The first lemma deals with the shape of the
+sensitivity function.
+Lemma A.1. The sensitivity function ψ(x, ξ) is a polynomial of degree 2q with
+positive leading term.
+Proof of Lemma A.1. For a continuous design ξ with density fξ(x), the information
+matrix M(ξ) and, hence, its inverse M(ξ)−1 is positive definite. Thus the last
+diagonal element m(pp) of M(ξ)−1 is positive and, as f(x) = (1, x, . . . , xq)⊤, the
+sensitivity function ψ(x, ξ) = f(x)⊤M(ξ)−1f(x) is a polynomial of degree 2q with
+coefficient m(pp) > 0 of the leading term.
+□
+The second lemma reveals a distributional property of the sensitivity function
+considered as a function in the covariates Xi.
+Lemma A.2. The random variable ψ(Xi, ξ) has a continuous cumulative distribu-
+tion function.
+Proof of Lemma A.2. As the sensitivity function ψ(x, ξ) is a non-constant poly-
+nomial by Lemma A.1, the equation ψ(x, ξ) = s has only finitely many roots
+x1, . . . , xℓ, say, by the fundamental theorem of algebra. Hence, P(ψ(Xi, ξ) = s) =
+�ℓ
+k=1 P(Xi = xk) = 0 by the continuity of the distribution of Xi which proves the
+continuity of the cumulative distribution function of ψ(Xi, ξ).
+□
+With the continuity of the distribution of ψ(Xi, ξ∗) the following equivalence
+theorem can be obtained from Corollary 1(c) in Sahm and Schwabe (2001) for
+the present setting by transition from the directional derivative to the sensitivity
+function.
+Theorem A.3 (Equivalence Theorem). The design ξ∗ is D-optimal if and only if
+there exist a threshold s∗ and a subset X ∗ of the design region such that
+(i) the D-optimal design ξ∗ is given by
+fξ∗(x) = fX(x)1X ∗(x)
+(ii) ψ(x, ξ∗) ≥ s∗ for x ∈ X ∗, and
+(iii) ψ(x, ξ∗) < s∗ for x ̸∈ X ∗.
+As P(ψ(Xi, ξ∗) ≥ s∗) = P(Xi ∈ X ∗) =
+�
+fξ∗(x) dx = α, the threshold s∗ is the
+(1 − α)-quantile of the distribution of ψ(Xi, ξ∗).
+Proof of Theorem 3.1. By Lemma A.1 the sensitivity function ψ(x, ξ) is a polyno-
+mial in x of degree 2q with positive leading term. Using the same argument as in
+the proof of Lemma A.2 we obtain that there are at most 2q roots of the equation
+ψ(x, ξ∗) = s∗ and, hence, there are at most 2q sign changes in ψ(x, ξ∗) − s∗. As
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+15
+ψ(x, ξ∗) is a polynomial of even degree, also the number of (proper) sign changes has
+to be even, and they occur at a1 > · · · > a2r, say, r ≤ q. Moreover, for 0 < α < 1,
+X ∗ is a proper subset of the design region and, thus, there must be at least one sign
+change, r ≥ 1. Finally, as the leading coefficient of ψ(x, ξ∗) is positive, ψ(x, ξ∗) gets
+larger than s∗ for x → ±∞ and, hence, the outmost intervals [a1, ∞) and (−∞, a2r]
+are included in the support X ∗ of ξ∗. By the interlacing property of intervals with
+positive and negative sign for ψ(x, ξ∗) − s∗, the result follows.
+□
+Proof of Theorem 3.2. First note that for any µ and σ > 0, the location scale
+transformation z = σx + µ is conformable with the regression function f(x), i. e.
+there exists a non-singular matrix Q such that f(σx + µ) = Qf(x) for all x. Then,
+for any design ξ bounded by fX(x), the design ζ has density fζ(z) = 1
+σfξ( z−µ
+σ )
+bounded by fZ(z) = 1
+σfX( z−µ
+σ ). Then, by the transformation theorem for measure
+integrals, it holds that
+M(ζ) =
+�
+f(z)f(z)⊤ζ(dz)
+=
+�
+f(σx + µ)f(σx + µ)⊤ξ(dx)
+=
+�
+Qf(x)f(x)⊤Q⊤ξ(dx)
+= QM(ξ)Q⊤.
+Therefore det(M(ζ)) = det(Q)2 det(M(ξ)). Thus ξ∗ maximizes the D-criterion over
+the set of designs bounded by fX(x) if and only if ζ∗ maximizes the D-criterion
+over the set of designs bounded by fZ(z).
+□
+Proof of Corollary 3.3. The checkerboard structure of the information matrix M(ξ∗)
+carries over to its inverse M(ξ∗)−1. Hence, the sensitivity function ψ(x, ξ∗) is an
+even polynomial, which has only non.zero coefficients for even powers of x, and is
+thus symmetric with respect to 0, i. e. ψ(−x, ξ∗) = ψ(x, ξ∗). Accordingly, also the
+roots of ψ(x, ξ∗) = s∗ are symmetric with respect to 0.
+□
+Proof of Theorem 5.2. Suppose there exists an α ∈ (0, 1) such that a = ∞. Then
+b = z(1+α)/2, obviously and it must hold that ψ(z(1−α)/2, ξ∗) ≥ limx→∞ ψ(x, ξ∗).
+Since M(ξ∗) is positive definite, the leading term of the polynomial ψ(x, ξ∗) in x is
+positive and subsequently ψ(z(1−α)/2, ξ∗) < limx→∞ ψ(x, ξ∗). This is a contradiction
+and therefore a < ∞ for all α ∈ (0, 1).
+Suppose there exists an α ∈ (0, 1) such that b = 0. Then a = z1−α/2, obviously.
+Further, it must hold that ψ(z1−α/2, ξ∗) ≥ ψ(0, ξ∗). We will show that this inequality
+is in fact false. Because ξ∗ is invariant to the sign change we have
+M(ξ∗) =
+�
+�
+α
+0
+m2(ξ∗)
+0
+m2(ξ∗)
+0
+m2(ξ∗)
+0
+m4(ξ∗)
+�
+�
+and thus
+M(ξ∗)−1 =
+�
+�
+�
+m4(ξ∗)
+αm4(ξ∗)−m2(ξ∗)2
+0
+−m2(ξ∗)
+αm4(ξ∗)−m2(ξ∗)2
+0
+1
+m2(ξ∗)
+0
+−m2(ξ∗)
+αm4(ξ∗)−m2(ξ∗)2
+0
+α
+αm4(ξ∗)−m2(ξ∗)2
+�
+�
+� ,
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+16
+where
+m2(ξ∗) =
+�
+R
+x2fξ∗(x) dx = α +
+�
+2/πz1−α/2 exp
+�
+−z2
+1−α/2
+2
+�
+,
+m4(ξ∗) =
+�
+R
+x4fξ∗(x) dx =
+�
+2/πz3
+1−α/2 exp
+�
+−z2
+1−α/2
+2
+�
++ 3m2(ξ∗).
+We will write a instead of z1−α/2 from here on out for readability. For the directional
+derivatives we have
+ψ(0, ξ∗) = α(1 0 0)M(ξ∗)−1(1 0 0)⊤
+=
+αm4(ξ∗)
+αm4(ξ∗) − m2(ξ∗)2
+and
+ψ(a, ξ∗) = α(1 a a2)M(ξ∗)−1(1 a a2)⊤
+=
+αm4(ξ∗)
+αm4(ξ∗) − m2(ξ∗)2 − c,
+where
+c = αa2
+�
+2m2(ξ∗)
+αm4(ξ∗) − m2(ξ∗)2 −
+a2α
+αm4(ξ∗) − m2(ξ∗)2 −
+1
+m2(ξ∗)
+�
+.
+c is continuous in α ∈ (0, 1) and does not have any roots in (0, 1). We can easily
+check, that c > 0 for e.g. α = 0.1 and thus c > 0 for all α ∈ (0, 1). This yields
+ψ(z1−α/2, ξ∗) < ψ(0, ξ∗) for all α ∈ (0, 1), which is a contradiction.
+□
+Proof of Theorem 5.4. Firstly, we check if limα→0 b(α)/α = 1/3, as b(α)/α =
+� b
+−b ξ∗(dx)/
+� 1
+−1 ξ∗(dx) describes the percentage of mass on [−b, b].
+Note that
+limα→0 b(α)/α is by definition the derivative of b(α) at the point α0 = 0. Thus we
+consider the derivative of b.
+db(α)
+dα
+= 1
+2 + 1
+2
+�
+u′(α)v(α) − u(α)v′(α)
+v(α)2
+1
+2
+�
+u(α)/v(α)
+�
+,
+where
+u(α) = 45 − 15α + 15α2 − 45α3 + 20α4
+− 4
+√
+5
+�
+45α2 − 90α3 + 90α4 − 75α5 + 57α6 − 27α7 + 5α8,
+c(α) = 4
+√
+5(90α − 270α2 + 360α3 − 375α4 + 342α5 − 189α6 + 40α7)
+2
+√
+45α2 − 90α3 + 90α4 − 75α5 + 57α6 − 27α7 + 5α8
+u′(α) = −15 + 30α − 135α2 + 80α3 − c(α),
+v(α) = 45 − 45α,
+v′(α) = −45.
+We have u(α0) = v(α0) = 45 and v′(α0) = −45. Note that c(α) > 0 for α ∈ (0, 0.85),
+as the polynomial in the numerator has roots in α = 0, α ≈ 0.85316 with no roots
+and positive values in between. Similarly, the polynomial in the denominator is
+positive for all α ∈ (0, 1). To find u′(α0) we study the limit of c(α)2.
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+17
+c(α)2 = 16 · 5(90α − 270α2 + 360α3 − 375α4 + 342α5 − 189α6 + 40α7)2
+4(45α2 − 90α3 + 90α4 − 75α5 + 57α6 − 27α7 + 5α8)
+= 80 · 902α2 + O(α3)
+4 · 45α2 + O(α3)
+= 80 · 902 + O(α)
+4 · 45 + O(α) .
+Therefore
+lim
+α↘0 c(α)2 = 80 · 902
+4 · 45
+= 3600.
+This yields limα↘0 c(α) = 60, as c(α) > 0 for positive values of α close to 0. We
+have u′(α0) = −75 and consequently limα→0
+b(α)
+α
+= 1/3.
+□
+
+OPTIMAL SUBSAMPLING DESIGN FOR POLYNOMIAL REGRESSION
+18
+Acknowledgments
+The work of the first author is supported by the Deutsche Forschungsgemeinschaft
+(DFG, German Research Foundation) within GRK 2297 MathCoRe.
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+Otto von Guericke University Magdeburg. Universit¨atsplatz 2, 39106 Magdeburg,
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+Email address: [email protected]
+

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+ 
+1 
+BUCKLING-INDUCED TRANSMISSION SWITCHING IN 
+PHONONIC WAVEGUIDES IN THE PRESENCE OF DISORDER 
+Ali Kanj1, Alexander F. Vakakis1, Sameh Tawfick1,2 
+1Department of Mechanical Science and Engineering, University of Illinois at Urbana-
+Champaign, Illinois 61801, United States 
+2The Beckman Institute of Advanced Science and Technology, University of Illinois at 
+Urbana-Champaign, Illinois 61801, United States 
+On-chip phononic circuits tailor the transmission of elastic waves, which can couple to 
+electronic and photonic systems, enabling new signal manipulation capabilities. Phononic 
+circuits rely on waveguides that transmit elastic waves within desired frequency 
+passbands, typically designed based on the Bloch modes of the waveguide constitutive 
+cell, assuming linearity and periodicity. MEMS waveguides composed of coupled 
+drumhead (membrane) resonators offer tunable MHz operation frequencies for 
+applications in nonlinear optomechanics, topological insulators, phononic cavities, and 
+acoustic switching. Here, we construct a reduced-order model (ROM) to demonstrate the 
+switching of signal transmission in drumhead-resonator waveguides due to thermoelastic 
+buckling. The ROM shows that buckling amplifies existing structural disorders, breaking 
+the periodicity required for waveguide transmission through the first passband. This 
+periodicity breaking manifests in the localization of the first-passband modes, like 
+classical Anderson localization caused by disorders. The proposed ROM is essential to 
+study the investigated phenomena since Bloch mode analysis fails for weakly-disordered 
+(< 5%) finite waveguides due to the disorder amplification caused by the thermoelastic 
+buckling. The illustrated transmission control should be useful for logical acoustic 
+operations, like switching, and can be extended to 2D circuits in the future.  
+I. Introduction 
+Phononic circuits are gaining increased interest because they tailor the propagation of 
+elastic and acoustic waves, which is advantageous for signal manipulation. For example, 
+phononic circuits are useful for cellular phone duplexers by serving as acoustic isolators and 
+mirrors [1] [2] [3]. In medical ultrasound applications and acoustic nondestructive tests, 
+phononic circuits promise to miniaturize the imaging aperture [4] [5] [6], decouple the electro-
+acoustic transduction [7] [8], and slow the signal for smaller delay lines [9] [10] [11]. 
+Moreover, nanostructural phononics operating in the hypersonic (GHz to THz) frequencies 
+enable thermal management [12] [13] [14], photonic-phononic interactions [15] [16], and 
+quantum information control [17] [18]. Phononic structures offer readily-achievable 
+nonlinearities allowing for strong optomechanical nonlinearities [19] [20], targeted-energy 
+transfer [21] [22], and passive structural nonreciprocity [23] [24] [25]. 
+Phononic circuits require accurately designed and fabricated waveguides to spatially 
+constrain the acoustic transmission within a specific frequency range referred to as the 
+passband (or the transmission band). In the passband, the temporal frequencies are linked to 
+
+ 
+2 
+the spatial frequencies (i.e., the wavenumbers) through the dispersion relation of the medium, 
+providing additional control over the acoustic transmission [1] [26]. This temporal and spatial 
+selectiveness stems from the dynamic characteristics of the unit cells whose periodic repetition 
+forms the waveguide. Therefore, the unit cell design is directly linked to the waveguide 
+characteristics via the Bloch modes of the unit cell. The Bloch modes are the vibrational modes 
+that the unit cell exhibits under Floquet boundary conditions with a wavenumber spanning the 
+irreducible Brillouin zone (IBZ) [26]. This approach calculates the possible wavenumber-
+frequency relationship known as the band structure of the phononic crystal (i.e., the unit cell). 
+This band structure matches the transmission in an infinite periodic waveguide of the same 
+repeated unit cell [26]. 
+Bloch modes predict the transmission of sufficiently long and weakly-disordered 
+waveguides [5] [6] [12] [16] [27] [28], although fabricated waveguides are neither infinite nor 
+perfectly periodic. In these cases, the finite-structure modal frequencies lie within (or close to) 
+the Bloch modes passbands [26]. For example, such a waveguide of 𝑁 cells possesses at most 
+𝑁 finite-structure modes for every passband; increasing 𝑁 makes the 𝑁 modes more densely 
+packed within the passband leading to the continuous Bloch-modes band structure as 𝑁 → +∞. 
+The dense packing of modes originates from the structural periodicity whose absence (i.e., 
+aperiodicity) generates frequency-distinct modes that cannot approximate the passbands. In 
+addition, the periodicity causes (spatially-) extended mode shapes that permit the transmission 
+of a signal between the ends of the structure [26]. These features – the approximate passband 
+and the extended mode shapes – are acoustically attractive and enable a finite periodic structure 
+to operate as a waveguide. The Bloch mode approach is computationally efficient in linear 
+periodic systems because it enables the tailoring of a single unit cell to estimate the behavior 
+of the entire waveguide. On the other hand, it significantly deviates from experimental results 
+when the number of unit cells is limited, when there is aperiodicity (structural asymmetry) in 
+the devices (whether intentional or uncontrolled), and when nonlinearities are profound. 
+Considering repetitive arrays of drumhead resonators composed of coupled flexible micro-
+membranes, we have recently shown that thermoelastic buckling of the membranes can switch 
+the acoustic transmission [29]. Waveguides made from coupled drumhead resonators were first 
+proposed by Hatanaka et al. in 2013 [30], who showed that they sustain megahertz-to-gigahertz 
+mechanical vibrations with high quality factors (high Qs) and optical finesse, features which 
+are valuable in mechanical, electrical, and optical applications [31, 32, 33]. For instance, 
+optomechanical interactions favor large surface-area structures (like the drumhead resonators) 
+
+ 
+3 
+over beams/cantilevers [32, 33, 34]. Another advantage of the drumhead resonators is their 
+manufacturability via conventional micro/nanofabrication [32, 33], while allowing for in-situ 
+structural tunability and actuation via piezoelectric [30, 34], electrostatic [35], and thermal 
+control [29]. Therefore, drumhead resonators were applied in tunable optical cavities [36] and 
+low-loss nonlinear optomechanical coupling [37]. Moreover, coupling drumhead resonators in 
+the form of arrays, like the devices studied in this article, served in realizing phononic 
+transistors [30], tunable 1D phononic waveguides [35], cavity-switchable waveguides [38], and 
+on-chip 2D topological insulators [39].  
+In this work, we study the mechanism of transmission switching in the drumhead-resonator 
+waveguides reported in [29], a phenomenon that has previously been attributed to buckling-
+induced aperiodicity. Specifically, we develop a reduced-order model (ROM) that mimics the 
+experiments observed in [29] (section II). The ROM accounts for out-of-plane translation, 
+rotation, and coupling to accurately predict the first and second passbands of the waveguides 
+as functions of the buckling state. The ROM uses the concept of the von Mises truss [40] to 
+capture the effect of buckling on drumhead-resonator waveguides, as illustrated in the electro-
+thermoelastic tunability of individual drumhead resonators in [41]. In turn, the von Mises 
+trusses permit modeling and predictive analysis of the drumhead-resonator waveguides via 
+lumped springs and rigid bodies, presenting simpler models that are amenable to analytical 
+studies compared to finite element models (e.g., continuous beams on elastic foundations). 
+With the von Mises ROM, we calculate the Bloch modes (section III), and compare them 
+to the transmission of (60-cell) finite waveguides in cases of perfect periodicity and (< 5%) 
+weak disorder (section IV). We investigate the acoustics of the finite waveguides by subjecting 
+their first cell to nonzero initial velocities and monitoring the resulting free responses in the 
+time and frequency domains as functions of the spatial propagation of wave packets in the 
+waveguide. We find that when the weakly-disordered finite waveguides are close to their 
+critical buckling state, the transmission through the first passband vanishes. Stronger disorder 
+results in a larger range of temperatures where the first passband does not transmit elastic 
+waves. This contrasts with the corresponding perfectly-periodic finite waveguide (i.e., with no 
+disorder), where the first passband transmits elastic waves at all considered temperatures, even 
+at the onsite of critical buckling. As for the second passband, the acoustic transmission persists 
+for all considered disorders and temperatures.  
+To thoroughly explain the effect of buckling on the transmission, we inspect the 
+dependencies of the mode shapes of the considered waveguides on temperature (section V). 
+The results show that the transmission-switching is associated with converting the mode shapes 
+
+ 
+4 
+from extended over the entire waveguide to localized at some cells. This localization of mode 
+shapes with disorders conforms to Anderson's localization originally discovered in 
+electromagnetic waves [42] [43] [44] and then applied in elastic settings [27] [28] [45]. Finally, 
+we present an evaluation of this buckling-switchable transmission on a finite element model 
+(FEM) of the experimental waveguide studied in [29] with 5% disorder far from, or close to 
+critical buckling (supplemental Video.S2). The FEM simulations agree with the predictions of 
+the ROM, thus conclusively proving that weak disorder leads to loss of transmission in the 
+repetitive array of drumhead resonators due to buckling.  
+II. Description of the waveguide and the reduced-order model (ROM) 
+In this work, we study the phononic waveguides shown in  Fig. 1a. This waveguide consists 
+of repetitive cells capable of transmitting flexural acoustic waves [29] [38] [46] [47]. This 
+waveguide was studied in [29], where the cells are drumhead-like membranes composed of 
+Silicone Nitride (SiNx) suspended by an etched Silicone Oxide (SiO2) layer on top of a Silicone 
+(Si) substrate. The involved materials and fabrication methods induce residual stresses in the 
+waveguide, whose cells buckle as depicted in Fig. 1b by atomic force microscopy (AFM) 
+conducted in [29].  
+In Fig. 1c, we show the effect of buckling on the elastic transmission of the waveguide of 
+Fig. 1a [29]. The temperature in Fig. 1c controls the state of buckling in the waveguide, where 
+lower temperature increases compressions between cells to provoke stronger buckling. At each 
+temperature, the colormap in Fig. 1c corresponds to the frequency response measured at the 
+middle cell of the waveguide due to the electrostatic actuation of the gold (Au) pad covering 
+the first cell (cf. Fig. 1a). At high temperatures in Fig. 1c (i.e., above ~230 K), the waveguide 
+exhibits three frequency regimes of effective transmission corresponding to the first three 
+passbands (labeled as I, II, and III). A decrease in temperature from 280 K down to ~230 K 
+decreases the mean frequency of all the passbands, indicating a softening behavior. During this  
+ 
+ 
+
+ 
+5 
+ 
+Fig. 1. Thermally buckled elastic waveguide: (a) Schematic drawing of a MEMS phononic 
+waveguide made of coupled drumhead-resonators [29]; (b) 3D topography map of 3 cells of 
+the waveguide [29] measured using atomic force microscopy (AFM) – the colormap shows the 
+out-of-plane deflections resulting from buckling in the structure; (c) measured transmission in 
+the waveguide [29] as a function of temperature and frequency of excitation applied to the first 
+cell in the waveguide – the colormap depicts the amplitude of oscillations at the middle of the 
+waveguide, which shows that the temperature change eliminates the transmission in passband 
+I and detunes the frequency of passbands I, II, and III; schematic drawings of the proposed 
+reduced-order model (ROM) of the waveguide undergoing thermal buckling showing (d) 
+undeformed and (e) deformed states. 
+ 
+softening, passband I diminishes its bandwidth until collapsing at ~230 K, whereas passbands 
+II and III maintain an almost constant bandwidth. Reducing the temperature to below ~230 K  
+completely eliminates passband I and increases the mean frequencies of passbands II and III 
+while possessing almost constant bandwidths. The observed temperature-dependent changes 
+in the frequency passbands and the switch in frequency detuning imply that the waveguide at 
+~230 K is in a critical buckling state associated with the softest structural configuration (since 
+buckling indicates minimum linearized stiffness). Accordingly, the waveguide is pre-buckled 
+for temperatures > ~230 K and post-buckled for temperatures < ~230 K, based on [29]. In both 
+buckling regimes, the frequency detuning of passbands II and III are direct consequences of 
+the buckling state of the waveguide. However, the frequency detuning of passband I and its 
+
+SiNx
+Au
+Transmission (a.u.)
+a
+c
+0
+40
+I=I
+Frequency (MHz)
+SiO2
+30
+b
+II
+20
+13
+10
+80
+180
+280
+0
+Temperature (K)
+d
+mi-1, Ji-1
+mi, Ji
+mi+1, Ji+1
+e
+Qi
+Qi-1
+Qi+1
+k-² T2 / k-1 T=1
+kTC
+Q
+9
+9
+Q
+0
+Q
+TEi
+ui
+ui+1
+ki-1
+QB
+QB
+77
+kB
+qs
+qs
+L
+L
+L
+L 
+6 
+transmission loss in the post-buckled regime necessitates both buckling and disorder in the 
+waveguide [29].  
+To further investigate this relationship between disorder, buckling, and elastic transmission 
+in the considered waveguide, we propose the reduced-order model (ROM) depicted in Figs. 
+1d-e. This ROM captures the thermally-mediated elastic buckling based on the ROM of a 
+single cell introduced in [41], exhibiting very good predictive capacity. Here, we extend the 
+ROM of [41] to account for the coupling between the cells in the waveguide and model the 
+acoustics of the entire phononic waveguide. Accordingly, we allocate to each cell a 
+translational degree-of-freedom (DoF) (as in [41]) and a rotational DoF to capture passbands I 
+and II, respectively.  
+As shown in Fig. 1d, each cell of index 𝑖 in the waveguide consists of a rigid mass 𝑚! with 
+a moment of inertia 𝐽!. Cell 𝑖 undergoes the motion illustrated in Fig. 1e with translational 
+coordinate 𝑢! and rotation angle 𝜃!. The translation deforms the grounding springs of 
+stiffnesses 𝑘!
+" and 𝑘!
+# representing the restoring forces for bending and stretching, respectively. 
+As in [41], these translational bending and stretching springs are confined at distances 𝑑" and 
+𝑑# (see Fig. 1e) while possessing free (undeformed) lengths 𝐿" and 𝐿#, respectively; clearly, a 
+free length larger than the confinement distance (i.e., 𝐿" > 𝑑" and 𝐿# > 𝑑#) introduces 
+compressive strains (precompression) in the cell. We assume that the remaining springs in the 
+ROM are undeformed at their undeformed positions. For example, the springs with stiffnesses 
+𝑇!
+",	𝑘!$%
+& , 𝑘!
+&, 𝑇!$%
+& , and 𝑇!
+& attached to the cell of index 𝑖 don’t apply any forces or torques in 
+Fig. 1d. The grounding torsional spring of stiffness 𝑇!
+" lumps the bending effects that oppose 
+the rotation 𝜃! due to the grounded boundary of the drumhead. The coupling springs with 
+stiffnesses	𝑘!
+& and 𝑇!
+& account for the force and torque, respectively, applied by cell 𝑖 to cell 
+(𝑖 + 1) due to the deformations illustrated in Fig. 1e. Lastly, we represent the lattice length 
+separating two successive cells by the length 𝐿 (see Figs. 1d-e).   
+III. Bloch modes of a single cell 
+In a previous article [41], we described the static equilibrium and the equations of motion 
+of a single drumhead resonator and identified its system parameters, which we refer to as the 
+reference cell parameters. These parameters are the translating mass 𝑚'() and springs 𝑘'()
+"
+ 
+and 𝑘'()
+#
+ (we use the subscript “𝑅𝑒𝑓” to label the reference cell), which are reproduced in 
+Table I. We start by studying the Bloch modes of an infinite waveguide based on a repetition 
+
+ 
+7 
+of this reference unit cell, as shown in Fig. 1d-e. The grounding translating springs exert the 
+force 𝐹!
+"*+, at the cell 𝑖 ∈ {	1, 2, … } and the temperature 𝑇 expressed for any translational 
+displacement 𝑢! as: 
+𝐹!
+"*+,(𝑢!; 𝑇) = 𝑘!
+"𝑑" ?𝑢!
+𝑑" − 𝛿"(𝑇)B + 𝑘!
+-𝑢!
+⎣
+⎢
+⎢
+⎡
+1 − 1 + 𝛿#(𝑇)
+F1 + G𝑢!
+𝑑#H
+.
+⎦
+⎥
+⎥
+⎤
+. 
+(1) 
+In (1), 𝛿"(𝑇) and 𝛿#(𝑇) are the temperature-dependent bending and stretching strains, 
+respectively, stemming from the thermal expansion and the fabrication-residual stresses in the 
+cells. We characterize these strains by the following temperature dependencies (as discussed 
+in [41]), 
+𝛿"(𝑇) ≝ 𝑑" − 𝐿"
+𝐿"
+= 𝛽/ + 𝛽%𝑇 + 𝛽.𝑇. 
+(2a) 
+𝛿#(𝑇) ≝ 𝑑# − 𝐿#
+𝐿#
+= 𝛾/ + 𝛾%𝑇, 
+(2b) 
+with the values of 𝛽/, 𝛽%, 𝛽., 𝛾/, and 𝛾% listed in Table I. We assume that these temperature 
+dependencies govern the strains of all the cells in the waveguide. Moreover, we 
+nondimensionalize the forces in this work by 𝑘'()
+"
+𝑑" leading to the following nondimensional 
+buckling force, 
+𝐹P!
+"*+,(𝑢P!; 𝑇) = 𝑢P! − 𝛿"(𝑇) + 𝜅!
+-𝑢P!
+⎣
+⎢
+⎢
+⎢
+⎢
+⎡
+1 − 1 + 𝛿#(𝑇)
+R1 + G𝑢P!
+𝑑̅#H
+.
+⎦
+⎥
+⎥
+⎥
+⎥
+⎤
+, 
+(3) 
+where 𝜅!
+- ≝ 𝑘!
+-/𝑘!
+", 𝑑̅# ≝ 𝑑#/𝑑", and 𝑢P! ≝ 𝑢!/𝑑" with the overbar denoting a 
+nondimensionalized entity. 
+Focusing on the reference cell which undergoes only translation while connected to 𝑘'()
+"
+ 
+and 𝑘'()
+#
+, we find its equilibrium displacement 𝑢P'()
+012(𝑇) by solving the following equation: 
+ 
+𝛿" 
+𝛿# 
+𝑑̅# 
+𝜅'()
+-
+ 
+%
+.3 FΛ'()
+"
+ [MHz] 
+𝜒 
+𝛽/ 
+𝛽% [K-1] 
+𝛽. [K-2] 
+𝛾/ 
+𝛾% [K-1] 
+7.65 -3.47E-2 3.81E-5 
+1.9 
+-4.07E-3 
+1 
+1 
+9.40 
+1/12 
+ 
+Table I. Parameters of the reference cell [41].  
+
+ 
+8 
+ 
+Fig. 2. Thermal buckling of the infinite perfectly periodic waveguide (i.e., Bloch modes): (a) 
+Static equilibrium of a single cell as a function of temperature based on (4); (b) frequency 
+dispersion curves as a function of the nondimensional wavenumber of the Bloch modes at a 
+temperature of 390 K, 370 K, and 350 K; (c) Bloch-modes frequency extrema as a function of 
+temperature illustrating the transmission detuning in an infinite perfectly-periodic waveguide 
+– we depict the Bloch-modes frequency extrema with 𝑘4/𝐿 close to 0 rad by the filled circles, 
+whereas the extrema with 𝑘4/𝐿 close to 𝜋 rad by open circles; also the blue and green colors 
+in (b, c) represent the Bloch-mode passbands I and II, respectively. 
+ 
+𝐹P'()
+"*+,X𝑢P'()
+012; 𝑇Y = 0 with the maximum satisfying 
+567!"#
+$%&'
+5*8!"# X𝑢P'(); 𝑇Y[
+*8!"#9*8!"#
+()* > 0 
+(4) 
+The maximum condition in (4) ensures 𝑢P'()
+012 to be the most stable equilibrium solution, which 
+should be favored experimentally. In Fig. 2a, we plot the values of 𝑢P'()
+012 as a function of 
+temperature based on (4), (3), (2), and the reference parameters in Table I. We observe that the 
+single cell translates upwards due to cooling, which increases the internal compressions leading 
+to buckling of the cell [41].  
+To study the effect of buckling on wave transmission, we evaluate the Bloch modes of the 
+cell at each temperature shown in Fig. 2a. The Bloch modes correspond to the infinite 
+waveguide of Fig. 1e-d made of cells whose parameters are identical to the considered single 
+cell. In this perfectly periodic infinite waveguide, all the cells attain at 𝑇 the equilibrium state 
+of 𝑢P!
+012 = 𝑢P'()
+012(𝑇) and 𝜃!
+012 = 0 rad for all 𝑖 ∈ {1,2,3, … , +∞}. At every instant 𝑡, we track 
+the oscillations of the 𝑖th cell about its equilibrium state via the perturbation coordinates: 
+𝑣̅!(𝑡) ≝ 𝑢P!(𝑡) − 𝑢P!
+012, 
+(5a) 
+ℎP!(𝑡) ≝ 𝐿P𝜃!(𝑡) − 𝐿P𝜃!
+012, where 𝐿P ≝ 𝐿/𝑑". 
+(5b) 
+The above coordinates allow writing Newton’s second law on any cell of index 𝑖 > 1 in 
+Fig. 1d-e as, 
+
+a
+350 K
+b
+c
+350 K
+Static
+390 K
+370 K
+350 K
+0.4
+(MHz)
+12
+(MHz)
+12
+390 K
+B
+370 K
+0.2
+370K
+10
+10
+,EQM
+0
+390 K
+8
+8
+-0.2
+6
+9
+400
+375
+350
+0
+π O
+O 
+T
+400
+375
+350
+Temperature (K)
+k. / L (rad)
+Temperature (K) 
+9 
+𝜇! a1
+0
+0
+𝜒b c
+5+:7,
+5;+
+5+<8,
+5;+
+d + 𝜇!$% e
+−Λ!$%
+&
+−
+=,-.
+/
+.
+=,-.
+/
+.
+=,-.
+/
+> − Γ!$%
+& g h𝑣̅!$%
+ℎP!$%
+i +
+j𝜇!$% e
+Λ!$%
+&
+$=,-.
+/
+.
+$=,-.
+/
+.
+=,-.
+/
+> + Γ!$%
+& g + 𝜇! e
+Λ!
+& + Λ!
+"*+,
+=,
+/
+.
+=,
+/
+.
+=,
+/
+> + Γ!
+& + Γ!
+"gk h𝑣̅!
+ℎP!
+i +
+𝜇! e
+−Λ!
+&
+=,
+/
+.
+−
+=,
+/
+.
+=,
+/
+> − Γ!
+&g h𝑣̅!?%
+ℎP!?%
+i = l0
+0m, 
+(6) 
+where 𝜇! ≝ 𝑚!/𝑚'(), 𝜒 ≝ 𝐽!/𝑚!𝐿.,	Λ!
+"*+,(𝑇) ≝ Λ!
+" 567,
+$%&'
+5*8, [
+*8,9*8,
+()*(-)
+, Λ!
+" ≝ 𝑘!
+"/𝑚!, Λ!
+& ≝
+𝑘!
+&/𝑚!, Γ!
+& ≝ 𝑇!
+&/(𝑚!𝐿.), and Γ!
+" ≝ 𝑇!
+"/(𝑚B𝐿.). Equation (6) only considers the linearized 
+dynamics of the undamped 𝑖th cell. Note that the nondimensionalization in (6) results in 
+(squared) frequency-like parameters (i.e., Λ!
+"*+,, Λ!
+&, Γ!
+", and Γ!
+&). This parameter conversion 
+offers an advantage when comparing the model to experiments because frequencies are easier 
+to identify than stiffnesses and directly affect the performance of the waveguides. For instance, 
+we deduce the value of Λ'()
+"
+ listed in Table I from the experiments of a single cell in [41]. For 
+the remaining frequency-like parameters, we assume the following relationships for all 𝑖 ∈
+{𝑅𝑒𝑓, 1, 2,3, … }: 
+Λ!
+&(𝑇) = 0.2 ?Λ!
+C(𝑇) −
+min
+DE/→>//	H Λ!
+C(𝑇)B 
+(7a) 
+Γ!
+" = 1
+12 Λ!
+" 
+(7b) 
+Γ!
+&(𝑇) = 1
+12 a3Λ!
+&(𝑇) − 3
+4 Γ!
+"b. 
+(7c) 
+To calculate the Bloch modes of a single cell, we apply the Floquet boundary conditions of 
+h𝑣̅!
+ℎP!
+i = 𝒑!𝑒IJ'0
+1 !?K;L with a normalized wavenumber 𝑘4/𝐿, a modal frequency of 𝜔, a modal 
+vector 𝒑!, and the imaginary number 𝑗. = −1. Additionally, we assume that all cells are 
+identical to the single cell with parameters listed in Table I, transforming (6) into the following 
+boundary value problem: 
+j−𝜔. a1
+0
+0
+𝜒b + e
+2 G1 − cos
+,0
+M 	H Λ'()
+&
++ Λ'()
+"*+,
+−Λ'()
+&
+sin
+,0
+M
+… 
+(8) 
+
+ 
+10 
+…
+Λ'()
+&
+sin
+,0
+M
+%
+. G1 + cos
+,0
+M H Λ'()
+&
++ 2 G1 − cos
+,0
+M H Γ'()
+&
++ Γ'()
+" gk 𝒑! = l0
+0m. 
+For all 𝑘4/𝐿 ∈ [0, 𝜋] rad, the irreducible Brillouin zone (IBZ) is defined by the respective 
+pair of eigenfrequencies 𝜔 that zero the determinant of the matrix operating on 𝒑! in (8). These 
+eigenfrequencies are the Bloch modes’ frequencies forming the dispersion curves in Fig. 2b at 
+390 K, 370 K, and 350 K for the single cell. The lower (blue) and upper (green) curves in Fig. 
+2b correspond to passbands I and II of the transmission in the perfectly periodic infinite 
+waveguide, respectively. We depict the transmission of this waveguide in Fig. 2c, where we 
+collect the extrema (maxima and minima) of passbands I and II (like in Fig. 2b) for the 
+temperature 𝑇 ∈ [350, 400] K.  
+Fig. 2c shows that cooling from 400 to ~370 K reduces the mean frequencies of both 
+passbands while narrowing the bandwidth of passband I. Cooling below ~370 to 350 K 
+increases again the mean frequencies of both passbands while widening the bandwidth of 
+passband I. The first cooling phase from 400 to ~370 K in Fig. 2c resembles the cooling phase 
+in Fig 1c between 280 and ~230 K. However, the second cooling phase between ~370 to 350 
+K in Fig. 2c diverges fundamentally from the experimental transmission in Fig. 1c between 
+~230 and ~80 K, where the transmission in passband I does not reemerge. Therefore, the ROM 
+buckling cannot eliminate the transmission of passband I in a perfectly periodic infinite 
+waveguide. This loss of transmission with buckling necessitates the consideration of disorder 
+(i.e., the break of perfect periodicity) in the waveguide as previously established in [29].  
+Note that we adopt the relationships in (7) to emulate the experimental transmission in Fig 
+1c between 280 and ~230 K. For this reason, we select Λ!
+&(𝑇) in (7a) to decrease until the 
+transmission vanishes at the point of minimum frequency, 
+min
+DE/→>//	H Λ!
+C(𝑇), leading to the 
+shrinkage of passband I between 400 and ~370 K in Fig. 2c. In (7b), we assume that the rotation 
+of the cell centerline (of length	𝐿) deflects an elastic foundation of stiffness density 𝑘!
+"/𝐿. In 
+(7c), we impose a temperature-constant bandwidth for passband II like the measurements in 
+Fig. 1c. The temperature-detuning of the mean frequencies  of the passbands in Fig. 2c is 
+considered for  the identified parameters (i.e, Λ'()
+"
+, 𝛽/, 𝛽%, 𝛽., 𝛾/, and 𝛾%) of the ROM in [41], 
+which slightly deviate from those in the devices used in Fig. 1c (extracted in [29]).   
+
+ 
+11 
+IV. Temporal transmission in finite waveguides 
+We focus now on the waveguide disorder resulting from the thickness variation between 
+the cells. We assume a thickness variation of the form, 
+ℎ! − ℎ'()
+ℎ'()
+= 𝜎<
+4
+⎩
+⎪
+⎨
+⎪
+⎧
+j2
+€𝑁 + 1
+2
+− 𝑖€
+𝑁 + 1
+2
+− 1
+− 1k
+•‚‚‚‚‚ƒ‚‚‚‚‚„
+N,
++ 𝑟!([−1, 1])
+⎭
+⎪
+⎬
+⎪
+⎫
+, 
+(9) 
+for 𝑖 ∈ {1,2, … , 𝑁}, where we denote by ℎ! the thickness of the 𝑖th cell, ℎ'() the thickness of 
+the reference cell discussed in the previous section, 𝜎< the level of thickness disorder, and 
+𝑟!([−1, 1]) a random rational number ∈ [−1, 1] generated at each 𝑖. We introduce the random 
+number 𝑟! to account for the random errors of the fabrication process. The 𝑠! term in (9) 
+represents the systematic errors resulting from wet etching that forms the waveguide cells as 
+explained in [29] [38] [46] [47].  
+The holes at the center of the cells in Fig. 1a-b are etching holes through which the etchant 
+attacks the underlying layer and suspends the cells. Thus, there is a higher (linear) density of 
+etching holes at the middle of the waveguide (of index 
+O?%
+. ) compared to the ends (of indices 
+1 and 𝑁). This higher etching-holes density increases the etching rate leading to over-etching 
+at the middle of the waveguide compared to its ends [29]. We model this over-etching by	𝑠! in 
+(9) as a linear distribution of the cell position from the middle of the waveguide. Figs. 3a-b 
+show two examples of thickness variation with 𝜎< = 0% (i.e., perfectly periodic waveguide) 
+and 𝜎< = 5%, respectively.  
+The thickness variation implies a corresponding variation in the dynamical properties of 
+the cells. Based on the theory of the mechanics circular plates [48] [49] and the assumption in 
+[41], the thickness affects the parameters of the 𝑖th cell in Fig. 1d-e as follows: 
+𝜅!
+#
+𝜅'()
+#
+	 = ‹ ℎ!
+ℎ'()
+Œ
+$.
+, 
+(10a) 
+Λ!
+"
+Λ'()
+"
+= ‹ ℎ!
+ℎ'()
+Œ
+.
+. 
+(10b) 
+The scaling relationships (10) with the expressions in (7) quantify the effect of the cell’s 
+thickness on the ROM parameters. 
+With the ROM of Fig. 1d-e, we apply Newton’s 1st law to calculate the static equilibrium 
+(𝑢P!
+012, 𝐿P	𝜃!
+012) of each cell	𝑖 ∈ {1, 2, … , 𝑁} in the 𝑁 cells waveguide using, 
+
+ 
+12 
+⎩
+⎪
+⎪
+⎪
+⎪
+⎪
+⎨
+⎪
+⎪
+⎪
+⎪
+⎪
+⎧𝐹P%
+"*+,X𝑢P%
+012; 𝑇Y
+0
+	
+⋮
+	
+𝐹P!
+"*+,X𝑢P!
+012; 𝑇Y
+0
+	
+⋮
+	
+𝐹PO
+"*+,X𝑢PO
+012; 𝑇Y
+0
+⎭
+⎪
+⎪
+⎪
+⎪
+⎪
+⎬
+⎪
+⎪
+⎪
+⎪
+⎪
+⎫
+•‚‚‚‚‚ƒ‚‚‚‚‚„
+𝑸8$%&'
++ 𝐾•#;Q;
+⎩
+⎪
+⎪
+⎪
+⎪
+⎪
+⎨
+⎪
+⎪
+⎪
+⎪
+⎪
+⎧ 𝑢P%
+012
+𝐿P𝜃%
+012
+	
+⋮
+	
+𝑢P!
+012
+𝐿P𝜃!
+012
+	
+⋮
+	
+𝑢PO
+012
+𝐿P𝜃O
+012⎭
+⎪
+⎪
+⎪
+⎪
+⎪
+⎬
+⎪
+⎪
+⎪
+⎪
+⎪
+⎫
+•‚‚ƒ‚‚„
+𝒒8	()*
+= 𝟎, 
+(11) 
+where 𝐹P!
+"*+,X𝑢P!
+012; 𝑇Y is the thermo-elastic buckling force expressed in (3). In (11), we 
+assume small angles of deformation allowing the approximation sin 𝜃!
+012 ≈ 𝜃!
+012 while 
+neglecting the longitudinal displacement of the cells’ ends. Under this approximation, we 
+express the nondimensional static stiffness 𝐾•#;Q; as, 
+𝐾•#;Q; =
+⎣
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎡
+𝒦•%
+	
+	
+	
+0.×.(O$.)
+	
+	
+	
+	
+	
+	
+⋱	
+	
+	
+	
+	
+	
+	
+	
+	
+0.×.(!$.)
+	
+𝒦•!
+	
+0.×.(O$!$%)
+	
+	
+	
+	
+	
+	
+	
+	
+⋱
+	
+	
+		
+	
+	
+	
+0.×.(O$.)
+	
+	
+	
+𝒦•O
+⎦
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎤
+, 
+(12a) 
+with 0T×U denoting a zero-filled matrix of 𝑀 rows by 𝑃 columns, 
+𝒦•% = e
+𝜇%Λ%
+&
+	
+V.=./
+.
+	
+−𝜇%Λ%
+&
+	
+V.=./
+.
+V.=./
+.
+	
+𝜇% G
+=./
+> + Γ%
+& + Γ%
+"H
+	
+−
+V.=./
+.
+	
+𝜇% G
+=./
+> − Γ%
+&H
+g, 
+(12b) 
+𝒦•! = –
+−𝜇!$%Λ!$%
+&
+−
+V,-.=,-.
+/
+.
+𝜇!$%Λ!$%
+&
++ 𝜇!Λ!
+&
+V,-.=,-.
+/
+.
+𝜇!$% —
+=,-.
+/
+> − Γ!$%
+& ˜
+$V,-.=,-.
+/
+?V,=,
+/
+.
+  … 
+… 
+$V,-.=,-.
+/
+?V,=,
+/
+.
+−𝜇!Λ!
+&
+V,=,
+/
+.
+𝜇!$% —
+=,-.
+/
+> + Γ!$%
+& ˜ + 𝜇! —
+=,
+/
+> + Γ!
+& + Γ!
+"˜
+−
+V,=,
+/
+.
+𝜇! —
+=,
+/
+> − Γ!
+&˜
+™, 
+(12c) 
+for 2 ≤ 𝑖 ≤ 𝑁 − 1, and, 
+ 
+ 
+
+ 
+13 
+ 
+Fig. 3. Effect of weak thickness disorder on the static equilibrium of finite waveguides: (a, b) 
+Thickness profile relative to the reference thickness ℎ'(), and (c, d) static deflections at 400, 
+390, 380, 370, 360, and 350 K of (a), and (b, d) the weakly (𝜎< = 5%) disordered 60-cell 
+waveguide ROM of (b); refer to (9) for the definition of the disorder parameter 𝜎<; in addition, 
+in (c, d), the thick segments represent the rigid masses of the cells in the ROM of Fig. 1d-e 
+translating and rotating according to the computed equilibria from (11).   
+ 
+𝒦•O = e
+−𝜇O$%ΛO$%
+&
+−
+V2-.=2-.
+/
+.
+V2-.=2-.
+/
+.
+𝜇O$% G
+=2-.
+/
+>
+− ΓO$%
+&
+H
+ … 
+… 
+𝜇O$%ΛO$%
+&
+−
+V2-.=2-.
+/
+.
+−
+V2-.=2-.
+/
+.
+𝜇O$% G
+=2-.
+/
+>
++ ΓO$%
+&
+H + 𝜇OΓO
+"g, 
+(12d) 
+where Λ!
+&, Γ!
+", and Γ!
+& are defined in (7). 
+We solve (11) via “fsolve” (gradient descent method) in MATLAB®. In the numerical 
+solver, the starting guesses for 𝑢P!
+012 in (11) correspond to the equilibria of individual cells in 
+(1), see Fig. 2a. We assign the differences X𝑢P!?%
+012 − 𝑢P!
+012Y as starting guesses for 𝐿P𝜃!
+012 for 
+𝑖 ∈ {1, 2, … , 𝑁 − 1}, and 𝐿P𝜃O$%
+012 as the guess for 𝐿P𝜃O
+012. Fig. 3c-d display the computed 
+equilibria of (11) at different temperatures in the perfectly periodic and weakly disordered 
+waveguides of Fig. 3a-b, respectively. In Fig. 3c-d, each segment (thick dashed line) represents 
+a cell in the ROM of Fig. 1d-e translated and rotated according to the equilibrium of (11). 
+In Fig. 3c, the cells at equilibrium undergo the same translational deflections without 
+rotation, which results from the perfect periodicity of the waveguide of Fig. 3a. For instance, 
+the weakly-disordered waveguide of Fig. 3b attains equilibrium with different cell translations 
+
+Perfect periodicity: oh = 0%
+Weak disorder: n = 5%
+a
+b
+(%)
+102.5
+102.5
+100
+100
+97.5
+97.5
+1
+20
+40
+60
+1
+20
+40
+60
+Cell number
+Cell number
+c
+d
+0.4
+0.4
+350 K
+B
+a
+0.2
+0.2
+360 K
+EQM
+370 K
+0
+0
+380 K
+390 K
+-0.2
+-0.2
+400K
+1
+20
+40
+09
+1
+20
+40
+60
+Cell number
+Cell number 
+14 
+and rotations, as shown in Fig. 3d. For both waveguides of Figs. 3c-d, the cooling increases the 
+cells' baseline translational deflections going from negative to positive values between 400 K 
+and 350 K(like the individual cell in Fig. 2a). Moreover, each 10 K of cooling induces larger 
+deflections at lower temperatures in Figs. 3c-d, which mirrors the buckling susceptibility 
+observed in Fig. 2a. 
+At this point, we linearize the dynamics around the calculated equilibria to study the 
+transmission in the finite waveguides for varying temperatures. Using the perturbation 
+coordinates in (5), the linearized equations yield, 
+𝑀• 5+𝒒8	
+5;+ + (𝐾•"*+, + 𝐾•#;Q;)
+•‚‚‚‚ƒ‚‚‚‚„
+W8
+	𝒒• = 𝟎, 
+(13) 
+where 𝒒• = œ𝑣̅%, ℎP%, … , 𝑣̅!, ℎP!, … , 𝑣̅O, ℎPO•
+-, 𝐾•#;Q; is defined in (12), 𝐾•"*+, corresponds to the 
+2𝑁 × 2𝑁 matrix whose diagonal contains the linear stiffnesses of the buckling forces, 
+𝐾•"*+, = diagX𝜇%Λ%
+"*+,, 0, … , 𝜇!Λ!
+"*+,, 0	, … , 𝜇OΛO
+"*+,, 0Y, 
+(14) 
+and 𝑀• denotes the nondimensional mass matrix expressed as: 
+ 
+ 
+𝑀• =
+⎣
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎡
+ℳ•%
+	
+	
+	
+0.×.(O$%)
+	
+	
+	
+	
+	
+	
+⋱	
+	
+	
+	
+	
+	
+	
+	
+	
+0.×.(!$%)
+	
+ℳ•!
+	
+0.×.(O$!)
+	
+	
+	
+	
+	
+	
+	
+	
+⋱
+	
+	
+		
+	
+	
+	
+0.×.(O$%)
+	
+	
+	
+ℳ•O
+⎦
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎤
+, 
+(15) 
+ where ℳ•! = 𝜇! a1
+0
+0
+𝜒b for 𝑖 ∈ {1, 2, 3, … , 𝑁}. In this work, we solve for the acoustics of the 
+waveguides by direct integration of (13) using MATLAB® “ode45” function. 
+To this end, we consider the solution of (13) subject to a nonzero initial translational 
+velocity at cell 1 and all other initial conditions set to zero: 
+𝒒𝟎 ≝ 𝒒•(𝑡 = 0) = 𝟎 and 𝒒̇ 𝟎 ≝
+5𝒒8
+5; (𝑡 = 0) = c
+10$D𝜇%
+0
+𝟎.(O$%)×%
+d. 
+(16) 
+The initial conditions (16) induce a motion that propagates in the waveguides as depicted in 
+Fig. 4-5 and the supplemental Video.S1. This motion enables the study of wave transmission 
+in the considered finite waveguides. In Figs. 4-5 and supplemental Video.S1, we consider the 
+responses of the waveguides of Figs. 3a-b at 390 K, to address the effect of weak disorder at a 
+ 
+ 
+
+ 
+15 
+ 
+Fig. 4. Elastic wave transmission through the perfectly periodic finite waveguide of Fig. 3a at 
+390 K: Temporal responses in terms of (a) the translations 𝑣̅Y and (b) the rotational ℎPY 
+perturbation coordinates of (from left to right) cell 𝑛 ∈ {1, 20, 40, 60} due to the initial 
+conditions in (16) (the rightmost plots in (a, b) show zoomed-in views of the responses of cell 
+60 in the period [16, 20] μs); (c) spatiotemporal evolution of the normalized mechanical energy 
+𝐸!
+T(+</𝐸BY of (17) for 𝑖 ∈ {1, 2, …, 60} in the ROM waveguide – the wavepacket labels 
+highlight the instants at which the fast and slow wavepackets reach cell 60 and reflect. 
+ 
+fixed temperature corresponding to (relatively) broad passbands I and II based on the Bloch 
+modes of Figs. 2b-c.  
+Figs. 4a-b show the translational and rotational time-responses, respectively, for 
+wavepackets propagating through the waveguide of Fig. 3a at a temperature of 390 K. We 
+observe that the initial velocity imposed at cell 1 by (16) provokes a wavepacket that propagates 
+to the last cell (with index 60) (cf. also supplemental Video.S1). The propagating front of this 
+wave consists mainly of two distinct wavepackets with different wave speeds, namely, a “fast” 
+wavepacket reaching cell 60 within < 20 𝜇s, and a “slow” wavepacket reaching that boundary 
+cell within ~40 𝜇s. Fig. 4a shows the fast wavepacket characterized by low translational 
+
+Perfectly uniform: n = 0% at 390 K
+a
+80
+20
+60
+19
+40
+18
+20
+17
+0
+16
+-20
+0
+20
+-5
+0
+5
+-5
+0
+5
+-5
+0
+5
+-0.1
+0.1
+V, (pm/m)
+V20 (pm/m)
+V40 (pm/m)
+V6o (pm/m)
+V6o (pm/m)
+b
+80
+20
+19
+18
+20
+17
+0
+16
+-10
+0
+10
+-4
+0
+4
+-4
+0
+4
+-4
+0
+4
+-4
+0
+4
+h, (pm/m)
+h2o (pm/m)
+h4o (pm/m)
+h6o (pm/m)
+h6o (pm/m)
+c
+80
+0.35
+60
+Slow
+E
+40
+wavepacket
+0.5
+Mech
+20
+Fast
+wavepacket
+0
+0
+1
+20
+40
+60
+Cell number 
+16 
+amplitudes compared to the slow wavepacket. Fig. 4b shows both wavepackets possessing 
+similar rotational amplitudes. The slow and fast wavepackets in the finite waveguide are 
+analogous to waves transmitted in passbands I and II, respectively, of the infinite perfectly 
+periodic waveguide (cf. Figs. 2b-c). We conclude that at the temperature considered, the finite 
+waveguide supports the propagation of spatially extended wavepackets with frequency-
+wavenumber contents lying inside passbands predicted in the waveguide of infinite extent. To 
+visualize the waves propagation over every cell, in Fig. 4c, we plot the spatiotemporal 
+normalized energy evolution in the corresponding perfectly periodic 60-cell waveguide. In 
+particular, we depict the contour plot of the normalized mechanical energy 𝐸!
+T(+</𝐸BY of each 
+cell 𝑖 ∈ {1, 2, … , 𝑁} defined by, 
+𝐸!
+T(+<
+𝐸BY
+= ¦𝜇! —𝑑𝑣̅!
+𝑑𝑡 ˜
+.
++ 𝜇!𝜒 ‹𝑑ℎP!
+𝑑𝑡 Œ
+.
++ Λ!
+"*+,𝑣̅!
+. + Γ!
+"𝑣̅!
+.
++ 1
+2 ?Λ!$%
+&
+(𝑣̅! − 𝑣̅!$%). + Λ!
+&(𝑣̅!?% − 𝑣̅!). + Γ!$%
+& XℎP! − ℎP!$%Y
+.
++ Γ!
+&XℎP!?% − ℎP!Y
+.B§	/	X𝒒̇ 𝟎
+-𝑀•𝒒̇ 𝟎 + 𝒒𝟎
+-𝐾•𝒒𝟎Y, 
+(17) 
+with Λ/
+& = ΛO?%
+&
+= Γ/
+& = ΓO?%
+&
+= 0. Fig. 4c demonstrates the two waves propagating in the 
+primary front, reaching the waveguide boundary at cell 60.  Moreover, Fig. 4c shows the slow 
+wavepacket transmitting a significantly larger portion of the input energy since it mainly 
+corresponds to the translational motion (cf. Fig. 4a) that is more effectively excited via the 
+initital conditions in (16).  
+For comparison, in Fig. 5 we depict the corresponding wave transmission in the (𝜎< = 5%) 
+weakly disordered waveguide at 390 K, forced by the same excitation in (16). A drastically 
+different acoustics are observed for the disordered system. We observe that only the early (fast) 
+wavepacket propagates to cell 60 in the presence of disorder. Moreover, as Fig. 5c shows, the 
+slow wavepacket (which carries the major portion of the available energy) becomes spatially 
+localized in the first 30 cells, a result which indicates that the weakly disordered waveguide at 
+390 K cannot transmit the slow wavepacket corresponding to passband I of Figs. 2b-c. This 
+transmission loss for wavepackets in passband I is in full agreement with the experimental 
+findings reported in [29] for a similar waveguide (and summarized in Figs. 1a-c). Therefore, 
+the ROM developed in this work captures the experimental transmission loss mediated by 
+ 
+ 
+Label needs fixing 
+
+ 
+17 
+ 
+Fig. 5. Elastic wave transmission through the weakly disordered finite waveguide of Fig. 3c at 
+390 K: Temporal responses in terms of (a) the translations 𝑣̅Y and (b) the rotational ℎPY 
+perturbation coordinates of (from left to right) cell 𝑛 ∈ {1, 20, 40, 60} due to the initial 
+conditions in (16) (the rightmost plots in (a, b) show zoomed-in views of the responses of cell 
+60 in the period [16, 20] μs; (c) spatiotemporal evolution of the normalized mechanical energy 
+𝐸!
+T(+</𝐸BY of (17) for 𝑖 ∈ {1, 2, …, 60} in the ROM waveguide – the “Fast wavepacket” label 
+highlights the instant at which the fast wavepacket reaches cell 60 and reflect; the “Slow 
+wavepacket” label highlights the cells where the slow wavepacket is confined. 
+ 
+buckling and provides conclusive proof regarding the important role that structural disorder 
+plays for the transmission loss in buckled waveguides at certain temperature ranges. 
+This transmission loss mechanism is also observed by the finite element model (FEM) of 
+the waveguide studied in [29], further illustrated in supplemental Video.S2 presenting the time-
+series deformations of the centerlines of two waveguides based on the COMSOL simulations 
+of [29]. In particular, we consider two identical 20-cell weakly disordered waveguides at -20 
+K (i.e., far from critical buckling) and -120 K (close to critical buckling), respectively. In 
+supplemental Video.S2, we assess the waveguides capacity to transmit propagating 
+
+Weakly disordered: on = 5% at 390 K
+a
+80
+20
+60
+19
+40
+18
+20
+17
+0
+16
+-20
+0
+20
+-5
+0
+5
+-5 
+0
+5
+-5 
+0
+5
+-0.1
+00.1
+V (pm/m)
+V20 (pm/m)
+V40 (pm/m)
+V6o (pm/m)
+V6o (pm/m)
+b
+80
+20
+T
+(sr)
+60
+19
+Time (
+40
+18
+20
+17
+0
+16
+-10
+0
+10
+-4
+0
+4
+-4 
+0
+4
+-4
+0
+4
+-4
+0
+4
+h, (pm/m)
+h20o (pm/m)
+h4o (pm/m)
+h6o (pm/m)
+ho (pm/m)
+c
+Slow wavepacket
+80
+0.35
+Time (μus)
+60
+E
+ 40
+0.5
+Mech
+20
+Fast
+wavepacket
+0
+0
+1
+20
+40
+60
+Cell number 
+18 
+wavepackets with frequency contents inside passband I of their corresponding infinite 
+waveguide (i.e. the passbands based on the Bloch modes of the constitutive unit cell). These 
+FEM simulations show that an elastic wavepacket can propagate only in the weakly disordered 
+waveguide far from critical buckling (cf. [29] for the FEM geometry and methods). Hence, 
+with the ROM developed in this work, we confirm that buckling-induced transmission loss for 
+waves in passband I is associated with weak disorder and thermo-elastic effects, confirming 
+the experimental and FEM results reported in [29].  
+V. Frequency transmission in the finite waveguides 
+To further study the frequency transmission in the considered waveguides at 390 K, Fig. 6 
+presents the amplitude of the Fast Fourier transforms (FFT) of the displacements at selected 
+cells subject to the initial conditions (16). For comparison, we overlay the FFT plots on top of 
+the Bloch modes’ passbands of the infinite waveguide (cf. Figs. 2a-c) and the modal 
+frequencies 𝜔TZ5( = √ΛTZ5( of the finite waveguide calculated by the eigenvalue problem, 
+(−ΛTZ5(𝑀• + 𝐾•)	𝚽 = 𝟎, 
+(18) 
+where 𝚽 denotes the mode shape vector. For clarity, we do not show in Fig. 6 the modal 
+frequencies that lie within the passbands.  
+Figs. 6a-b illustrate that all modal frequencies of the perfectly periodic finite waveguide 
+are inside the passbands (there exist 60 modes in each passband); whereas Figs. 6c-d shows 
+certain modes of the (𝜎< = 5%) weakly disordered finite waveguide are lying outside these 
+passbands, i.e., in stopbands. Hence, the Bloch modes of the infinite waveguide constitute a 
+perfect estimator of wave transmission only in the perfectly periodic finite waveguide.  
+In Figs. 6a-b, the perfectly periodic finite waveguide corresponds to strong translational 
+and rotational responses at cell 60, respectively, only when their frequency contents are inside 
+the passbands. Outside of the passbands, however, the frequency responses of the responses at 
+cell 60 are minimal compared to the response of the excited cell 1, indicating a lack of 
+transmission throughout the waveguide (i.e., stopband). In Figs. 6c-d, cell 60 of the (𝜎< = 5%) 
+weakly disordered finite waveguide admits weak responses compared to the response of cell 1 
+in the Bloch modes defining passband I. These frequency responses verify that the transmission 
+loss shown in Fig. 5 corresponds to the transmission loss of wavepackets inside passband I. 
+Notably, concerning wavepackets initiated in passband II (cf. Fig. 6d), the rotational response 
+of cell 60 compares in magnitude to cell 1 due to the persistence of the transmission in this 
+passband, as previously illustrated in Fig. 5.  
+
+ 
+19 
+ 
+Fig. 6. Effect of weak thickness disorder on the frequency responses of finite waveguides: Fast 
+Fourier Transforms (FFT) of the (a,c) translational and (b,d) rotational temporal responses of  
+of cells 1 (red line), and 60 (purple line), at 390 K of the perfectly periodic 60-cell waveguide 
+in (a,b), and the weakly disordered 60-cell waveguide in (c, d) (refer to the ROM of Figs. 3a,c); 
+the blue and green shaded background regions correspond to the frequencies of passbands I 
+and II of the corresponding perfectly periodic waveguide at 390K, respectively(cf. Figs. 2b-c), 
+whereas the black vertical lines denote the modal frequencies of the finite waveguides 
+calculated using (18) whose values are not inside the Bloch modes passbands. 
+ 
+Similar performance to the results of Figs. 4-6 is observed for different temperatures 
+between 400 and 350 K, as shown in Fig. 7. In particular, the perfectly periodic finite 
+waveguide of Fig. 3a does not lose transmission for the considered temperatures; the weakly 
+disordered finite waveguide of Fig. 3c cannot transmit waves in passband I between 390 and 
+352 K.  
+In Figs. 7-8, we summarize the results for all measured temperatures by considering the 
+frequency contents of the wavepackets transmitting throughout the spatial extent of the 
+perfectly periodic or weakly disordered waveguides. We relate the frequency transmission to 
+the nondimensional kinetic energy 𝐸P!
+W!Y(𝑡) attained by the cell of index 𝑖 ∈ {1, 2, …, 𝑁} and 
+numerically approximated by, 
+𝐸P!
+W!Y(𝑡) ≅ « 𝐸¬!,2 cosX𝜔-2𝑡 + 𝜃¬2Y
+Y334
+29%
+, 
+(19a) 
+
+h = 0% at 390 K
+Oh = 5% at 390 K
+a
+c
+10-11
+Cell 1
+Cell 60
+10-13
+10-13
+FT(v/
+10-15
+10-15
+10~17E
+10~17
+6
+8
+10
+12
+6
+8
+10
+12
+Frequency (MHz)
+Frequency (MHz)
+b
+d
+10-11
+10-11
+MB
+a
+IFFT(h / 
+10-13
+10~13
+10-15
+10-17
+6
+8
+10
+12
+6
+8
+10
+12
+Frequency (MHz)
+Frequency (MHz) 
+20 
+with	𝐸¬!,2 = 1
+2 𝜇!𝜔-2
+. ?𝑣±!,2
+. + 𝜒ℎ¬!,2
+.B, 
+(19b) 
+where we denote by 𝑛66- the output number of FFT sampled frequencies 𝜔-2 ≥ 0 rad/s with 
+phase shifts 𝜃¬2, and 𝑣±!,2 and ℎ¬!,2 the FFT amplitudes of 𝑣̅!(𝑡) and ℎP!(𝑡), respectively, at 
+frequencies 𝜔-2 for 𝑚 ∈{1, 2, …, 𝑛66-}. Equation (19) assumes that the FFT of both 𝑣̅!(𝑡) and 
+ℎP!(𝑡) have identical phase shifts 𝜃¬2 at 𝜔-2 leading to 𝑞P!(𝑡) ≅ ∑
+𝑞±!,2 cosX𝜔-2𝑡 + 𝜃¬2Y
+Y334
+29%
+ for 
+𝑞 ∈{𝑣, ℎ}.  
+In Figs. 7-8, the contour plots depict the 𝐸¬!,2 normalized by max 𝐸¬I,\ over all 𝑗 ∈ {2, 3, …, 
+𝑁} and all 𝑝 ∈{1, 2, …, 𝑛66-} of the respective response. For better visualization, we coerce 
+the contour values to 0 and 1 if they fall below the minimum threshold ℰ-<N ≤ 2×10-3, and 
+above the saturation limit ℰ#Q; ≥ 0.5, respectively. In essence, in Figs. 7-8, we assess the binary 
+behavior of the considered waveguide by checking whether the energy can transmit to cell	𝑖 at 
+frequency 𝜔-2 under the studied temperature and disorder conditions.  
+In particular, we investigate the acoustics of three waveguides with disorders 𝜎< ∈ {0%, 
+2.5%, 5%} as defined in (9); the corresponding thickness profiles of the waveguides are 
+provided in Fig. 3a, the supplemental Video.S3, and Fig. 3c, respectively. Moreover, to 
+efficiently excite passband II in addition to passband I, we modify in this section the initial 
+conditions in (16) into: 
+𝒒𝟎 ≝ 𝒒•(𝑡 = 0	s) = 𝟎 and 𝒒̇ 𝟎 ≝
+5𝒒8
+5; (𝑡 = 0	s) = c
+10$D𝜇%
+10$D𝜇%√𝜒
+𝟎.(O$%)×%
+d. 
+(20) 
+Note that by the new initial conditions (20), we deliver the same initial kinetic energy for both 
+translational and rotational coordinates of cell 1. We use in (20) the same value of initial kinetic 
+energy delivered in (16) in order to provide a fair analogy to the previous results.  
+The plots of Figs. 7a-e display the wave transmission in the three finite waveguides at 400, 
+390, 370, 353, and 350 K, respectively. For all these temperatures, the perfectly periodic finite 
+waveguide (𝜎< = 0%) transmits energy to the last cell 60, with frequency contents in both 
+passbands I and II – cf. Figs. 7(i). This transmission for all temperatures is unique for the 
+perfectly periodic waveguide and does not occur in the weakly disordered finite waveguides 
+considered in Figs. 7(ii)-(iii). For example, we observe transmission loss of waves with 
+frequency content in passband I for the waveguide with 𝜎< = 2.5% at 370 K  in Fig. 7c(ii), and 
+for the waveguide with 𝜎< = 5% at 390, 370, and 353 K in Figs. 7b-d(iii), respectively. Hence, 
+larger disorders correspond to a more extended range of temperatures with transmission loss 
+in passband I.  
+
+ 
+21 
+ 
+Fig. 7. Effects of thickness disorder and thermal buckling on the frequency content of 
+transmitted waves through finite waveguides: Contour plots of normalized transmitted energy 
+vs. cell number and wave frequency at (a) 400 K, (b) 390 K, (c) 370 K, (d) 353 K, and (e) 350 
+K through the 60-cells waveguides with thickness disorders 𝜎< of (i) 0%, (ii) 2.5%, and (iii) 
+5%. ; black and yellow colors correspond to the limiting values of 0 and 1, respectively. 
+ 
+ 
+
+%0 = 40
+Oh = 2.5%
+Oh = 5%
+a
+(MHz)
+12
+12
+12
+K
+10
+10
+10
+400 
+8
+8
+8
+(i)
+(i)
+(ili)
+6
+6
+6
+20
+40
+60
+1
+20
+40
+60
+1
+20
+40
+60
+b
+12
+12
+K
+10
+10
+390
+requency
+8
+8
+8
+(i)
+(i)
+(ili)
+6
+6.
+20 
+40
+60
+1
+20
+40
+60
+20
+40
+60
+c
+12
+12
+K
+10
+10
+10
+370
+requency
+8
+8
+8
+(i)
+(i)
+(ii)
+6
+20 
+40
+60
+1
+20
+40
+60
+1
+20
+40
+60
+d
+12
+12
+K
+10
+10
+10
+353
+E
+8
+8
+8
+(i)
+(i)
+(ii)
+6
+6.
+6
+1
+20
+40
+60
+1
+20
+40
+60
+1
+20
+40
+60
+e
+12
+12
+K
+10
+10
+10
+350
+8
+(i)
+(i)
+(ii)
+6
+6
+20 
+40
+60
+1
+20
+40
+60
+20
+40
+60
+Cell number
+Cell number
+Cell number 
+22 
+ 
+Fig. 8. Effects of thickness disorder and thermal buckling on the frequency content of 
+transmitted waves reaching  75% into the finite waveguides: Contour plots of the transmitted 
+energy to cell 45 vs. the temperature and the wave frequency in the 60-cell waveguides with 
+thickness disorders 𝜎< of (a) 0%, (b) 2.5%, and (c) 5%; the displayed values are extracted from 
+cell 45 similarly to the plots of Fig. 7 but over the temperature domain (350K, 400K); the black 
+and yellow colors correspond to the limiting values of 0 and 1 for the normalized energy, 
+respectively, the blue lines to the frequency extrema of passbands I and II, respectively, cf. Fig. 
+2c, and the solid/dashed lines to the filled and open circles in Fig. 2c, respectively. 
+ 
+Moreover, we notice that the frequency width of passband I is smaller in the transmitting 
+scenarios of the weakly disordered waveguides compared to the perfectly periodic waveguide. 
+This width-narrowing of passband I accompanies a similar narrowing of passband II in the 
+weakly disordered waveguides of Figs. 7(ii)-(iii). However, contrary to passband I, the weakly 
+disordered waveguides keep transmitting energy to the last cell 60 in passband II at all 
+temperatures considered, as depicted in Figs. 7. Therefore, we conclude that passband II is less 
+susceptible to structural disorder than passband I, which fully agrees with what was 
+experimentally witnessed in [29]. Hence, the ROM developed herein accurately captures these 
+acoustic aspects of the phononic lattice under investigation. 
+To further clarify the dependency of wave transmission on temperature, in Figs. 8a-c, we 
+study the frequency content of transmitted waves reaching cell 45 in the three finite waveguides 
+with disorder 𝜎< ∈ {0%, 2.5%, 5%}, respectively. On top of the finite waveguides results, we 
+overlay the corresponding passbands of the infinite waveguides, cf. Fig. 2c. The results in Fig. 
+8a prove that, at all the considered temperatures, the wave transmission in the perfectly periodic 
+finite waveguide have frequency contents solely inside the passbands. Thus, the passbands of 
+the infinite waveguide perfectly estimate the wave transmission in the perfectly periodic finite 
+waveguide at all temperatures, as concluded in the previous section from Figs. 6a-b at 390 K. 
+However, this perfect estimation does not hold for the wave transmission in the weakly 
+disordered finite waveguides considered in Figs. 8b-c, especially around the critical-buckling 
+temperature (i.e., 370 K). Although, as discussed previously, there is a band-narrowing of both 
+
+a
+b
+c
+%0 = 40
+O h = 2.5%
+Oh = 5%
+12
+12
+12
+Cell 45
+Frequency (
+10
+10
+10
+8
+8
+8
+6
+6
+375
+350
+400
+375
+350
+400
+375
+350
+Temperature (K)
+Temperature (K)
+Temperature (K) 
+23 
+passbands I and II, wave transmission loss is only realized for passband I, cf. Figs. 8b-c. This 
+behavior confirms that passband I is more susceptible to disorders than passband II, confirming 
+the analogous conclusion illustrated in Fig. 1c of the experimental work in [29]. Lastly, by 
+comparing Figs. 8b to 8c, we deduce that stronger disorders result in more severe wave 
+transmission losses with thermal-mediated buckling.  
+VI. Buckling-induced localized modes 
+To explain the causes of transmission loss due to disorder, we plot in Figs. 9-10 the spatial 
+distributions (modeshapes) of two modes of the finite waveguides, specifically modes 30 and 
+90, at 400, 390, 370, 353, and 350 K. Thick dash lines represent the normalized translational 
+and rotational deformations of individual cells calculated by the eigenvector 𝚽 of (18) at the 
+considered temperature and waveguide. For better visualization, we join the cells with cubic 
+splines depicted as thin lines to imitate the continuous deformation of the waveguide’s 
+centerline. Mode 30 considered in Fig. 9 is inside passband I of the infinite perfectly periodic 
+waveguide (which contains also the first 60 modes of the perfectly periodic finite waveguide 
+with no disorder), whereas mode 90 in Fig. 10 is located inside passband II (which also contains 
+modes 61 to 120 of the corresponding perfectly periodic finite waveguide with no disorder). 
+Figs. 9a-e(i) show that mode 30 of the perfectly periodic finite waveguide deforms with 
+comparable amplitudes over the entire spatial extent of the system, i.e., from cell 1 (the excited 
+cell) up to cell 60, at all the considered temperatures. Such modes with spatially extended 
+amplitude distributions over the entire waveguide are called “extended modes” that are 
+necessary for wave propagation in the finite waveguides. For instance, this type of extended 
+modes enable energy initially applied to cell 1 to appreciably deform the remaining cells 
+resulting in detectable mechanical energy propagation through the entire spatial length of the 
+waveguide. The extended modes in Figs. 9a-e(i) verifies the persistence of wave transmission 
+via passband I of the perfectly periodic finite waveguide as depicted in Figs. 7a-e(i) and 8a for 
+all temperatures, even very close to the critical-buckling temperature. 
+Conversely, in the weakly disordered finite waveguides not all modes are extended for all 
+temperatures. For example, for disorder level 𝜎< = 5%, mode 30 is an extended mode only at 
+400 and 350 K, respectively – cf. Figs. 9(ii)a,e. This extended shape directly affects wave 
+transmission through the waveguide: A nonzero initial deformation of cell 1 due to the applied 
+excitation deforms the cells throughout the waveguide as shown by the modeshapes of Figs. 
+9(ii)a and 9(ii)e, allowing energy to propagate throughout the waveguide (cf. Figs. 7(iii)a,e and 
+
+ 
+24 
+ 
+Fig. 9. Effect of thickness disorder and thermal buckling on mode 30: Modeshape at (a) 400 
+K, (b) 390 K, (c) 370 K, (d) 353 K, and (e) 350 K in (i) the perfectly periodic finite 
+waveguide (where mode 30 is in passband I), and (ii) the (𝜎< = 5%) weakly disordered finite 
+waveguide. 
+ 
+ 8c). However, cell 1 exhibits minimal vibrations in Figs. 9(ii)b-d at 390, 370, and 353 K, 
+respectively, justifying the inability to transmit energy by mode 30 in Figs. 7(iii)b-d and 8c. 
+The modeshapes in Figs. 9(ii)b-d are “localized modes” where large deformations are confined 
+only locally without extending over the entire waveguide (like in extended modes).  
+Localized modes characterize aperiodic/disordered structures because extended modes 
+necessitate the periodicity between the constitutive cells in the waveguides. In other words, 
+cells of similar geometry and material configurations form a periodic structure with cells of 
+similar (isolated) modal frequencies, which we refer to as modal periodicity. This modal 
+periodicity is necessary to form the extended modes that enable transmission throughout the 
+structure. In this work, we conjecture that the modal periodicity breaks in the weakly disordered 
+
+Oh = 0%, Mode #30
+Oh = 5%, Mode #30
+a
+(i)
+(ii)
+400 K
+b
+(i)
+(ii)
+390 K
+B
+c
+(i)
+(ii)
+370 K
+d
+(i)
+(ii)
+K
+d
+B
+353
+V/
+e
+(i)
+(ii)
+350 K
+B
+a
+20
+40
+60
+1
+20
+40
+60
+Cell number
+Cell number 
+25 
+waveguides under the effect of buckling because the cells in the waveguides develop different 
+(isolated) modal frequencies in passband I due buckling-induced changes in the grounding and 
+coupling stiffnesses (cf. [41]). The buckling-induced differences in modal frequencies lead to 
+localized modes, like in Fig. 9(ii)b-d, inhibiting energy transmission throughout the 
+waveguides.  
+ As a general conclusion, buckling and thermoelastic effects lead to shifts of modeshapes 
+in the frequency domain due to disorder, which, in turn, “transforms” certain modes from 
+extended to localized. Therefore, thermal effects and buckling magnify the “modal” disorder 
+in the disordered waveguides, leading to energy localization and confinement, similar to 
+Anderson localization [42]. Due to this effect, many modes between #1 and #60 (inside 
+passband I of the perfectly periodic finite waveguide) become localized in the weakly 
+disordered waveguides, as seen at 390 K in the supplemental Video.S3. Indeed, at 390 K, all 
+the leading 60 modes are localized in the waveguide with 𝜎< = 5%, prohibiting passband I 
+transmission, cf.  Figs. 7(iii)b and 8c. In addition, supplemental Video.S3 demonstrates the 
+existence of some extended modes between mode #1 and #60 in the waveguide with 𝜎< = 2.5% 
+at 390 K, which explains the observed passband I transmission at 390 K of Figs. 7(ii)b and 8b. 
+Lastly, in Supplemental Video.S3, all the leading 60 modes of the perfectly periodic waveguide 
+are extended modes, resulting in broader passband I at 390 K than the waveguide with 𝜎< = 
+2.5% – compare Figs. 7b(i) to 7b(ii). Note that all 60 leading modes of the perfectly periodic 
+waveguide are extended modes even at the critical-buckling temperature of 370 K, as shown 
+in supplemental Video.S4. 
+ 
+ 
+
+ 
+26 
+ 
+Fig. 10. Effect of thickness disorder and thermal buckling on the mode 90: Modeshape at (a) 
+400 K, (b) 390 K, (c) 370 K, (d) 353 K, and (e) 350 K; in (i) the perfectly periodic finite 
+waveguide (where mode 30 is in passband II), and (ii) the (𝜎< = 5%) weakly disordered finite 
+waveguide.  
+ 
+The buckling-induced localization does not occur in mode 90 of Fig. 10, not even in the 
+weakly disordered waveguide of 𝜎< = 5%. Therefore, mode 90 remains an extended mode at 
+all temperatures and weak disorders, enabling the propagation of a wave at the modal frequency 
+of mode 90. Most modes between 61 and 120 possess similar extended modeshapes, as 
+illustrated in supplemental Video.S5 and Video.S6 at 390 K and the critical-buckling 
+temperature of 370 K, respectively. In addition, as expected, all modes between #61 and #120 
+are extended in the perfectly periodic finite waveguide at all temperatures. In contrast, some 
+modes away from the median mode 90 are localized in the weakly disordered finite 
+waveguides, resulting in the narrowing of passband II in Figs. 8b-c. Supplemental Video.S5 
+
+Oh = 0%, Mode #90
+Oh = 5%, Mode #90
+a
+(i)
+(ii)
+400 K
+!
+b
+(i)
+(ii)
+390 K
+AAA
+c
+(i)
+(ii)
+370 K
+d
+(i)
+(ii)
+K
+353
+V/ 
+e
+(i)
+(ii)
+350 K
+A/
+20
+40
+60
+1
+20
+40
+60
+Cell number
+Cell number 
+27 
+and Video.S6 show fewer localized modes for weaker disorders (i.e., 𝜎< of 2.5% vs. 5%), 
+verifying the wider passband II of Fig. 8b compared to Fig. 8c. 
+VII. Conclusions 
+We developed a reduced-order model (ROM) for on-chip phononic waveguides made from 
+coupled drumhead resonators. In particular, we investigated the effect of thermal-induced 
+buckling on eliminating transmission over a low-frequency passband in weakly disordered 
+waveguides. The considered disorders are very small and typically result from fabrication 
+errors. We show that buckling magnifies the effect of weak geometric aperiodicity by 
+amplifying the modal disorders between constitutive cells of the waveguide. The resulting 
+effective aperiodicity yields transmission loss in the first passband of the waveguide due to 
+spatial localization of subsets of modes, similar to Anderson localization. This localization is 
+ineffective for the higher-frequency passbands of the disordered waveguides, which support 
+robust transmission to disorder and buckling.  
+Notably, the developed ROM can capture the dynamics of a two-passband waveguide 
+under thermal buckling. We adopt the buckling model from the experimental results in [41] by 
+introducing the thermal expansion of the plate microstructure of the individual cells 
+(undergoing stretching and bending) and the fabrication-residual stresses. Moreover, we 
+control the level of buckling in the ROM waveguide by assigning different temperatures. We 
+study the transmission as a function of temperature by considering the Bloch modes and the 
+free response of the perfectly periodic finite waveguides. Hence, we present a method to relate 
+the free response to the frequency content of transmitted waves in the waveguide, which saves 
+computational effort compared to simulating the frequency response of the ROM.  
+The present study highlights the important role of validated ROMs in the design of 
+phononic or acoustic waveguides that undergo buckling phase transitions. In these cases, Bloch 
+mode analysis fails to capture the experimental results even for weakly disordered finite 
+waveguides. We highlight the fact that transmission in finite waveguides is achieved via 
+extended modes of the waveguides, whereas the existence of localized modes inhibits wave 
+transmission, as energy becomes spatially confined and does not transmit throughout the extent 
+of the waveguide. These results are fully captured by the developed ROM, which offers a 
+reliable and robust alternative predictive design tool for on-chip phononic waveguides, 
+compared to experimental and/or finite element computational methods which are not as 
+versatile or computationally inexpensive.  
+
+ 
+28 
+VIII. Acknowledgment 
+This work was supported in part by NSF Emerging Frontiers in Research and Innovation 
+(EFRI) Grant 1741565. This support is greatly acknowledged by the authors. 
+ 
+ 
+ 
+
+ 
+29 
+IX. References 
+ 
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+ArXiv 1–24
+Certified Invertibility in Neural Networks
+via Mixed-Integer Programming
+Tianqi Cui
+[email protected]
+Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD, USA
+Thomas Bertalan
+[email protected]
+Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD, USA
+George Pappas
+[email protected]
+Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA
+Manfred Morari
+[email protected]
+Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA
+Yannis Kevrekidis
+[email protected]
+Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD, USA
+Mahyar Fazlyab
+[email protected]
+Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, USA
+Abstract
+Neural networks are notoriously vulnerable to adversarial attacks – small imperceptible perturba-
+tions that can change the network’s output drastically. In the reverse direction, there may exist
+large, meaningful perturbations that leave the network’s decision unchanged (excessive invariance,
+nonivertibility). We study the latter phenomenon in two contexts: (a) discrete-time dynamical sys-
+tem identification, as well as (b) calibration of the output of one neural network to the output of
+another (neural network matching). We characterize noninvertibility through the lens of mathemat-
+ical optimization, in which the global solution quantifies the “safety” of the network predictions:
+their distance from the noninvertibility boundary. For ReLU networks and Lp norms (p = 1, 2, ∞),
+we formulate these optimization problems as mixed-integer programs (MIPs) that apply to neural
+network approximators of dynamical systems. We also discuss the applicability of our results to
+invertibility certification in transformations between neural networks (e.g. at different levels of
+pruning).
+1. Introduction
+Despite achieving high performance in a variety of classification and regression tasks, neural net-
+works are not always guaranteed to satisfy certain desired properties after training. A prominent
+example is adversarial robustness. Neural networks can be overly sensitive to carefully designed
+input perturbations (Szegedy et al. (2013)). This intriguing property holds in the reverse direction
+too. In classification problems, neural networks can also be excessively insensitive to large pertur-
+bations, causing two semantically different inputs (e.g., images) to be classified in the same category
+(Jacobsen et al. (2018)). Indeed, a fundamental trade-off has been shown between adversarial ro-
+bustness and excessive invariance (Tram`er et al. (2020)), which is mathematically related to the
+noninvertibility of the map defined by the neural network.
+© T. Cui, T. Bertalan, G. Pappas, M. Morari, Y. Kevrekidis & M. Fazlyab.
+arXiv:2301.11783v1  [cs.LG]  27 Jan 2023
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+To mitigate noninvertibility, and hence excessive invariance, one can consider invertible-by-
+design architectures. Invertible neural networks (INNs) have been used to design generative models
+(Donahue and Simonyan (2019)), implement memory-saving gradient computation (Gomez et al.
+(2017)), and solve inverse problems (Ardizzone et al. (2018)). However, commonly-used INN ar-
+chitectures suffer from exploding inverses; in this paper, we therefore consider the problem of cer-
+tifying the (possible) nonivertibility of conventional neural networks after training. Specifically, we
+study two relevant invertibility problems: (i) local invertibility of neural networks: given a dynami-
+cal system whose time-τ map is parameterized by a neural network, we verify whether it is locally
+invertible around a certain input (or trajectory) and compute the largest region of local invertibil-
+ity; and (ii) local invertibility of transformations between neural networks: we certify whether two
+(assumed “equivalent”) neural networks (e.g., related through pruning) can be transformed (i.e. cal-
+ibrated) to each other locally via an invertible transformation. We develop mathematical tools based
+on mixed-integer linear/quadratic programming for the characterization of noninvertibility that are
+applicable to both (a) neural network approximators of dynamics, as well as to (b) transformations
+between neural networks.
+Related work
+Noninvertibility in neural networks was studied in the 1990s (Gicquel et al. (1998);
+Rico-Martinez et al. (1993)); more recently, several papers focus on the global invertibility property
+in neural networks (see Chang et al. (2018); Teshima et al. (2020); Chen et al. (2018); MacKay et al.
+(2018); Jaeger (2014)). Analyzing invertibility of neural networks (Behrmann et al. (2018)) and
+constructing invertible architectures arises in many contexts, such as generative modeling (Chen
+et al. (2019)), inverse problems (Ardizzone et al. (2019)) or probabilistic inference (Radev et al.
+(2020)). Neural networks invertible by design have been developed for these applications. Some
+of the these networks (e.g. RevNet (Gomez et al. (2017)), NICE (Dinh et al. (2015)), real NVP
+(Dinh et al. (2017))) partition the input domains and use affine or coupling transformations as the
+forward pass, keeping the Jacobians (block-)triangular with nonzero diagonal elements, resulting in
+nonzero determinants; others, like i-ResNet (Behrmann et al. (2019)) have no analytical forms for
+the inverse dynamics, yet their finite bi-Lipschitz constants can be derived: both methods can guar-
+antee global invertibility. A comprehensive analysis is found in (Behrmann et al. (2021); Song et al.
+(2019)). However, a theoretical understanding of the expressiveness of these architectures, as well
+as of their universal approximation properties, is still incomplete. Compared to standard networks
+like multi-layer perceptrons (MLPs) or convolutional neural networks (CNNs), the novel invertible
+neural networks (INNs) become computationally demanding. Neural ODE (Chen et al. (2018)) use
+an alternative method to compute gradients for backward propagation; i-ResNet (Behrmann et al.
+(2019)) has restrictions on the norm of every weight matrix to be enforced during the training pro-
+cess. In most cases, the input domain of interest is a small subset of the whole space. For example,
+the grey-scale image domain in computer vision problems is [0, 1]H×W (where H and W are height
+and width of images), and it is unnecessary to consider the whole space RH×W . We thus focus on
+local invertibility: how do we know if our network is invertible on a given finite domain, and if not,
+how do we quantify noninvertibility?
+Beyond classification problems, noninvertibility can also lead to catastrophic consequences in
+regression, and more specifically in dynamical systems prediction. The flow of smooth differential
+equations is invertible when it exists; yet traditional numerical integrators used to approximate them
+can be noninvertible. Neural network approximations of the corresponding time-τ map also suffer
+2
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+from this potential pathology. In this paper, we initially study noninvertibility in the context of
+dynamical systems predictions.
+2. Local invertibility of dynamical systems and neural networks
+Continuous-time dynamical systems, in particular autonomous ordinary differential equations (ODEs)
+have the form dX(t)/dt = f(X(t)), X(t = t0) = X0, where X(t) ∈ Rm are the state variables of
+interest; f : Rm �→ Rm relates the states to their time derivatives and X0 ∈ Rm is the initial con-
+dition at t0. If f is uniformly Lipschitz continuous in X and continuous in t, the Cauchy-Lipschitz
+theorem guarantees the existence and uniqueness of the solution.
+In practice, we observe the states X(t) at discrete points in time, starting at t0 = 0. For a fixed
+timestep τ ∈ R+, and ∀n ∈ N, tn = nτ denotes the n-th time stamp, and Xn = X(t = tn) the
+corresponding state values. Now we will have:
+Xn+1 := F(Xn) = Xn +
+� tn+1
+tn
+f(X(t))dt; Xn = F −1(Xn+1).
+(1)
+This equation also works as the starting point of many numerical ODE solvers.
+For the time-τ map in (1), the inverse function theorem provides a sufficient condition for its
+invertibility: If F is a continuously differentiable function from an open set B of Rm into Rm, and
+the Jacobian determinant of F at p is non-zero, then F is invertible near p. Thus, if we define
+the noninvertibility locus as the set J0(F) = {p ∈ B : det(JF (p)) = 0}, then the condition
+J0(F) = ∅ guarantees global invertibility of F (notice that this condition is not necessary: the scalar
+function F(X) = X3 provides a counterexample). If F is continuous over B but not everywhere
+differentiable, then the definition of J0 set should be altered to:
+J0(F) = {p ∈ B : ∀N0(p), ∃ p1, p2 ∈ N0(p), p1 ̸= p2, s.t. det(JF (p1)) det(JF (p2)) ≤ 0} ., (2)
+the set of points where the determinant discontinuously changes sign.
+Numerical integrators are (often) noninvertible
+Numerically approximating the finite integral in
+(1) can introduce noninvertibility in the transformation. Here is a simple one-dimensional illustra-
+tive ODE example: dX/dt = f(X) = X2 + bX + c,
+X(t = 0) = X0, where b, c ∈ R are two
+fixed parameters. The analytical solution (1) is invertible; however a forward-Euler discretization
+with step τ gives
+Xn+1 = F(Xn) = Xn + τ(X2
+n + bXn + c) ⇒ τX2
+n + (τb + 1)Xn + (τc − Xn+1) = 0.
+(3)
+Given a fixed Xn+1, Equation (3) is quadratic w.r.t. Xn; this determines the local invertibility
+of F based on ∆ = (τb + 1)2 − 4τ(τc − Xn+1): no real root if ∆ < 0; one real root with
+multiplicity 2 if ∆ = 0; and two distinct real roots if ∆ > 0. In practice, one uses small timesteps
+τ ≪ 1 for accuracy/stability, leading to the last case: there will always exist a solution Xn close
+to Xn+1, and a second preimage, far away from the region of our interest, and arguably physically
+irrelevant (this second Xn → −∞ as τ → 0). On the other hand, as τ grows, the two roots move
+closer to each other, J0(F) moves close to the regime of our simulations, and noninvertibility can
+have visible implications on the predicted dynamics. Thus, choosing a small timestep in explicit
+integrators guarantees desirable accuracy, and simultaneously practically mitigates noninvertibility
+pathologies in the dynamics.
+3
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Invertibility in transformations between neural networks
+Training two neural networks for the
+same regression or classification task practically never gives identical network parameters. Numer-
+ous criteria exist for comparing the performance of different models (e.g. accuracy in classification,
+or mean-squared loss in regression). Here we explore whether two different models can be cal-
+ibrated to each other (leading to a de facto implicit function problem). Extending our analysis
+provides invertibility guarantees for the transformation from the output of network 1 to the output
+of network 2 (and vice versa).
+3. Invertibility certification of neural networks and of transformations between them
+Here we pose the verification of local invertibility of continuous functions as an optimization prob-
+lem. We then show that for ReLU networks, this leads to a mixed-integer linear/quadratic program.
+For an integer q ≥ 1, we denote the Lq-ball centered at xc by Bq(xc, r) = {x ∈ Rn | ∥x−xc∥q ≤ r}
+(the notation also holds when q → +∞).
+Problem 1 (Local Invertibility of NNs)
+Given a neural network f : Rm �→ Rm and a point
+xc ∈ Rm in the input space, we want to find the largest radius r > 0 such that f is invertible on
+Bq(xc, r), i.e., f(x1) ̸= f(x2) for all x1, x2 ∈ Bq(xc, r), x1 ̸= x2.
+Another relevant problem is to verify whether, for a particular point, a nearby point exists with
+the same forward image. This is of particular interest in assessing invertibility of discrete-time
+dynamical systems around a given trajectory. We formally state the problem as follows:
+Problem 2 (Pseudo Local Invertibility of NNs)
+Given a neural network f : Rm �→ Rm and a
+point xc ∈ Rm in the input space, we want to find the largest radius R > 0 such that f(x) ̸= f(xc)
+for all x ∈ Bq(xc, R), x ̸= xc.
+If r and R are the optimal radii in problems 1 and 2 respectively, we must have r ≤ R. For
+Problem 1, the ball Bq(xc, r) just “touches” the J0 set; for Problem 2, the ball Bq(xc, R) extends
+to the “other” closest preimage of f(xc). Figure 1 illustrates both concepts in the one-dimensional
+case. For the scalar function y = f(x) and around a particular input xc, we show the nearest bounds
+of local invertibility and pseudo invertibility. The points Q1 = (xQ1, yQ1) and Q2 = (xQ2, yQ2)
+are the two closest turning points (elements of the J0 set) to the point C = (xc, yc); f is uniquely
+invertible (bi-Lipschitz) on the open interval (xQ1, xQ2), so that the optimal solution to Problem 1
+is: r = min{|xQ1 − xc|, |xQ2 − xc|} = |xQ1 − xc|. Noting that M1 = (xM1, yM1) and M2 =
+(xM2, yM2) are the two closest points that have the same y-coordinate as the point C = (xc, yc), the
+optimal solution to Problem 2 is R = min{|xM1 − xc|, |xM2 − xc|} = |xM1 − xc|.
+4
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Figure 1: Illustration of problems 1 (distance to invertibility boundary, red) and 2 (distance to pseudo invert-
+ibility boundary, blue).
+We now state our first result, posing the local invertibility of a function (such as a neural net-
+work) as a constrained optimization problem.
+Theorem 1 (Local Invertibility of Continuous Functions)
+Let f : Rm → Rm be a continuous
+function and B ⊂ Rm be a compact set. Consider the following optimization problem,
+p⋆ ←max
+∥x1 − x2∥
+subject to x1, x2 ∈ B,
+f(x1) = f(x2).
+(4)
+Then f is invertible on B if and only if p⋆ = 0.
+Theorem 2 (Pseudo Local Invertibility)
+Let f : Rm → Rm be a continuous function and B ⊂
+Rm be a compact set. Suppose xc ∈ B. Consider the following optimization problem,
+P ⋆ ← max
+∥x − xc∥
+subject to x ∈ B,
+f(x) = f(xc).
+(5)
+Then we have f(x) ̸= f(xc) for all x ∈ B \ {xc} if and only if P ⋆ = 0.
+Note that by adding the equality constraints x = x1, xc = x2 to the optimization problem (4),
+we obtain the optimization problem (5). Hence, we will only focus on (4) in what follows.
+Invertibility certification of ReLU networks via mixed-integer programming
+We now show
+that for a given ball B∞(xc, r) in the input space, and piecewise linear networks with ReLU activa-
+tions, the optimization problem in (4) can be cast as an MILP.
+A single ReLU constraint y = max(0, x) with pre-activation bounds x ≤ x ≤ ¯x can be
+equivalently described by the following mixed-integer linear constraints (Tjeng et al. (2017)):
+y = max(0, x), x ≤ x ≤ ¯x ⇐⇒ {y ≥ 0, y ≥ x, y ≤ x �� x(1 − t), y ≤ ¯xt, t ∈ {1, 0}},
+(6)
+where the binary variable t ∈ {1, 0} is an indicator of the activation function being active (y = x) or
+inactive (y = 0). Now consider an ℓ-layer feed-forward fully-connected ReLU network with input
+x given by the following recursions,
+x(k+1) = max(W (k)x(k) + b(k), 0) for k = 0, · · · , ℓ − 1; f(x(0)) = W (ℓ)x(ℓ) + b(ℓ),
+(7)
+5
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+where x(k) ∈ Rnk gives the input to the (k + 1)-th layer (specifically, we have x = x(0) and
+n0 = m), W (k) ∈ Rnk+1×nk, b(k) ∈ Rnk+1 are the weight matrices and bias vectors of the affine
+layers. We denote n = �ℓ
+k=1 nk the total number of neurons. Suppose l(k) and u(k) are known
+elementwise lower and upper bounds on the input to the (k + 1)-th activation layer, i.e., l(k) ≤
+W (k)x(k) + b(k) ≤ u(k). Then the neural network equations are equivalent to a set of mixed-integer
+constraints as follows:
+x(k+1) = max(W (k)x(k) + b(k), 0) ⇔
+�
+�
+�
+�
+�
+x(k+1) ≥ W (k)x(k) + b(k)
+x(k+1) ≤ W (k)x(k) + b(k) − l(k) ⊙ (1nk+1 − t(k))
+x(k+1) ≥ 0,
+x(k+1) ≤ u(k) ⊙ t(k),
+(8)
+where t(k) ∈ {1, 0}nk+1 is a vector of binary variables for the (k + 1)-th activation layer and 1nk+1
+denotes vector of all 1’s in Rnk+1. We note that the element-wise pre-activation bounds {l(k), u(k)}
+can be precomputed by, for example, interval bound propagation or linear programming, assuming
+known bounds on the input of the neural network (Weng et al. (2018); Zhang et al. (2018); Hein and
+Andriushchenko (2017); Wang et al. (2018); Wong and Kolter (2018)). Since the state-of-the-art
+solvers for mixed-integer programming are based on branch & bound algorithms (Land and Doig
+(1960); Beasley (1996)), tight pre-activation bounds will allow the algorithm to prune branches
+more efficiently and reduce the total running time.
+Local invertibility certificates via mixed-integer programming
+Having represented the neural net-
+work equations by mixed-integer constraints, it remains to encode the objective function ∥x1 − x2∥
+of (4) as well as the set B. We assume that B is an L∞ ball around a given point xc, i.e., B =
+B∞(xc, r). Furthermore, for the sake of space, we only consider L∞ norms for the objective func-
+tion. Specifically, consider the equality w = ∥x1 − x2∥∞. This equality can be encoded as mixed-
+integer linear constraints by introducing 2n0 mutually exclusive indicator variables, which leads to
+the following MILP:
+p⋆ ← max w subject to ∥x1 − xc∥∞ ≤ r, ∥x2 − xc∥∞ ≤ r
+(I) :
+�
+�
+�
+�
+�
+(x1 − x2) ≤ w1n0 ≤ (x1 − x2) + 4r(1n0 − f)
+−(x1 − x2) ≤ w1n0 ≤ −(x1 − x2) + 4r(1n0 − f′)
+f + f′ ≤ 1n0, 1⊤
+n0(f + f′) = 1, f, f′ ∈ {0, 1}n0
+(II) : W (ℓ)x(ℓ)
+1
+= W (ℓ)x(ℓ)
+2
+(9)
+for k = 0, · · · , ℓ − 1 :
+(III) :
+�
+�
+�
+�
+�
+x(k+1)
+1
+≥ W (k)x(k)
+1
++ b(k), x(k+1)
+2
+≥ W (k)x(k)
+2
++ b(k)
+x(k+1)
+1
+≤ W (k)x(k)
+1
++ b(k) − l(k) ⊙ (1 − t(k)), x(k+1)
+2
+≤ W (k)x(k)
+2
++ b(k) − l(k) ⊙ (1 − t(k))
+x(k+1)
+1
+≥ 0, x(k+1)
+2
+≥ 0, x(k+1)
+1
+≤ u(k) ⊙ t(k), x(k+1)
+2
+≤ u(k) ⊙ t(k); t(k), s(k) ∈ {0, 1}nk+1,
+where the set of constraints in (I) model the objective function ∥x1−x2∥∞, and the set of constraints
+(III) encode the network x(k+1)
+1
+= max(W (k)x(k)
+1 +b(k), 0) and x(k+1)
+2
+= max(W (k)x(k)
+2 +b(k), 0).
+The constraint (II) enforces that f(x1) = f(x2). This optimization problem (4) has 2(n0 + n)
+integer variables.
+6
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Remark 3 If we instead use the ℓ2 norm both for the objective function and the ball B2(xc, r),
+we will arrive at a mixed-integer quadratic program (MIQP). However, (9) remains an MILP if we
+change them to ℓ1 norms.
+Largest region of invertibility
+For a fixed radius r ≥ 0, the optimization problem (9) either verifies
+whether f is invertible on B∞(xc, r) or it finds counterexamples x1 ̸= x2 such that f(x1) = f(x2).
+Thus, we can find the maximal r by performing a bisection search on r (Problem 1).
+To close this section, we consider the problem of invertibility certification in transformations
+between two functions (and in particular two neural networks).
+Problem 3 (Transformation Invertibility) Given two functions f1, f2 : Rm → Rm and a partic-
+ular point xc ∈ Rm in the input space, we would like to find the largest ball Bq(xc, r) over which
+the output of f2 is a function of the output of f1 (and vice versa).
+Theorem 4
+Let f1 : Rm → Rn, f2 : Rm → Rn be two continuous functions and B ⊂ Rm be a
+compact set. Consider the following optimization problem,
+p⋆
+12 ← max
+∥f2(x1) − f2(x2)∥
+subject to x1, x2 ∈ B,
+f1(x1) = f1(x2).
+(10)
+Then the output of f2 is a function of the output of f1 on B if and only if p⋆
+12 = 0.
+Similar to Problem 1, we can pose Problem 3 as a mixed-integer program. Furthermore, we can
+also de���ne p⋆
+21, whose zero value determines whether output of f1 is a function of output of f2 over
+B. It is straightforward to see that p⋆
+12 = p⋆
+21 = 0 if and only if output of f2 is an invertible function
+of output of f1.
+4. Numerical Experiments
+We now present experiments with ReLU multi-layer perceptrons (MLPs) in both (a) regression
+problems, and also in (b) transformations between two ReLU networks.
+1D Example
+We use a 1-10-10-1 randomly generated fully-connected neural network f(x) with
+ReLU activations. We find the largest interval around the points x = −1.8; −1; −0.3 on which f is
+invertible (Problem 1); we also find the largest interval around the point x = −1 for which no other
+interior points map to f(−1) (Problem 2). The results are plotted in Figure 2, where intervals in
+red and blue respectively represent the optimal solutions for the two problems. The largest certified
+radii are 0.157, 0.322 and 0.214 for Problem 1 and 0.553 for Problem 2.
+7
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Figure 2:
+Solutions to Problem 1 (left, red) and Problem 2 (right, blue) for the MLP corresponding to a
+randomly-generated ReLU network (see text).
+2D Example: a disrete-time integrator.
+The Brusselator (Tyson (1973)) is a system of two ODEs
+for the two variables (x, y), depending on the parameters (a, b); it describes oscillatory dynamics in
+a theoretical chemical reaction scheme. We use its forward-Euler discretization with step τ,
+xn+1 = xn + τ(a + x2
+nyn − (b + 1)xn), yn+1 = yn + τ(bxn − x2
+nyn).
+(11)
+Rearranging and eliminating yn+1 in (11) we obtain:
+τ(1 − τ)x3
+n + τ(τa − xn+1 − yn+1)x2
+n + (τb + τ − 1)xn + (xn+1 − τa) = 0.
+(12)
+Equation (12) is a cubic for xn given (xn+1, yn+1) when τ ̸= 1. By varying the parameters a, b and
+τ, we see the past states (xn, yn)T of point (xn+1, yn+1)T (also called “inverses” or “preimages”)
+may be multi-valued, so that this discrete-time system is, in general, noninvertible. We fix a = 1
+and consider how inverses will be changing (a) with b for fixed τ = 0.15; and (b) with τ, for fixed
+b = 2.
+We are interested in training a neural network that learns this time-τ mapping; for a fixed set
+of parameter values, this is a network from 3D to 2D: (xn+1, yn+1)T ≈ N(xn, yn; p)T , where
+p ∈ R is the parameter. The network dynamics will be parameter-dependent if we set p ≡ b, or
+timestep-dependent if p ≡ τ. The first layer of such an MLP reads
+W (0)
+�
+�
+xn
+yn
+p
+�
+� + b(0) = (W (0)(e1 + e2))
+�xn
+yn
+�
++ (pW (0)e3 + b(0)),
+(13)
+where e1,2,3 ∈ R3 are indicator vectors. Here we trained two separate MLPs, ione with b and one
+with τ dependence. For fixed p (either b or τ) each of these two networks N can be thought of as a
+MLP mapping from R2 to R2, by slightly modifying the weights and biases in the first linear layer.
+Parameter-dependent Inverses
+It is useful to start with a brief discussion of the dynamics and
+noninvertibilities in the ground-truth system (see Figure 3). Consider a state located on the invariant
+circle (IC, shown in orange), for we therefore know there exists at least one preimage also on this
+IC. In Figure 3 we indeed see that every point on the IC has three preimages: one still on the IC, and
+two extra inverses (in green and purple) after one iteration, all three loops map to the orange one,
+8
+
+-4
+-2
+1.5
+1
+-0.5
+0
+c0
+-4
+-2
+1.5
+1
+-0.5
+0
+cCERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+and then remain forward invariant. The phase space, upon iteration, folds along the two branches
+of the J0 curve (sets of red points). For lower values of b, these three closed loops do not intersect
+each other. As b increases the (orange) attractor will become tangent to, and subsequently intersect
+J0, leading to an interaction with the other (green) preimage branch. At this point the dynamics
+predicted by the network become unphysical (beyond just inaccurate).
+Figure 3: Attractors (and their multiple inverses) for several parameter values of the discrete Brusselator
+neural network for τ = 0.15. Notice the relative positions of the J0 curves (red), the “main” preimage locus
+(yellow), and the “extra” preimages (green, purple). When the attractor starts interacting with the J0 curve
+and, therefore, with these extra preimages, the dynamic behavior degenerates quantitatively and qualitatively
+(see also Rico-Martinez et al. (1993)).
+After convergence of training, we employ our algorithm to obtain noninvertibility certificates
+for the resulting MLP, and plot results for b = 2.1 in Figure 4. In Figure 4, we arbitrarily select one
+representative point, marked by triangle (△), on the attractor (the orange invariant circle); we know
+there exists one inverse also located on the attractor, see the nearby cross (+); we call this the primal
+inverse. Our algorithm will produce two regions for this point, one for each of our problems (squares
+of constant L∞ distance in 2D). As a sanity check, we also compute the J0 sets (the red point), as
+well as a few additional inverses, beyond the primal ones with the help of a numerical root solver
+and automatic differentiation (Baydin et al. (2017)). Clearly, the smaller square neighborhood just
+hits the J0 curve, while the larger one extends to the closest non-primal inverse of the attractor.
+Timestep-dependent Inverses
+In the right two subfigures of Figure 4, we explore the effect of
+varying the time horizon τ. We compare a single Euler step of the ground truth ODE to the MLP
+approximating the same time τ map, and find that, for both of them, smaller time horizons lead to
+larger regions of invertibility.
+9
+
+(a,b)= (1, 2)
+(a,b)= (1, 2.5)
+(a,b)= (1, 3.2)
+『, F-1()
+f-1(r)
+f-1(r)"
+0
+X
+X
+xCERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+5
+0
+5
+10
+x
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+y
+J0
+Attractor
+Image
+Inverses
+5.0
+2.5
+0.0
+2.5
+5.0
+x
+4
+2
+0
+2
+4
+6
+y
+the Brusselator Integrator
+J0( =  0.05)
+J0( =  0.30)
+5.0
+2.5
+0.0
+2.5
+5.0
+x
+the Brusselator Network
+Figure 4: Left: illustration of our solution to Problems 1 and 2 for the Brusselator network with (a, b) =
+(1, 2.1). For a particular reference point on the attractor, we show the neighborhoods found by our algorithms.
+They clearly locate the closest point on the J0 curve / the closest “extra preimage” of the point of interest. Last
+two: plots of J0 curves at different τ with (a, b) = (1, 2), for both the Euler integrator and our Brusselator
+ReLU network. Small timesteps lead to progressively more remote J0 curves. Notice also the piecewise linear
+nature of the J0 curve for the ReLU network; its accurate computation constitutes an interesting challenge by
+itself.
+Network Transformation Example: Learning the Van der Pol Equation
+Here, to test our al-
+gorithm on the problem of transformations between networks 3, we trained two networks on the
+same regression task. Our data comes from the 2D Van der Pol equation dx1/dt = x2, dx2/dt =
+µ(1 − x2
+1)x2 − x1, where the input and output are the initial and final states of 1000 short solution
+trajectories of duration 0.2 for µ = 1, when a stable limit cycle exists. The initial states are uni-
+formly sampled in the region [−3, 3]×[−3, 3]. The neural network A used to learn the time-τ = 0.2
+map is a 2-32-32-2 MLP, while the neural network B is a retrained sparse version of A, where half
+of the weight entries are pruned (set to zero) based on Zhu and Gupta (2018). To visualize the per-
+formances of the two networks, two trajectories, generated by respectively iterating each network
+function for a fixed number of times starting from a common given initial state have been plotted in
+the left subplot of Figure 5. The ODE solution trajectory starting at the same initial state with same
+overall time duration is also shown. We see that both network functions A and B exhibit long term
+oscillations; the shapes of both attractors appear to only have small visual differences from the true
+ODE solution (the red curve).
+These two network functions were then used to illustrate the algorithm for Problem 3. Here we
+chose a center point xc = (0, 0)T , computed and plotted the mappable regions (the regions over
+which there is a one-to-one mapping between the output of one network and the output of the other,
+i.e. where one network can be calibrated to the other). This was done for two subcases (see the right
+subfigure of Figure 5): (a) where the output of network B is a function of the output of network A
+(the square with white bounds centered at the red point, radius 3.0820), and vice versa, where the
+output of network A is a function of the output of the network B (the square with black bounds
+centered at the red point, radius 3.6484). This also gives us the “common” region (the interior
+of the white square) where both networks can be calibrated to each other. For validation we also
+computed the Jacobian values of network A and network B on every grid point of the input domain,
+and shown that the white square touches the J0 curve of network A, while the black square touches
+the J0 curve of network B. Inside the black square the Jacobian of network B remains positive, so
+that network B is invertible (i.e. there exists a mapping from fB(x) to x, or equivalently, f−1
+B (x));
+10
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+therefore we can find the mapping from fB(x) to fA(x) by composing the mapping from fB(x) to
+x with the mapping from x to fA(x) (the function fA(x) itself). The size of the white square can be
+similarly rationalized, validating our computation.
+2
+0
+2
+x1
+3
+2
+1
+0
+1
+2
+3
+x2
+ode solution
+NN A (original)
+NN B (pruned)
+5
+0
+5
+x1
+4
+2
+0
+2
+4
+x2
+rAB = 3.0820 (white), rBA = 3.6484 (black)
+det(JA) < 0, det(JB) < 0
+det(JA) < 0, det(JB) > 0
+det(JA) > 0, det(JB) < 0
+det(JA) > 0, det(JB) > 0
+Figure 5: Left: Trajectories of the ODE solution for the Van der Pol system (red), and their discrete-time
+neural network approximations (blue and green). All three trajectories begin at the same initial state. While
+the ODE solution curve is smooth due to its continuous-time nature, the others are just straight line segments
+connecting consecutive states (discrete-time dynamics). However, it is clear that all three systems have visu-
+ally nearby long-time dynamic attractors, corroborating the good performance of the network and its pruned
+version. Right: visualization of MILP computation results, along with signs of Jacobian values of networks
+on the grid points of the input domain. Here, the center of the region is shown in red, while the white and
+black boundaries quantify the mappable region between outputs of network A and network B.
+Sparsity
+40 %
+50 %
+60 %
+Network B
+B1
+B2
+B3
+B4
+B5
+B6
+B7
+B8
+B9
+rAB
+3.0820
+3.0820
+3.0820
+3.0820
+3.0820
+3.0820
+3.0820
+3.0820
+3.0820
+rBA
+3.4609
+3.1055
+3.8555
+3.6484
+2.6523
+3.8203
+3.6328
+3.9727
+4.5547
+Table 1: The radii of the mappable regions between the original network A and its pruned versions B. rAB
+relates to the region within which fB(x) is a function of fA(x).
+As a sanity check, we consructed eight more pruned networks; two of them have 50% sparsity
+(networks B5 and B6), three have 40% sparsity (networks B1, B2 and B3) and the others have 60%
+sparsity (networks B7, B8 and B9). Above we discussed network B4 For each pruned network, we
+computed the radii of the regions of interest (aka rAB and rBA). The results are listed in Table 1.
+All pruned networks {Bi} share the same radii rAB, consistent with the invertibility of A itself.
+Since rA = 3.0820, A is invertible in the ball we computed, and the existence of the mapping
+fA(x) �→ fB(x) follows by composition of fA(x) �→ x and x �→ fB(x). Based on these few
+computational experiments one might very tentatively surmise a trend: the higher the pruning (e.g.
+60%) the larger the invertibility guarantee for the pruned network. In our work the input and output
+11
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+dimensions are the same (e.g. m = n in Problem 3). However, this condition is not necessary, and
+our algorithm can be conceptually extended to classification problems, where in general m ≫ n.
+5. Conclusions
+In this paper, we revisited noninvertibility issues that arise in discrete-time dynamical systems inte-
+grators) as well as in neural networks that perform approximations of the same (time-series related)
+task. We argued that such noninvertibility may have dramatic pathological consequences, going
+beyond just inaccuracies, in the dynamics predicted by the networks. We also extended the analysis
+to transformations between different neural networks. We formulated three problems that provide a
+quantifiable assessment of “local” invertibility for any given, arbitrarily selected input. Specifically,
+for functions like MLPs with ReLU activations, these problems were formulated as mixed-integer
+programs. We then performed experiments on regression tasks. An extension of our algorithm to
+ResNets. can be found in the Appendix.
+Future directions include developing structure-exploiting methods to globally solve these MIPs
+more efficiently, and for larger networks. On the other hand, given that convolution and aver-
+age pooling are linear operations, while max pooling is piecewise linear, it is natural to adapt our
+algorithms to convolutional neural networks like AlexNet (Krizhevsky et al. (2017)) or VGG (Si-
+monyan and Zisserman (2015)). The successful application of our algorithm to ResNet architectures
+(He et al. (2016)) holds promise for applicability also to recursive architectures (Lu et al. (2018);
+E (2017)), such as fractal networks (Larsson et al. (2017)), poly-inception networks (Zhang et al.
+(2016)), and RevNet (Gomez et al. (2017)). We are working on making the algorithm practical for
+continuous differentiable activations like tanh or Swish (Ramachandran et al. (2017)), and for other
+piecewise activations like gaussian error linear units (GELUs, Hendrycks and Gimpel (2016)). We
+are particularly interested in the case when the input and output domains are of different dimension
+(e.g., classifiers).
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+16
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Appendix A. Further Discussions
+A.1. Invertibilty in View of Lipschitz Constants
+One can consider the neural network inversion problem in terms of Lipschitz continuity and the
+Lipschitz constant. Indeed, quantifying invertibility of a neural network (more generally, a function)
+is intimately connected with its Lipschitz constant.
+Definition 5 (Lipschitz continuity and Lipschitz constant) A function F : B ⊆ Rm �→ Rm is
+Lipschitz continuous on B if there exists a non-negative constant L ≥ 0 such that
+||F(x1) − F(x2)||
+||x1 − x2||
+≤ L,
+∀x1, x2 ∈ B, x1 ̸= x2.
+(14)
+The smallest such L is called the Lipschitz constant of F, L = Lip(F).
+A generalization for Definition 5 is the bi-Lipschitz map defined as follows.
+Definition 6 (bi-Lipschitz continuity and bi-Lipschitz constant) Suppose F : B ⊆ Rm �→ Rm
+is globally Lipschitz continuous with Lipschitz constant L. Now we define another nonnegative
+constant L′ ≥ 0 such that
+L′ ≤ ||F(x1) − F(x2)||
+||x1 − x2||
+,
+∀x1, x2 ∈ Rm, x1 ̸= x2.
+(15)
+If the largest such L′ is strictly positive, then (15) shows F is invertible on B due to F(x1) ̸= F(x2)
+given x1 ̸= x2. Moreover, one could easily derive (1/L′) = Lip(F −1), where F −1 is the inverse
+function of F. We also say F is bi-Lipschitz continuous in this case, with bi-Lipschitz constant
+L∗ = max {L, 1/L′}.
+A.2. Structure of Preimages for the Learned Map of the Brusselator Flow
+As discussed in the main paper, we trained a network to approximate the time-τ Euler map (16) for
+the Brusselator. The attractor (locus of long-term image points) is a small amplitude, stable invariant
+circle (IC), the discrete time analog of the ODE stable limit cycle. We mark four representative
+points on it (Q, R, S, and T) and divide it into parts A, B1, B2, and C between these points, so
+as to facilitate the description of the dynamics and its multiple (due to noninvertibility) preimages.
+The locus of red points (the locus on which the determinant of the Jacobian of the network changes
+sign, or, in the language of noninvertible systems, the J0 curve) separates state space here into five
+distinct regions I, . . . , V, each with different preimage behavior, as illustrated in Figure A.1. For
+smooth maps, like the Brusselator forward Euler discretization or a tanh activation neural network,
+J0 is the locus of points for which the determinant of the map Jacobian is zero (and therefore, the
+map is singular). In those cases, the curve is easy to compute through continuation algorithms.
+For ReLU activations, however, this locus is nontrivial to compute through algebraic solvers, and
+piecewise smooth computational techniques or brute force exploration must be used to locate it; see
+the inset in Figure, where the color intensity indicates the magnitude, red for positive and blue for
+negative, of the map Jacobian determinant. After we locate the J0 points however, we see that they
+17
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+define the I through V (and implicitly, through forward iteration of the J0 curve, regions A through
+C on the IC):
+𝐵1
+𝑄
+𝑅
+𝑆
+𝑇
+𝑄′′ = 𝑄′′′
+𝑅′′ = 𝑅′′′
+𝑆′′ = 𝑆′′′
+𝑇′′ = 𝑇′′′
+I
+II
+III
+IV
+V
+𝐶
+𝐵2
+𝐴
+𝑅′
+Figure A.1: Top: Structure of Preimages and (top inset; positive is red, and negative is blue) magnitude
+of the map Jacobian determinant for the Brusselator network with b = 2.1. Bottom: Labeling of key
+representative points and important regions; see text. This is a qualitative rendering of the relevant regions in
+the top figure, deformed to enhance visualization.
+• Each point in part A (shown in yellow), has three inverses, located in regions I, II and III
+respectively. The “physically meaningful inverse”, the one in III, is contained in the IC itself.
+18
+
+Attractor
+Inverses
+Jo
+Determinant of
+Jacobian
+B1
+B2
+C
+IVCERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+• Points in part C (shown in cyan) similarly have inverses located in regions III, IV and V.
+• Finally, we label two segments of the IC, located between the A and C segments, as B1
+and B2 (shown in dark green, with solid line and dashed lines, respectively). Points in these
+portions of the IC only have a single inverse each (that we could find within the picture): the
+one located on the attractor itself in region III.
+It is informative to study the location and behavior of preimages as a phase point is moved along
+the invariant circle. At the transitions from either Bi part into A (or C), two preimages (initially
+one preimage with multiplicity two) are born touching the J0 curve, at the junction between I and
+II (or IV and V). Notice also the “extra preimages” of the points R and Q (R′′, R′′′, Q′′, Q′′′)
+off the invariant circle, on J0. The physically meaningful preimages (R′, Q′) lie on the invariant
+circle itself; one of them, R′, close to R, in shown in the figure. As we move further into the A
+(or C) parts of the attractor, the “extra” two preimages separate, traverse the two blue wings of the
+preimage isolas, and then collide again on the J0 curve as the phase point transitions from A (or C)
+into the other Bi part.
+A.3. Noninvertibility in Partially Observed Dynamic Histories
+Recall that the forward Euler discretization of the Brusselator is a two-dimensional map
+� xn+1 = xn + τ(a + x2
+nyn − (b + 1)xn),
+yn+1 = yn + τ(bxn − x2
+nyn).
+(16)
+In (16), we have two equations, but five unknowns (xn, yn, xn+1, yn+1, τ), so the system is in
+principle solvable only if three of them are given. This leads to
+�5
+3
+�
+= 10 possible cases, enu-
+merated below, which can be thought of as generalizations of the inversion studied in depth in the
+representative paper Adomaitis and Kevrekidis (1991); Frouzakis et al. (1997).
+1. (xn, yn, τ) ⇒ (xn+1, yn+1). (This is the usual forward dynamics case.) The evolution is
+unique (by direct substitution into (16)).
+2. (xn+1, yn+1, τ) ⇒ (xn, yn). (This is the case studied in depth in the paper.) The backward-
+in-time dynamic behavior is now multi-valued. Substituting equation (18) into the equation
+for yn+1 in system (16) we obtain
+τ(1 − τ)x3
+n + τ(τa − xn+1 − yn+1)x2
+n + (τb + τ − 1)xn + (xn+1 − τa) = 0.
+(17)
+(17) is a cubic equation w.r.t. xn if τ ̸= 0 and τ ̸= 1, which may lead to three distinct
+real roots, two distinct real roots (with one of them multiplicity 2), or one real root (with
+multiplicity 3, or with two extra complex roots). We can then substitute the solution of xn
+into (18) to obtain yn.
+3. (xn, xn+1, τ) ⇒ (yn, yn+1). Here we know the x history, and want to infer the y history:
+create an observer of y from x. This is very much in the spirit of the Takens embedding theo-
+rem Takens (1981), where one uses delayed measurements of one state variable as surrogates
+of other, unmeasured state variables. For our particular Brusselator example, the y dynamics
+inferred are unique. For the system (16), we rearrange the equation of xn+1 to obtain:
+yn = xn+1 − xn + τ(b + 1)xn − τa
+τx2n
+,
+(18)
+19
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+which shows the solution for yn is unique. Substituting (18) into (16) gives yn+1.
+4. (yn, yn+1, τ) ⇒ (xn, xn+1). Now we use history observations for y in order to infer the x
+history. The inference of the x dynamic behavior is now multi-valued. From the system (16),
+we can rearrange the equation of yn+1 to obtain
+τynx2
+n − τbxn + (yn+1 − yn) = 0.
+(19)
+(19) is a quadratic equation w.r.t. xn if τ ̸= 0 and yn ̸= 0, which may lead to two distinct
+real roots, one real root with multiplicity 2, or two (nonphysical) complex roots. We can then
+substitute (19) into (18) to obtain yn.
+5. (xn, yn+1, τ) ⇒ (yn, xn+1). We now work with mixed, asynchronous history observations.
+For this particular choice of observations the inferred dynamic behavior is unique. For the
+system (16), we can rearrange the equation of yn+1 and obtain
+yn = yn+1 − τbxn
+1 − τx2n
+,
+(20)
+which shows that the solution for yn is unique. Then we can substitute (20) into (16) to obtain
+xn+1.
+6. (yn, xn+1, τ) ⇒ (xn, yn+1). Interestingly, for this alternative set of asynchronous history
+observations, the inferred dynamic is multi-valued. From the system (16), we can rearrange
+the equation of xn+1 and obtain
+τynx2
+n + (1 − τ − τb)xn + (τa − xn+1) = 0.
+(21)
+(21) is a quadratic equation w.r.t. xn if τ ̸= 0 and yn ̸= 0, which may lead to two distinct
+real roots, one real root with multiplicity 2, or two complex roots. We can then substitute (21)
+into (16) to obtain yn+1.
+7. (xn, yn, xn+1) ⇒ (τ, yn+1). This is an interesting twist: several asynchronous observations,
+but no time label. Is this set of observations possible ? Does there exist a time interval τ
+consistent with these observations ? And how many possible τ values and possible “history
+completions” exist ? For this example, the inferred possible history is unique. For the system
+(16), we can rearrange the equation of xn+1 and obtain
+τ =
+xn+1 − xn
+a + x2nyn − (b + 1)xn
+,
+(22)
+which shows that the solution for τ is unique. We can then substitute (22) into (16) to obtain
+yn+1. The remaining cases are alternative formulations of the same “reconstructing history
+from partial observations” setting.
+8. (xn, yn, yn+1) ⇒ (τ, xn+1). The inferred history is again unique. For the system (16), we
+can rearrange the equation of yn+1 and obtain
+τ = yn+1 − yn
+bxn − x2nyn
+,
+(23)
+which shows that the solution for τ is unique. We can then substitute (23) into (16) to obtain
+xn+1.
+20
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+9. (yn, xn+1, yn+1) ⇒ (τ, xn). The inferred history is now multi-valued. Substituting equation
+(22) in the equation for yn+1 in system (16) we obtain:
+ynx3
+n+((yn−yn+1)yn−b−xn+1yn)x2
+n+(bxn+1+(b+1)(yn+1−yn))xn+a(yn−yn+1) = 0.
+(24)
+(24) is a cubic equation w.r.t. xn if yn ̸= 0, which may lead to three distinct real roots, two
+distinct real roots (with one of them multiplicity 2), or one real root (with multiplicity 3, or
+with two extra complex roots). Then we could substitute the solution of xn into (22) to obtain
+τ.
+10. (xn, xn+1, yn+1) ⇒ (τ, yn). The inferred history is again multi-valued. Substituting equation
+(18) in the equation for yn+1 in system (16) we obtain:
+x2
+n(a − xn)τ 2 + (x3
+n − (xn+1 + yn+1)x2
+n + (b + 1)xn − a)τ + (xn+1 − xn) = 0.
+(25)
+(25) is a quadratic equation w.r.t. τ if xn ̸= 0 and xn ̸= a, which may lead to two distinct
+real roots, one real root with multiplicity 2, or two complex roots. We can then substitute (25)
+into (18) to obtain yn.
+As a demonstration, we select the last of these cases, in which τ is an unknown, and show
+that multiple consistent “history completions”, i.e. multiple roots can be found; see Table A.1.
+Roots with negative or complex τ are possible, while negative timestep could be considered as a
+backward-time integration, complex results have to be filtered out as nonphysical. The methodology
+and algorithms in our paper are clearly applicable in providing certifications for regions of existence
+of unique consistent solutions; we are currently exploring this computationally.
+Given
+Unknowns
+xn
+xn+1
+yn+1
+τ1
+τ2
+yn,1
+yn,2
+4.88766
+1.62663
+2.27734
+0.27018
+0.12996
+0.06670
+-0.47845
+2.36082
+3.27177
+2.13372
+-1.51470
+0.07929
+0.98342
+3.15257
+2.19914
+1.97336
+3.22943
+-1.51394
+-0.02572
+1.18823
+2.97282
+4.60127
+2.27780
+2.21088
+(0.09960 ± 0.14337i)
+(0.24609 ± 0.51630i)
+Table A.1: (xn, xn+1, yn+1) ⇒ (τ, yn), where a = 1, b = 2.
+A.4. Extensions to Residual Architectures
+We demonstrate that our algorithms are also applicable to residual networks with ReLU activations.
+The MILP method does not extend in a simple way to networks with tanh or sigmoid activation, but
+we show here that simple algebraic formulas, like the residual connection, are addressable in this
+framework. This follows from the fact that the identity function is equivalent to a ReLU multi-layer
+perceptron (MLP) with an arbitrary number of hidden layers,
+x = g(x) − g(−x) = g(g(x)) − g(g(−x)) = g(g(g(x))) − g(g(g(−x))) = · · · ,
+(26)
+where g(x) = max(0, x) is the ReLU function. Because ReLU is idempotent g(g(x)) = g(x), we
+are able to add more and more nested versions in the right side of (26). Thus one could transform a
+ReLU ResNet with fully-connected layers to a single ReLU MLP by applying the equivalence (26).
+21
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Proposition 7 A ReLU ResNet with ℓ fully-connected layers in its residual architecture is equiva-
+lent to an MLP with the same number of layers.
+Proof Suppose f : Rm �→ Rm is a ResNet. We can rewrite y = f(x) as
+y =W (ℓ)g(W (ℓ−1)g(· · · g(W (0)x + b(0)) + · · · ) + b(ℓ−1)) + b(ℓ) + x
+=W (ℓ)g(W (ℓ−1)g(· · · g(W (0)x + b(0)) + · · · ) + b(ℓ−1)) + b(ℓ)
++ Img(Img(· · · g(Imx) + · · · )) + (−Im)g(Img(· · · g(−Imx) + · · · ))
+=(W (ℓ), Im, −Im)g(
+�
+�
+W (ℓ−1)
+Im
+−Im
+�
+� g(· · · g(
+�
+�
+W (0)
+Im
+−Im
+�
+� x +
+�
+�
+b(0)
+0m
+0m
+�
+�) + · · · )
++
+�
+�
+b(ℓ−1)
+0m
+0m
+�
+�) + b(l).
+(27)
+Here, Im ∈ Rm×m is an identity matrix, and 0m ∈ Rm is a zero vector. If we denote
+W ′(0) =
+�
+�
+W (0)
+Im
+−Im
+�
+� , W ′(ℓ) = (W (ℓ), Im, −Im),
+W ′(j) =
+�
+�
+W (j)
+Im
+−Im
+�
+� for j = 1, 2, · · · , ℓ − 1,
+b′(k) =
+�
+�
+b(k)
+0m
+0m
+�
+� for k = 0, 1, · · · , ℓ − 1, and b′(ℓ) = b(ℓ),
+(28)
+then the function
+y = W ′(ℓ)g(W ′(ℓ−1)g(· · · g(W ′(0)x + b′(0)) + · · · ) + b′(ℓ−1)) + b′(ℓ)
+(29)
+is a ReLU MLP with ℓ layers.
+Structurally Invertible Networks. It is interesting to consider how our algorithm would per-
+form when the network under study is invertible by architectural construction (e.g. an invertible
+ResNet (“i-ResNet”, Behrmann et al. (2019)). Then there is only the trivial solution to the MILP in
+(9) for any r > 0 (two identical points). What we can do in such cases is to request a certificate of
+guarantee that we are sufficiently far from noninvertibility boundaries – e.g. by a threshold larger
+than, say, 106. This is suggestive of global invertibility of the i-ResNet, and serves as a sanity check
+of the algorithm.
+Computational Effort. In general, several key factors impact the computational time of the
+MILP: the input dimension n0, the number of layers ℓ, the total number of neurons �ℓ
+i=1 ni, and
+the radius parameter r. Because a multi-layer network can be approximated to desired accuracy by
+a single-layer network with enough neurons, we will perform our experiment with a single-layer
+perceptron (ℓ = 1) and observe the dependence of the running time on n0, n1 and r by optimizing
+22
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+starting from multiple randomly-generated i-ResNets. To reduce the influence of difficult i-ResNet
+parameters that might cause the optimizer to stall, diverge, converge very slowly, or (of most con-
+cern) halt by our 30-minute timeout, we track the median of the running times for replicate experi-
+ments. See Figure A.2 for these results.
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Radius
+10
+1
+100
+101
+102
+103
+Time(s)
+n0 = 4, n1 = 10
+n0 = 6, n1 = 10
+n0 = 8, n1 = 10
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Radius
+n0 = 6, n1 = 10
+n0 = 6, n1 = 20
+n0 = 6, n1 = 50
+Figure A.2: Running time of the algorithm on a single-layer invertible ResNet as the network size varies.
+We observe that the n0 hyperparameter has a greater impact on the running time than the n1 hyper-
+parameter.
+Appendix B. Proof of Theorems and Corollaries
+B.1. Proposition Regarding Solutions to Problem 1 and Problem 2
+Proposition
+For a given function f : Rm �→ Rm and a point xc ∈ Rm, if r and R are optimal
+solutions to problems 1 and 2 respectively, then we must have r ≤ R.
+Consider a point x ∈ Bq(xc, r)\{xc}. Since f is invertible on Bq(xc, r), we must have f(x′) ̸=
+f(x) for all x′ ∈ Bq(xc, r) \ {x}. In particular, by choosing x = xc, we have f(x′) ̸= f(xc) for all
+x′ ∈ Bq(xc, r) \ {x}. Thus, we must have r ≤ R.
+B.2. Proof of Theorem 1
+Theorem Let f : Rm → Rm be a continuous function and B ⊂ Rm be a compact set. Consider
+the following optimization problem,
+p⋆ ←max
+∥x − y∥
+subject to x, y ∈ B,
+f(x) = f(y).
+(30)
+Then f is invertible on B if and only if p⋆ = 0.
+23
+
+CERTIFIED INVERTIBILITY IN NEURAL NETWORKS VIA MIXED-INTEGER PROGRAMMING
+Suppose f is invertible on B. Then for all x, y ∈ B for which f(x) = f(y), we must have
+x = y. Therefore, the objective function for Problem 1 is zero on the feasible set. Hence, p⋆ = 0.
+Conversely, suppose p⋆ = 0. Then x = y for all x, y ∈ B such that f(x) = f(y), hence invertibility.
+B.3. Proof of Theorem 2
+Theorem
+Let f : Rm → Rm be a continuous function and B ⊂ Rm be a compact set. Suppose
+xc ∈ B. Consider the following optimization problem,
+P ⋆ ← max
+∥x − xc∥
+subject to x ∈ B,
+f(x) = f(xc).
+(31)
+Then we have f(x) ̸= f(xc) for all x ∈ B \ {xc} if and only if P ⋆ = 0.
+Suppose f(x) ̸= f(xc) for all x ∈ B \{xc}. Then, the only feasible point in the optimization of
+Problem 2 is x = xc. Hence, P ⋆ = 0. Conversely, start by assuming P ⋆ = 0. Suppose there exists
+a x′ ∈ B \ {xc} such that f(x′) = f(xc). Then, we must have 0 < ∥x′ − xc∥ ≤ P ⋆ = 0, which is
+a contradiction. Therefore, we must have f(x) ̸= f(xc) for all x ∈ B \ {xc}.
+B.4. Proof of Theorem 4
+Theorem
+Let f1 : Rm → Rn, f2 : Rm → Rn be two continuous functions and B ⊂ Rm be a
+compact set. Consider the following optimization problem,
+p⋆
+12 ← max
+∥f2(x(1)) − f2(x(2))∥
+subject to x(1), x(2) ∈ B,
+f1(x(1)) = f1(x(2)).
+(32)
+Then (a) f2 is a function of f1 on B if and only if (b) p⋆
+12 = 0.
+We first set up a definition (with a slight abuse of notation) of preimage set to simplify our proof.
+Definition 8 For a given function f : X �→ Y, X ⊆ Rm, Y ⊆ Rn, the preimage of y ∈ Y is
+f−1(y) = {x ∈ X | f(x) = y}.
+We then prove the following theorem.
+Theorem 9 For two functions fi : X �→ Yi, X ⊆ Rm, Yi ⊆ Rn, i = 1, 2, we have (a) output of f2
+is a function of output of f1 if and only if (c) output of f2 is constant over the preimage set f−1
+1 (y1)
+for all y1 ∈ Y1.
+Proof We will show the equivalence of (a) and (c).
+(c) ⇒ (a): If f−1
+1 (y1) is a singleton {x1}, then f2(x1) = y2 ∈ Y2 is the only value correspond-
+ing to y1. Otherwise, we could arbitrarily choose two different values xA, xB ∈ f−1
+1 (y1), and we
+must have f2(xA) = f2(xB) = y2 ∈ Y2. Therefore, we can find a unique y2 ∈ Y2 that corresponds
+to the given y1, which infers the existence of a mapping from Y1 to Y2.
+(a) ⇒ (c): We prove this by contradiction. Suppose f2 is a function of output of f1, and
+∃y1 ∈ Y1 and ∃xA, xB ∈ f−1
+1 (y1) such that f2(xA) ̸= f2(xB) (i.e. f2 is constant over f−1
+1 (y1)).
+Therefore, we can find a y1 ∈ Y1 simultaneously corresponding to two different values f2(xA) and
+f2(xB) in Y2, showing the contradiction with (a).
+It is not hard to show (b) “p∗
+12 = 0” in (32) is equivalent with the statement that f2(x) is constant
+for ∀x ∈ f−1
+1 (f1(x)), which is just rephrasing of (c) by denoting y1 = f1(x), and therefore, we
+show the equivalence of (a) and (b).
+24
+

8dE2T4oBgHgl3EQflgcs/content/tmp_files/2301.03988v1.pdf.txt

@@ -0,0 +1,1518 @@
+Preprint
+SANTACODER: DON’T REACH FOR THE STARS!
+Loubna Ben Allal*
+Hugging Face
+Raymond Li*
+ServiceNow Research
+Denis Kocetkov*
+ServiceNow Research
+Chenghao Mou
+Independent
+Christopher Akiki
+Leipzig University and ScaDS.AI
+Carlos Munoz Ferrandis
+Hugging Face
+Niklas Muennighoff
+Hugging Face
+Mayank Mishra
+IBM Research
+Alex Gu
+MIT
+Manan Dey
+SAP
+Logesh Kumar Umapathi
+Saama Technologies
+Carolyn Jane Anderson
+Wellesley College
+Yangtian Zi
+Northeastern University
+Joel Lamy Poirier
+ServiceNow Research
+Hailey Schoelkopf
+EleutherAI
+Sergey Troshin
+University of Amsterdam
+Dmitry Abulkhanov
+Huawei Noah’s Ark Lab
+Manuel Romero
+Independent
+Michael Lappert
+Berner Fachhochschule
+Francesco De Toni
+UWA
+Bernardo Garc´ıa del R´ıo
+Flowrite
+Qian Liu
+Sea AI Lab
+Shamik Bose
+Independent
+Urvashi Bhattacharyya
+Discover Dollar Pvt Ltd
+Terry Yue Zhuo
+CSIRO’s Data61 and Monash University
+Ian Yu
+PIISA
+Paulo Villegas
+Telefonica I+D
+Marco Zocca
+Unfold ML
+Sourab Mangrulkar
+Hugging Face
+David Lansky
+Independent
+Huu Nguyen
+Ontocord, LLC
+Danish Contractor
+IBM Research
+Luis Villa
+Independent
+Jia Li
+Independent
+Dzmitry Bahdanau
+ServiceNow Research
+Yacine Jernite
+Hugging Face
+Sean Hughes
+ServiceNow
+Daniel Fried
+Carnegie Mellon University
+Arjun Guha
+Northeastern University and Roblox
+Harm de Vries‡
+ServiceNow Research
+Leandro von Werra‡∗
+Hugging Face
+ABSTRACT
+∗Corresponding authors (denoted by ‡) can be contacted at [email protected]
+1
+arXiv:2301.03988v1  [cs.SE]  9 Jan 2023
+
+Preprint
+The BigCode project is an open-scientific collaboration working on the responsi-
+ble development of large language models for code.1 This tech report describes
+the progress of the collaboration until December 2022, outlining the current state
+of the Personally Identifiable Information (PII) redaction pipeline, the experi-
+ments conducted to de-risk the model architecture, and the experiments investi-
+gating better preprocessing methods for the training data. We train 1.1B param-
+eter models on the Java, JavaScript, and Python subsets of The Stack (Kocetkov
+et al., 2022) and evaluate them on the MultiPL-E text-to-code benchmark (Cas-
+sano et al., 2022). We find that more aggressive filtering of near-duplicates can
+further boost performance and, surprisingly, that selecting files from repositories
+with 5+ GitHub stars deteriorates performance significantly. Our best model out-
+performs previous open-source multilingual code generation models (InCoder-
+6.7B and CodeGen-Multi-2.7B) in both left-to-right generation and infilling on
+the Java, JavaScript, and Python portions of MultiPL-E, despite being a sub-
+stantially smaller model. All models are released under an OpenRAIL license
+at https://hf.co/bigcode.
+1
+INTRODUCTION
+Over the last two years, we have witnessed tremendous progress in the development of code generat-
+ing AI assistants (Chen et al., 2021; Chowdhery et al., 2022; Nijkamp et al., 2022; Fried et al., 2022;
+Li et al., 2022; Athiwaratkun et al., 2022). Machine learning models are now capable of assisting
+professional developers through the synthesis of novel code snippets, not only from surrounding
+code fragments, but also from natural language instructions. The models powering these code com-
+pletion systems are usually referred to as Large Language Models for Code—or code LLMs—and
+are created by training large transformer neural networks (Vaswani et al., 2017) on big corpora of
+source code. However, with the exception of a few small-scale efforts (Xu et al., 2022), there is
+generally a lack of transparency on the development of code LLMs in the research community, in
+part due to their commercial value and the legal uncertainty around distributing training data and
+models. Some groups have released model weights (Fried et al., 2022; Nijkamp et al., 2022) or pro-
+vided access to the model through a paid API service (Chen et al., 2021; Athiwaratkun et al., 2022),
+but these works did not release the full training data or the preprocessing methods that were used.
+BigCode2 is an open scientific collaboration working on the responsible development of large lan-
+guage models for code, empowering the machine learning and open-source communities through
+open governance. BigCode was inspired by the BigScience project, an open-scientific collaboration
+which culminated in July 2022 with the release of a large multi-lingual language model (Scao et al.,
+2022). As in BigScience, various BigCode working groups focus on relevant subtopics such as
+collecting datasets, implementing methods for training code LLMs, developing an evaluation suite,
+and discussing ethical best practices for these powerful models. For example, the Legal, Ethics, and
+Governance working group has explored questions on data licensing, attribution of generated code to
+original code, the redaction of Personally Identifiable Information (PII), and the risks of outputting
+malicious code. In earlier work, the BigCode community released The Stack v1.1 (Kocetkov et al.,
+2022), a 6.4 TB dataset of permissively licensed source code in 384 programming languages. That
+work also introduced “Am I in The Stack”,3 a governance tool for developers to check whether their
+source is part of the dataset, and an opt-out form for those who wish to have their code removed
+from the dataset.4
+In this tech report, we summarize the learnings of the BigCode community in developing the Santa
+models, a set of 1.1B-parameter models trained on the Java, JavaScript, and Python subsets of The
+Stack and evaluated on MultiPL-E (Cassano et al., 2022). We describe the first steps of the commu-
+nity towards developing larger code models and report experiments to de-risk the model architecture
+and the data processing pipeline. Specifically, the contributions of this report can be summarized as
+follows:
+1See https://www.bigcode-project.org
+2See https://www.bigcode-project.org
+3https://huggingface.co/spaces/bigcode/in-the-stack
+4https://www.bigcode-project.org/docs/about/the-stack/
+2
+
+Preprint
+• We describe the current state of the PII redaction pipeline. We detail how we create a PII
+benchmark of 400 code files, describe the filters for detecting emails, ip addresses, and
+secret keys, and analyze its performance on the annotation benchmark. All experiments in
+this work are conducted on the PII-redacted version of The Stack.
+• We run ablations for Multi Query Attention (MQA) (Shazeer, 2019; Chowdhery et al.,
+2022; Li et al., 2022) and Fill-in-the-Middle (FIM) (Fried et al., 2022; Bavarian et al.,
+2022). MQA can significantly speed-up inference for larger batch sizes, while FIM en-
+ables code models to do infilling tasks. We find that both changes only slightly deteriorate
+downstream performance compared to baseline models.
+• We investigate the impact of 4 preprocessing methods on the training data: filtering files
+from repositories with 5+ GitHub stars, filtering files with a high comments-to-code ratio,
+more aggressive filtering of near-duplicates, and filtering files with a low character-to-token
+ratio. We observe modest impact of the new filters except for the stars filter, which deterio-
+rates performance on text2code benchmarks significantly. This is an interesting result given
+that previous work has explicitly filtered for GitHub Stars as a proxy for data quality (Gao
+et al., 2020; Xu et al., 2022).
+• Using the findings from these experiments, we train a final 1.1B parameter model, dubbed
+SantaCoder, on Python, JavaScript, and Java. This model obtains comparable or stronger
+performance than previous open-source multilingual models, InCoder-6.7B and CodeGen-
+Multi-2.7B, on code generation and infilling tasks on the MultiPL-E benchmark for these
+three languages, despite being substantially smaller.
+2
+RELATED WORK
+Code LLMs
+Recently, there has been an increasing amount of research on using large-scale trans-
+former models to analyze or generate source code. Many studies have focused on using decoder-only
+models with a causal language modeling objective (Chen et al., 2021; Austin et al., 2021; Nijkamp
+et al., 2022; Christopoulou et al., 2022; Izadi et al., 2022; Xu et al., 2022; Athiwaratkun et al., 2022),
+while other studies have investigated encoder (Feng et al., 2020a; Kanade et al., 2020) and encoder-
+decoder architectures (Li et al., 2022; Ahmad et al., 2021; Wang et al., 2021; Roziere et al., 2021).
+Bavarian et al. (2022); Fried et al. (2022) propose to use decoder-only models for code-infilling
+tasks using a causal masking mechanism, and Bavarian et al. (2022) argues that training with such
+a fill-in-the middle (FIM) objective does not harm the model’s ability to do left-to-right generation.
+Shazeer (2019) proposes Multi Query Attention (MQA), an architectural change to the transformer
+neural network in which key and value embeddings are shared across attention heads, resulting in
+lower memory requirements and faster inference for large batch settings. Multi Query Attention was
+implemented in AlphaCode (Li et al., 2022) and PaLM (Chowdhery et al., 2022).
+Evaluating text-to-code
+The correctness of generated code can be tested using unit tests, a method
+for approximating semantic equivalence. Textual similarity metrics have also been used to evaluate
+code (Feng et al., 2020b; Ren et al., 2020); however, they have been shown to correlate only weakly
+with code correctness (Austin et al., 2021; Chen et al., 2021).
+Many single-language benchmarks for evaluating code completion exist (Kulal et al., 2019; Iyer
+et al., 2018; Zhong et al., 2017; Yu et al., 2018; Austin et al., 2021; Hendrycks et al., 2021; Chen
+et al., 2021; Austin et al., 2021; Athiwaratkun et al., 2022; Lai et al., 2022). Two of the most popular
+benchmarks for Python are HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021), which
+consist of a natural language description of a function and a set of unit tests.
+MultiPL-E (Cassano et al., 2022) extends two popular benchmarks for code completion, MBPP
+and HumanEval, to 18 additional languages. The doctests, function signatures, and unit tests for
+each benchmark suite are automatically compiled to new languages. Python-specific terminology
+in the prompt is automatically replaced with the equivalent terminology used for each programming
+language. MBXP (Athiwaratkun et al., 2022) is a concurrent benchmark that uses a similar approach,
+but differs in the details of type inference, prompt construction, and evaluation. In particular, MBXP
+uses the same set of assertions in the prompt that it uses to test the correctness of generated solutions.
+In contrast, MultiPL-E keeps the tests hidden from the model and only uses them to test correctness.
+3
+
+Preprint
+Evaluating other tasks
+Code generation models have also been used to solve a variety of tasks
+(Tufano et al., 2020; Feng et al., 2020b; Ahmed & Devanbu, 2022; Hellendoorn et al., 2018; Pradel
+et al., 2020). CodeXGLUE (Lu et al., 2021) is a set of 14 datasets for evaluating code generation
+models. The tasks include code-to-code tasks like clone detection, code repair, and code translation;
+text-to-code tasks like code search and code generation; and code-to-text tasks like generating doc-
+umentation. The programming languages included vary by task; the most common are Python and
+Java.
+3
+OPT-OUT PROCESS
+Developers who do not wish their source code to be used for training code LLMs are given the op-
+portunity to opt-out of The Stack (Kocetkov et al., 2022). We received 9 opt-out requests before the
+cut-off date for removing data (31 October 2022). These individuals accounted for 299 repositories.
+Of these, 161 were already excluded from The Stack v1.0 (because they did not have a permissive
+license), and 138 were in The Stack v1.0. We honored the requests to opt-out and removed these
+repositories from The Stack v1.1. After the cut-off date (31 October 2022), we have received more
+requests for requests and we will remove these repositories prior to releasing The Stack v1.2.
+4
+REDACTING PERSONALLY IDENTIFIABLE INFORMATION
+We describe our first efforts to redact PII from The Stack.
+4.1
+PII BENCHMARK
+We construct a PII benchmark by annotating the following entities on a small subset of The Stack:
+names, emails, usernames, passwords, IP addresses, API keys, and SSH keys. We pre-filtered 400
+samples from a total of 4000 code files that were likely to contain Personally Identifiable Information
+(PII). We first select 4000 code files from 11 programming languages, with a total of 800 samples
+for Python and C++, 400 samples for Java, JavaScript, TypeScript, and PHP, and 160 samples for
+C, C#, Markdown, Go, and Ruby. To detect keys in these samples, we used the detect-secrets tool5
+with all default plugins activated. In addition, we used regular expressions to detect emails, IPv4
+and IPv6 addresses, see Appendix C.1. Twelve members of the BigCode community annotated the
+files on the LightTag platform6, with one annotator assigned per file. After the annotation phase, one
+member reviewed all the annotation tags. To further increase annotation quality, we ran our initial
+PII detection tools on the annotated files and manually corrected any incorrect annotations identified
+as false positives or false negatives.
+4.2
+PII DETECTION AND REDACTION
+For the first iteration of the PII redaction pipeline, we focus on emails, IP addresses, and keys, and
+leave the detection of names, usernames, and passwords for future work.
+Emails
+We use a regular expression to detect emails, see Appendix C.1. We replace detected
+emails with [random 5 character string]@example.com.
+IP addresses
+We use regular expressions for IPv4 and IPv6 IP addresses, see Appendix C.1. In
+addition, we check if the detected IP addresses have a valid format using the ipaddress python
+package. We also do not select IP addresses of the format a.b.c.d where a, b, c and d are single digit
+numbers, except if the words “dns” or “server” appear in the neighboring context (100 characters
+before or after). These detected addresses were mostly false positives, consisting of package and
+release versions. Lastly, we do not anonymize private IP addresses7 and popular DNS servers, as we
+don’t consider them sensitive information. See Appendix C.2 for the full list.
+We replace detected IP addresses with one of 5 randomly generated IP addresses.
+5https://github.com/Yelp/detect-secrets
+6https://www.lighttag.io/
+7They are non-internet facing IP addresses used in internal networks
+4
+
+Preprint
+Figure 1: Precision and recall of PII de-
+tection tools.
+Figure 2: Distribution of PII detected
+in The Stack for Python, Java and
+JavaScript.
+Keys
+We employed the detect-secrets tool to identify secret keys in the code files. To this
+end, we kept all the regex and entropy based plugins, including the AWS key detector, the GitHub
+Token detector, the Azure storage key detector, and the Base64 High Entropy String detector. You
+can find the full list of plugins in Appendix C.4. We deactivated keyword detectors because they
+were detecting commonly used words like ”password” rather than actual secret keys. To remove
+false positives, we activated filters like UUIDs and string-like secret filtering, see the full list in
+Appendix C.3. We also observed that entropy detectors sometimes detected human-readable text
+like paths and URLs as secrets, even when adjusting the entropy threshold. To address this issue, we
+added a gibberish8 detector filter on top of detect-secrets to verify that the detected string was
+actually gibberish. Additionally, we noticed that hashes were sometimes falsely detected as secret
+keys. To mitigate this problem, we added a hash filter that verifies the size of the detected string
+and checks for the presence of keywords like “sha”, “md5”, “hash”, and “byte” in the neighboring
+context. Finally, to avoid corrupting any files, we prevent the removal of keys from files where
+words like “sha” or “hash” are mentioned in more than 2% of the number of lines.
+4.3
+PERFORMANCE ANALYSIS
+Evaluation on PII benchmark
+We evaluated our PII detection pipeline on the benchmark we
+annotated. The 400 files contained 214 emails, 99 IP addresses and 34 secret keys. Figure 1 shows
+the precision and recall for each PII entity. Email and IP address detection perform well, with a
+precision and recall above 90% for emails and above 80% for IP addresses. While key detection
+also achieves almost 80% precision, its recall is much lower (slightly above 50%). We found that
+the key detection pipeline was especially sensitive to the precision-recall trade-off, as including more
+plugins or disabling some filters detected more keys but also increased the number of false positives.
+PII detection on The Stack
+We run the PII pipeline on the Python, Java and JavaScript subsets
+of The Stack v1.1 (Kocetkov et al., 2022). Table 1 shows some statistics on the number of files
+containing PII and the total number of secrets found. Some files containing PII are not modified if
+they contain only private IP addresses or popular DNS servers, as explained in the previous section.
+The number of files containing PII is significantly lower for JavaScript compared to Python and
+Java, but this could be due to the fact that JavaScript files were filtered based on line length and
+percentage of alphanumeric characters before running PII detection. We also observe that Python
+and JavaScript have a higher number of secrets per file compared to Java.
+To better understand these results, we computed the relevant percentiles in Table 2. We can see that
+Java indeed has fewer secrets per file, and that almost 0.1% of the files contain a large number of
+secrets (about 100). We found that some of these files contained multiple instances of PII, such as
+emails stored in some form of database, or are files containing long encodings and key-like strings
+8https://github.com/domanchi/gibberish-detector
+5
+
+1.0
+Precision
+Recall
+0.8 
+0.6 
+0.4 -
+0.2 
+EMAIL
+IP_ADDRESS
+KEY2M
+Python
+Java
+JavaScript
+1M
+100k
+EMAIL
+KEY
+IP_ADDRESSPreprint
+Language
+# files
+# files with PII
+# secrets
+# modified files
+Python
+15,148,604
+1,224,632
+3,255,053
+1,040,809
+Java
+25,124,914
+1,588,453
+2,757,169
+1,506,766
+JavaScript*
+23,670,848
+835,198
+2,468,183
+744,842
+Table 1: Statistics from running PII detection on The Stack. JavaScript files initially went through
+line-length filtering. Modified files are those altered during PII redaction.
+Language
+mean
+median
+95th percentile
+99th percentile
+99.9th percentile
+Python
+2.7
+1
+6
+23
+135
+Java
+1.7
+1
+3
+11
+63
+JavaScript
+3.3
+1
+7
+30
+197
+Table 2: Statistics of the number of detected PII per file in The Stack.
+that are split into multiple keys. Finally, we also plot the distributions of detected secrets by entity
+type in Figure 2. For this graph, we filtered out files with more than 100 secrets, but this did not
+change the distribution of PII across languages. We observe that IP addresses are most often found
+in Python, keys in JavaScript and emails in Java.
+5
+EXPERIMENTS
+5.1
+DATASET, MODEL, AND TRAINING DETAILS
+Dataset
+The base training dataset for the experiments in this paper contains 268 GB of Python,
+Java and JavaScript files from The Stack v1.1 (Kocetkov et al., 2022) after removing data from opt-
+out requests, near-deduplication, PII-redaction (see Section 4), and filtering based on line-length
+and percentage of alphanumeric characters. This dataset was also decontaminated by removing
+files that contained test-samples from the following benchmarks: HumanEval (Chen et al., 2021),
+APPS (Hendrycks et al., 2021), MBPP (Austin et al., 2021) and MultiPL-E (Cassano et al., 2022).
+Tokenizer
+Seeing as the Santa models were the first models our community would train, our
+design choices for the tokenizer were modulated by a conservative approach, partly based on in-
+sights developed during the development of InCoder (Fried et al., 2022). We train a Hugging Face
+Tokenizer (MOI et al., 2022) using the Byte-Pair Encoding (BPE) algorithm on raw bytes with a
+vocabulary size of 49,152 tokens. This tokenizer was trained on 600,000 rows (Around 2.6 GB) of
+data—200,000 for each language—which were pre-tokenized using a digit splitter and the default
+GPT-2 pre-tokenizer regex before being converted to bytes.
+Training details
+Our base model is a 1.1B-parameter decoder-only transformer with FIM and
+MQA trained in float16. It has 24 layers, 16 heads and a hidden-size of 2048. The model is
+trained for 300K iterations with a global batch-size of 192 using Adam (Kingma & Ba, 2015) with
+β1 = 0.9, β2 = 0.95, ϵ = 10−8 and a weight-decay of 0.1. A total of 118B tokens are seen in
+training. The learning-rate is set to 2 × 10−4 and follows a cosine decay after warming up for 2% of
+the training steps. Each training run takes 3.1 days to complete on 96 Tesla V100 GPUs for a total
+of 1.05 × 1021 FLOPs. The final model described in Section 6.2 uses twice the amount of compute.
+5.2
+ARCHITECTURE ABLATIONS
+We perform ablation experiments to de-risk the model architecture and training objective. Specif-
+ically, we investigate Fill-in-the-Middle (Bavarian et al., 2022) and Multi Query Attention
+(MQA) (Shazeer, 2019).
+FIM vs No-FIM
+Recent works (Fried et al., 2022; Bavarian et al., 2022) have shown that autore-
+gressive language-models can learn to infill code snippets by random transformation of the training
+6
+
+Preprint
+Language
+Base
+Stars
+Comments-to-code
+Near-dedup
+Tokenizer fertility
+Python
+75.6 GB
+26.6 GB
+65.6 GB
+62.0 GB
+72.5 GB
+Java
+110 GB
+35.8 GB
+92.7 GB
+88.4 GB
+105.5 GB
+JavaScript
+82.7 GB
+20.8 GB
+57.5 GB
+65.1 GB
+76.4 GB
+Table 3: Data volume after additional filtering of the Python, Java, JavaScript subsets of The Stack.
+data. Bavarian et al. (2022) argue that such data transformations do not harm the left-to-right gen-
+erative capabilities of the model. Following Bavarian et al. (2022), we implement FIM as a random
+transformation of the input sequence and split each training document into three parts uniformly
+at random: prefix, middle and suffix. Each part is prepended with a corresponding sentinel token,
+then documents are rearranged to put the middle part at the end of the sequence. The autoregressive
+training objective is unchanged. We use context-level FIM, apply transformations at the character
+level, use a FIM-rate of 0.5 and SPM+PSM joint training. We compare our base model to a model
+that was trained with the standard left-to-right objective only (No-FIM).
+Multi Query Attention vs Multi Head Attention
+Shazeer (2019) proposes Multi Query Atten-
+tion (MQA), an architectural change to transformer that shares key and value embeddings across
+attention heads. Compared to Multi Head Attention (MHA), this lowers the memory bandwidth
+requirements at generation time and results in faster inference. We compare our base model to a
+similar model using MHA instead, with the same hyper-parameters otherwise. Note that the MHA
+model has more parameters (1.3B) than the base model in this setting.
+5.3
+DATA FILTERING ABLATIONS
+We experiment with a number of preprocessing methods applied to the base dataset, described in
+Section 5.1. Note that the filters are applied on top of the other filters such as near-deduplication,
+line length filtering, etc.
+GitHub stars
+Do popular repositories contain good quality code? We use GitHub stars as a proxy
+metric. We set the minimum threshold to 5 stars, as we believe that a lower number of stars would
+not be an indicator of popularity. This filter removes more than 60% of the data (in terms of volume),
+see Table 3. Note that more than 40% of the files do not have stars and that setting the threshold to
+10 stars would remove an additional 5% of the data.
+Comment-to-code ratio
+Good code should be well documented. With this assumption, we filter
+files with a high comments-to-code ratio. We use the ast and tokenize modules to extract
+docstrings and comments from Python files, and Pygments to extract comments from Java and
+JavaScript files. We then analyze the comment-to-code character ratio. We find that about 20% of
+Python and Java files and 45% of JavaScript files have no comments. We use a minimum threshold
+of 1%, removing an additional 3% of files in each language. We also find that files with a ratio above
+80% have poor quality, so we filter them out, eliminating 2% of data in all languages. Overall, this
+comment-to-code filter removes 20% of the data in terms of volume.
+More near-deduplication
+Previous work (Kocetkov et al., 2022) has demonstrated the effective-
+ness of deduplication in boosting the performance of code LLMs. Based on this finding, we investi-
+gate whether more aggressive near-deduplication can further improve performance. To this end, we
+conduct experiments on a 100K subset of the base dataset. In the original deduplication pipeline, we
+implemented a false positive check on top of the MinHash LSH9 output. This added processing
+time, but was necessary due to a high false positive rate of around 15%. To remove more duplicates
+while maintaining a low false positive rate and a low false negative rate, we switch to using 5-gram
+for min-hashing, and 0.7 for the Jaccard Similarity threshold, without any additional false positive
+checks after the initial near-deduplication. As a result, we see additionally 16%–20% fewer files
+than the original already-deduplicated base dataset (see Table 3), and a decrease in both the esti-
+9https://github.com/ekzhu/datasketch
+7
+
+Preprint
+Language
+Attention
+FIM
+HumanEval
+MBPP
+Java
+Multi Query Attention
+
+0.35
+0.54
+Multi Head Attention
+
+0.36
+0.55
+Multi Query Attention
+
+0.37
+0.55
+JavaScript
+Multi Query Attention
+
+0.33
+0.64
+Multi Head Attention
+
+0.37
+0.67
+Multi Query Attention
+
+0.37
+0.65
+Python
+Multi Query Attention
+
+0.36
+0.67
+Multi Head Attention
+
+0.38
+0.70
+Multi Query Attention
+
+0.39
+0.68
+Table 4: Pass@100 results for the architecture ablations on HumanEval and MBPP.
+Model
+Java
+JavaScript
+Python
+Baseline
+0.64
+0.61
+0.42
+GitHub stars
+0.54
+0.57
+0.37
+Comments-to-code
+0.62
+0.59
+0.44
+More near deduplication
+0.66
+0.57
+0.45
+Tokenizer fertility
+0.67
+0.65
+0.45
+Final
+0.62
+0.60
+0.44
+Table 5: Fill-in-the-middle results for the data filtering ablations on MultiPL-HumanEval. Each
+number reports the fraction of lines where the model exactly reproduces a single line of code that is
+held out from the body of a function in a held out problem.
+mated false positive rate (from 15% to 5%) and the estimated false negative rate for documents with
+high similarities (from 35% to 24%).
+Tokenizer fertility
+Can we use the tokenizer to remove low-quality files from the dataset? We
+experiment with filtering files with a low character-to-token ratio10. For each language, we find that
+files with a ratio below the 5th percentile are usually of poor quality, but increasing the threshold may
+eliminate some good-quality files. We therefore set the cutoff value for this ratio to the following
+values: 2.5 for Python, 2.9 for Java, and 2.6 for JavaScript. This filters out roughly 4% to 5% of
+data. Note that these values depend highly on the tokenizer and the data. This filter may also be
+biased against files with non-English comments.
+5.4
+EVALUATION
+Text2code evaluation
+The text2code task involves generating the body of a function from a
+prompt that includes a function description, the function signature (its name and arguments), and
+optionally a handful of example inputs and outputs. Every problem is accompanied by a set of
+hidden test cases, which are used to determine if the generated function is correct. We use the
+MultiPL-E text2code benchmark Cassano et al. (2022), which is derived from HumanEval Chen
+et al. (2021) and MBPP Austin et al. (2021) (the “sanitized” subset of MBPP.). Whereas the latter
+two benchmarks target Python, MultiPL-E has a suite of compilers that translate HumanEval and
+MBPP to 18 other programming languages. Since our models are only trained on Java, JavaScript,
+and Python, we only evaluate them on these three languages.
+We use the methodology of Chen et al. (2021) and we calculate pass@k rates for (k = 1, 10, 100)
+for every problem. Intuitively, pass@1 estimates the likelihood a model will generate a correct
+solution in a single attempt, whereas pass@10 and pass@100 estimate the likelihood that the model
+will generate a correct solution given 10 and 100 attempts respectively. Following the literature,
+10We slightly abuse the term tokenizer fertility in this work as it usually refers to the average number of
+subwords per token, where a token is determined by the true tokenizer of the programming language. See e.g.
+(Rust et al., 2021)
+8
+
+Preprint
+Figure 3: HumanEval pass@100 performance throughout training for all models. Note that evalua-
+tion shown here is based on OpenAI Python prompts and might differ (slightly) from the MultiPL-E
+prompts used in the rest of this paper.
+we sample 200 completions at temperatures 0.2 and 0.8 and use 0.2 to estimate pass@1 and 0.8 for
+pass@10 and pass@100.
+Fill-in-the-middle evaluation
+To evaluate fill-in-the-middle, we use the single-line exact match
+metric, which was introduced by Fried et al. (2022) and also employed by Bavarian et al. (2022). For
+every benchmark problem, we mask out a single line of text from the function body (i.e., not from
+the function description or signature), and prompt the model to fill in that line of code. We exclude
+blank lines and comments, and count the number of times the model produces exactly the masked out
+line. This benchmark requires working solutions for problems, which MultiPL-E does not have. (A
+text2code benchmark like MultiPL-E only needs hidden tests.) Instead, of writing solutions by hand,
+we use solutions generated by a code generation model, which is the approach of Athiwaratkun et al.
+(2022). Specifically, we use working solutions produced by code-davinci-002 at temperature
+0.8. Note that this approach does not produce solutions to every problem, since not all problems
+are solvable. Moreover, for uniformity, we use this approach for Python as well, even though hand-
+written Python solutions exist for our benchmarks. We only report fill-in-the-middle evaluations for
+the data filtering ablations.
+6
+RESULTS
+6.1
+ABLATIONS
+For the architecture ablations, we report the results on text2code benchmarks in Table 4. For the
+data filtering ablations, we show the text2code results in Figure 4 and report the fill-in-the middle
+evaluations in Table 5. We show the HumanEval performance throughout all training runs in Figure
+3. You can find the full results tables of the text2code experiments are Appendix A.
+Slight drop in performance for MQA
+We see a small drop in performance for Multi Query
+Attention (MQA) compared to Multi Head Attention (MHA). As shown in Table 4, the MHA model
+improves pass@100 with 1-4% on HumanEval and with 1-3% on MBPP. We specifically observe
+noticeable improvements for the JavaScript versions of the text2code benchmarks. However, it
+should be noted that the MHA model has more parameters (1.3B) than the MQA model (1.1B),
+and a head-to-head comparison might, therefore, not be entirely fair. We think that the inference
+speed-ups of MQA might outweigh the small drop in performance.
+9
+
+0.45
+0.40
+0.35
+0.30
+350M-theStackv1near-dedup-pass@100
+Base-pass@100
+0.25
+Arch: No Fim -pass@100
+Arch: MHA -pass@100
+Dataset:comments-pass@1oo
+0.20
+Dataset:stars-pass@1oo
+Dataset: fertility -pass@100
+0.15
+Dataset: near-dedup-pass@1o0
+Final (near-dedup + comments)-pass@100
+.
+50
+100
+150
+200
+Number of tokens seen in training (B)Preprint
+Multi−HumanEval Pass@100
+Multi−MBPP Pass@100
+Multi−HumanEval Pass@10
+Multi−MBPP Pass@10
+Multi−HumanEval Pass@1
+Multi−MBPP Pass@1
+Java
+JavaScript
+Python
+Java
+JavaScript
+Python
+0.0
+0.2
+0.4
+0.6
+0.8
+0.0
+0.2
+0.4
+0.6
+0.8
+0.0
+0.2
+0.4
+0.6
+0.8
+Language
+Estimate
+Model
+Baseline
+Comments
+Dedup Alt
+Fertility
+Stars
+Final
+Figure 4: Pass@k rates on Multi-HumanEval and Multi-MBPP by model and language
+Left-to-right pass@100
+Fill-in-the-middle ex. match
+Model
+Size
+Java
+JavaScript
+Python
+Java
+JavaScript
+Python
+InCoder
+6.7B
+0.36
+0.38
+0.47
+0.49
+0.51
+0.31
+CodeGen-multi
+2.7B
+0.42
+0.39
+0.39
+
+
+
+CodeGen-mono
+2.7B
+
+
+0.57
+
+
+
+Codex11
+2.5B
+
+
+0.60
+
+
+
+SantaCoder
+1.1B
+0.41
+0.47
+0.49
+0.62
+0.60
+0.44
+Table 6: Comparing the performance of the final version of SantaCoder with InCoder (Fried et al.,
+2022), CodeGen (Nijkamp et al., 2022), and Codex (Chen et al., 2021) on left-to-right (HumanEval
+pass@100) and fill-in-the-middle benchmarks (HumanEval line filling, exact match).
+FIM for cheap
+We observe a minor drop in performance of the FIM model compared to the
+No-FIM model. Specifically, we see that the pass@100 performance of the FIM model is 2-4%
+lower on HumanEval and 1% lower on MBPP. While Bavarian et al. (2022) presented evidence
+for the existence of a FIM-for-free property (i.e., arguing that autoregressive models can be trained
+with FIM without harming left-to-right capabilities), we do find a small but consistent drop of FIM
+models on left-to-right text2code benchmarks.
+11This is the performance of a Codex model reported by Chen et al. (2021). It is not clear if this model is
+available via the OpenAI API.
+10
+
+Preprint
+Modest impact of near-deduplication, comments, and fertility filter
+On text2code benchmarks,
+we observe small gains for the near-deduplication and comment-to-code filters and a neutral effect
+of the tokenizer filter. The near-deduplication filter improves HumanEval performance by 1-3% and
+MBPP by 1-4% across the three programming languages. The comment-to-code filter improves
+HumanEval performance by 0-2% but decreases MBPP performance in certain cases (Java). See
+Appendix A for the full results table. On fill-in-the-middle benchmarks, we see that the tokenizer
+fertility filter performs well, improving performance by 2-4% across the three languages. The near-
+duplication and comments filters have a mixed effect, improving fill-in-the-middle performance for
+Python but deteriorating performance for JavaScript.
+GitHub stars deteriorate performance
+Surprisingly, we find that the GitHub stars filter performs
+poorly. On HumanEval and MBPP, the pass@100 performance consistently drops by 3-6% across
+the three languages. On the fill-in-the-middle benchmark, the performance drops by 5-11% (Table
+5). Note that the stars filter removes the most data (over 60%) and, therefore, raises the question
+whether the performance difference is due to the smaller dataset. However, as can be seen in Figure
+3, HumanEval pass@100 diverged early on in training, indicating that the drop in performance is
+not only due to data size but also data quality.
+6.2
+FINAL MODEL
+Based on the insights from the architecture and dataset ablations, we train a final model, which we
+call SantaCoder, with MQA and FIM and the two data filters that yielded the best results: more near-
+deduplication and comments-to-code filter. We train this model for 600K iterations (236B tokens)
+and keep all other hyper-parameters the same.
+Improved text2code performance
+Doubling the training iterations leads to much stronger
+text2code performance on MultiPL-E, significantly boosting performance across all benchmarks
+and programming languages (see Figure 4). Looking at the performance throughout training (Figure
+3), it is likely that longer training can further increase performance. Surprisingly, we find that the
+final training run did not improve the fill-in-the-middle evaluations (see Table 5), at least on these
+single line infilling tasks.
+Comparison to InCoder, CodeGen, and Codex
+Table 6 compares our SantaCoder model to
+comparably-sized code generation models from previous work on the MultiPL-E benchmark, using
+the methodology described in Section 5.4. We find that our model generally outperforms previ-
+ous open-source multi-language code generation models despite being smaller, outperforming the
+InCoder 6.7B (Fried et al., 2022) model on both left-to-right generation and single line fill-in-the-
+middle infilling across languages, and obtaining comparable or stronger performance to CodeGen-
+multi 2.7B (Nijkamp et al., 2022).
+7
+CONCLUSION
+We described the progress of the BigCode project until December 2022. The community took its
+first steps towards redacting PII and demonstrated that regular expressions are reasonably effective
+at detecting emails and IP addresses. Future work should focus on increasing the precision and recall
+of secret keys, as well as detecting other sensitive information such as names, usernames, and pass-
+word. Using the PII-redacted version of The Stack, we conducted a series of architectural and data
+filtering ablations. One of our main findings was that filtering for Github stars consistently decreased
+performance across all benchmarks and programming languages. Using the findings of these abla-
+tion studies, we trained a final 1.1B model—dubbed SantaCoder—for 236B tokens and showed it
+is able to outperform previous multi-lingual code models (InCoder-6.7B and CodeGen-Multi-2.7B)
+on both left-to-right generation and infilling tasks. We anticipate that larger architectures and more
+training data will be able to produce stronger multilingual, infilling-capable models, and plan to
+continue to scale the findings from our investigations here.
+11
+
+Preprint
+8
+CONTRIBUTIONS
+Model license
+Carlos Munoz Ferrandis, Christopher Akiki, Danish Contractor, Harm de Vries,
+Huu Nguyen, Leandro von Werra, Luis Villa, Sean Hughes, Yacine Jernite, David Lansky
+PII redaction
+Loubna Ben Allal, Jia Li, Paulo Villegas, Harm de Vries, Leandro Von Werra,
+Christopher Akiki, Ian Yu, Michael Lappert, Urvashi Bhattacharyya, Shamik Bose, Bernardo Garc´ıa
+del R´ıo, Francesco De Toni, Terry Yue Zhuo, Qian Liu, Manuel Romero
+Dataset
+Denis Kocetkov, Chenghao Mou, Loubna Ben Allal, Leandro von Werra, Dmitry Ab-
+ulkhanov, Christopher Akiki, Raymond Li
+Tokenizer
+Christopher Akiki, Sergey Troshin, Dmitry Abulkhanov, Daniel Fried, Leandro von
+Werra, Harm de Vries
+Training and architecture
+Raymond Li, Daniel Fried, Hailey Schoelkopf, Joel Lamy Poirier,
+Qian Liu, Niklas Muennighoff, Loubna Ben Allal, Dzmitry Bahdanau, Harm de Vries, Leandro von
+Werra
+Opt out
+Sean Hughes, Carlos Munoz Ferrandis, Christopher Akiki, Denis Kocetkov, Harm de
+Vries, Huu Nguyen, Leandro von Werra, Luis Villa
+Evaluation
+Arjun Guha, Yangtian Zi, Carolyn Jane Anderson, Loubna Ben Allal, Raymond Li,
+Niklas Muennighoff, Manan Dey, Logesh Kumar Umapathi, Leandro von Werra, Harm de Vries,
+Marco Zocca
+Inference
+Mayank Mishra, Alex Gu, Joel Lamy Poirier, Leandro von Werra, Harm de Vries,
+Sourab Mangrulka
+Acknowledgement
+We thank ServiceNow and HuggingFace for the provided compute resources.
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+natural language using reinforcement learning, 2017.
+URL https://arxiv.org/abs/
+1709.00103.
+15
+
+Preprint
+A
+FULL TEXT2CODE RESULTS
+We report the full results of all experiments. Table 7 and 8 show the full results for the data filtering
+ablations on HumanEval and MBPP, respectively. Table 9 and 10 reports the full results for the
+architecture ablations on HumanEval and MBPP, respectively.
+Language
+Model
+Pass@1
+Pass@10
+Pass@100
+Java
+Baseline
+0.1
+0.19
+0.35
+GitHub stars
+0.08
+0.16
+0.3
+Comments-to-code ratio
+0.11
+0.2
+0.35
+More near deduplication
+0.13
+0.22
+0.38
+Tokenizer fertility
+0.11
+0.19
+0.35
+JavaScript
+Baseline
+0.12
+0.19
+0.33
+GitHub stars
+0.08
+0.15
+0.3
+Comments-to-code ratio
+0.12
+0.2
+0.35
+More near deduplication
+0.14
+0.2
+0.37
+Tokenizer fertility
+0.1
+0.19
+0.35
+Python
+Baseline
+0.12
+0.21
+0.36
+GitHub stars
+0.1
+0.18
+0.31
+Comments-to-code ratio
+0.14
+0.22
+0.38
+More near deduplication
+0.13
+0.22
+0.37
+Tokenizer fertility
+0.14
+0.21
+0.36
+Table 7: Full results for data filtering ablations on HumanEval
+16
+
+Preprint
+Language
+Model
+Pass@1
+Pass@10
+Pass@100
+Java
+Baseline
+0.23
+0.37
+0.54
+GitHub stars
+0.18
+0.33
+0.49
+Comments-to-code ratio
+0.22
+0.37
+0.52
+More near deduplication
+0.23
+0.38
+0.55
+Tokenizer fertility
+0.22
+0.38
+0.53
+JavaScript
+Baseline
+0.25
+0.43
+0.64
+GitHub stars
+0.19
+0.37
+0.59
+Comments-to-code ratio
+0.25
+0.44
+0.65
+More near deduplication
+0.26
+0.45
+0.66
+Tokenizer fertility
+0.24
+0.43
+0.65
+Python
+Baseline
+0.27
+0.47
+0.67
+GitHub stars
+0.24
+0.41
+0.63
+Comments-to-code ratio
+0.3
+0.48
+0.69
+More near deduplication
+0.31
+0.49
+0.71
+Tokenizer fertility
+0.28
+0.47
+0.68
+Table 8: Full results for data filtering ablations on MBPP
+Language
+Attention
+FIM
+Pass@1
+Pass@10
+Pass@100
+Java
+Multi Query Attention
+
+0.1
+0.19
+0.35
+Multi Head Attention
+
+0.12
+0.21
+0.36
+Multi Query Attention
+
+0.11
+0.21
+0.37
+JavaScript
+Multi Query Attention
+
+0.12
+0.19
+0.33
+Multi Head Attention
+
+0.13
+0.21
+0.37
+Multi Query Attention
+
+0.14
+0.21
+0.37
+Python
+Multi Query Attention
+
+0.12
+0.21
+0.36
+Multi Head Attention
+
+0.13
+0.24
+0.38
+Multi Query Attention
+
+0.14
+0.23
+0.39
+Table 9: Full results for architecture ablations on HumanEval
+17
+
+Preprint
+Language
+Attention
+FIM
+Pass@1
+Pass@10
+Pass@100
+Java
+Multi Query Attention
+
+0.23
+0.37
+0.54
+Multi Head Attention
+
+0.23
+0.38
+0.55
+Multi Query Attention
+
+0.23
+0.37
+0.55
+JavaScript
+Multi Query Attention
+
+0.25
+0.43
+0.64
+Multi Head Attention
+
+0.26
+0.46
+0.67
+Multi Query Attention
+
+0.23
+0.44
+0.65
+Python
+Multi Query Attention
+
+0.27
+0.47
+0.67
+Multi Head Attention
+
+0.31
+0.49
+0.7
+Multi Query Attention
+
+0.28
+0.47
+0.68
+Table 10: Full results for architecture ablations on MBPP
+Model Family
+Variant
+BLEU
+InCoder
+6.7B
+16.04
+CodeGen-Mono
+16B
+20.56
+SantaCoder
+Baseline
+17.67
+SantaCoder
+No-FIM
+17.71
+SantaCoder
+MHA
+17.72
+SantaCoder
+Bf16
+17.67
+SantaCoder
+GitHub Stars
+18.04
+SantaCoder
+Comments-to-code
+17.81
+SantaCoder
+More near deduplication
+17.65
+SantaCoder
+Tokenizer fertility
+17.64
+SantaCoder
+Final
+18.13
+Table 11: CodeXGLUE (Lu et al., 2021) Python Docstring generation smoothed 4-gram BLEU
+scores using the same methodology as Fried et al. (2022) (L-R single). Models are evaluated zero-
+shot, greedily and with a maximum generation length of 128.
+B
+DOCSTRING GENERATION
+In addition to code completion benchmarks, we also report results on docstring generation. To this
+end, we evaluate our models on CodeXGLUE code-to-text Lu et al. (2021), which is a benchmark
+constructed from CodeSearchNet Husain et al. (2019). We use the bigcode-evaluation-harness li-
+brary Ben Allal et al. (2022), which is derived from lm-evaluation-harness Gao et al. (2021). Models
+are prompted with a Python function signature and asked to output a corresponding docstring. Re-
+sults are shown in Table 11.
+Findings
+We find all BigCode Santa variants with 1.1B parameters to outperform the 6.7B In-
+Coder model (Fried et al., 2022), which we attribute to differences in the training datasets. Among
+BigCode models, variants trained on more Python perform better: The stars variant with 32% of
+Python in its training corpus outperforms the tokenizer fertility variant with only 28.5% of Python
+(see proportions in Table 3). The bfloat16 is the same as the no-fim variant, except for the lat-
+ter being trained in float16. There’s no notable performance difference between the two, likely
+because at our small scale of 1.1B parameters we did not face any training instabilites.
+Qualitative examples
+Below is an example prompt from CodeXGLUE. Model generations and
+the correct solution are in Table 12.
+def dailymotion_download(url, output_dir=’.’, merge=True,
+info_only=False, **kwargs):
+"""
+18
+
+Preprint
+Model Family
+Variant
+Generation
+InCoder
+6.7B
+Download a video from Dailymotion.
+CodeGen-Mono
+16B
+Downloads Dailymotion videos by URL.
+SantaCoder
+Baseline
+Download Dailymotion videos.
+SantaCoder
+FIM
+Download a video from a dailymotion video.
+SantaCoder
+MHA
+Download a video from a Dailymotion video.
+SantaCoder
+bf16
+Download video from dailymotion.com.
+SantaCoder
+GitHub stars
+Download media from dailymotion.com
+SantaCoder
+Comments-to-code
+Download a video from Dailymotion.
+SantaCoder
+More near deduplication
+Download a dailymotion video.
+SantaCoder
+Tokenizer fertility
+Download a video from Dailymotion.
+Correct solution
+Downloads Dailymotion videos by URL.
+Table 12: CodeXGLUE (Lu et al., 2021) Python Docstring generation examples.
+C
+PII
+C.1
+REGULAR EXPRESSIONS
+Email addresses
+We used the following regular expression to detect emails.
+email_pattern = r’’’
+(?<= ˆ | [\b\s@,?!;:)(’".\p{Han}<] )
+(
+[ˆ\b\s@?!;,:)(’"<]+
+@
+[ˆ\b\s@!?;,/]*
+[ˆ\b\s@?!;,/:)(’">.]
+\.
+\p{L} \w{1,}
+)
+(?= $ | [\b\s@,?!;:)(’".\p{Han}>] )
+’’’
+We replace detected emails with [random 5 character string]@example.com.
+IP addresses
+We used the following regular expressions to detect IPv4 and IPv6 addresses.
+ipv4_pattern = r"(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)
+(?:\.(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)){3}"
+ipv6_pattern = r"(?:[0-9a-fA-F]{1,4}:){7,7}[0-9a-fA-F
+]{1,4}|(?:[0-9a-fA-F]{1,4}:){1,7}:|(?:[0-9a-fA-F]{1,4}:)
+{1,6}:[0-9a-fA-F]{1,4}|(?:[0-9a-fA-F]{1,4}:){1,5}(?::[0-9a-fA-
+F]{1,4}){1,2}|(?:[0-9a-fA-F]{1,4}:){1,4}(?::[0-9a-fA-F]{1,4})
+{1,3}|(?:[0-9a-fA-F]{1,4}:){1,3}(?::[0-9a-fA-F]{1,4})
+{1,4}|(?:[0-9a-fA-F]{1,4}:){1,2}(?::[0-9a-fA-F]{1,4})
+{1,5}|[0-9a-fA-F]{1,4}:(?:(?::[0-9a-fA-F]{1,4}){1,6})
+|:(?:(?::[0-9a-fA-F]{1,4}){1,7}|:)|fe80:(?::[0-9a-fA-F]{0,4})
+{0,4}%[0-9a-zA-Z]{1,}|::(?:ffff(?::0{1,4}){0,1}:)
+{0,1}(?:(?:25[0-5]|(?:2[0-4]|1{0,1}[0-9]){0,1}[0-9])\.)
+{3,3}(?:25[0-5]|(?:2[0-4]|1{0,1}[0-9]){0,1}[0-9])|(?:[0-9a-fA-
+F]{1,4}:){1,4}:(?:(?:25[0-5]|(?:2[0-4]|1{0,1}[0-9]){0,1}[0-9])
+\.){3,3}(25[0-5]|(?:2[0-4]|1{0,1}[0-9]){0,1}[0-9])"
+ip_pattern = (
+r"(?:ˆ|[\b\s@?,!;:\’\")(.\p{Han}])("
++ r"|".join([ipv4_pattern, ipv6_pattern])
+19
+
+Preprint
++ ")(?:$|[\s@,?!;:’\"(.\p{Han}])"
+)
+Data pre-filtering
+This is the regular expression we used to pre-filter the annotation dataset for
+data containing emails.
+email_pattern = r’([ˆ\s@,?!;:\’\"=)(]+@[ˆ,\s!?;,\’\"=]{3,}[\.][ˆ\s
+\b\’\"@,?!;:)(.]+)’
+For IP addresses, we used the same regular expression as the one used for PII detection.
+C.2
+LIST OF PRIVATE IP ADDRESSES AND POPULAR DNS SERVERS
+• 8.8.8.8
+• 8.8.4.4
+• 1.1.1.1
+• 1.0.0.1
+• 76.76.19.19
+• 76.223.122.150
+• 9.9.9.9
+• 149.112.112.112
+• 208.67.222.222
+• 208.67.220.220
+• 8.26.56.26
+• 8.20.247.20
+• 94.140.14.14
+• 94.140.15.15
+C.3
+DETECT-SECRETS FILTERS
+• detect secrets.filters.heuristic.is potential uuid
+• detect secrets.filters.heuristic.is likely id string
+• detect secrets.filters.heuristic.is templated secret
+• detect secrets.filters.heuristic.is sequential string
+Implementation
+available
+at
+https://github.com/bigcode-project/
+bigcode-dataset/blob/6b3f54751b6e38e1ed70f2307331d6943ba39eae/
+pii/utils/keys_detection.py#L11.
+C.4
+DETECT-SECRETS PLUGINS
+• ArtifactoryDetector
+• AWSKeyDetector
+• Base64HighEntropyString
+• HexHighEntropyString
+• AzureStorageKeyDetector
+• CloudantDetector
+• DiscordBotTokenDetector
+• GitHubTokenDetector
+20
+
+Preprint
+• IbmCloudIamDetector
+• IbmCosHmacDetector
+• JwtTokenDetector
+• MailchimpDetector
+• NpmDetector
+• SendGridDetector
+• SlackDetector
+• SoftlayerDetector
+• StripeDetector
+• TwilioKeyDetector
+Implementation
+available
+at
+https://github.com/bigcode-project/
+bigcode-dataset/blob/6b3f54751b6e38e1ed70f2307331d6943ba39eae/
+pii/utils/keys_detection.py#L19.
+21
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+A note on constructions of bent functions
+Yanjun Li, Haibin Kan∗, Jie Peng, Lijing Zheng, and Changhui Chen
+Abstract
+Recently, Li et al. presented a generic construction of bent functions in [IEEE Trans.
+Inf. Theory, vol. 68, no. 4, pp. 2735-2751, 2022]. In this paper, we give a characterization
+for their construction from another perspective. This characterization enables us to obtain
+another efficient construction of bent functions. Based on that, we derive several infinite
+families of concrete bent functions and give their duals. Our results show that many known
+bent functions are particular cases of our bent functions.
+Index Terms: Bent function, duals, cryptography, Walsh-Hadamard transform, Gold function.
+Mathematics Subject Classification 2020: 06E30, 94A60, 94D10.
+1
+Introduction
+Bent functions, introduced in [19], are those Boolean functions in an even number of variables
+having the highest nonlinearity. Such functions have been extensively studied in the last four
+decades, because of their closely relationship with the theory of difference sets, and their sig-
+nificant applications in coding theory and cryptography [4]. Bent functions are not balanced,
+however, they often act as an important and efficient ingredient for finding some balanced func-
+tions with a higher nonlinearity. For instance, the authors of [25] used bent functions to construct
+some disjoint spectra plateaued functions with higher nonlinearities. Their results provided a
+positive answer to an open problem of [26]. The authors of [21] utilized bent functions to con-
+struct balanced Boolean functions with high nonlinearity and low absolute indicator.
+Their
+results partially disproved a conjecture of [27]. In the past, a large amount of work was done
+on the characterizations and constructions of bent functions. But until now, a complete classi-
+fication is not finished and it remains elusive. Along with the deep-going of the research, the
+progress on bent functions becomes more and more difficult even if a tiny progress is not easy.
+For a comprehensive survey and a book on bent functions, the interested readers are referred to
+[5] and [15], respectively.
+In this paper, we focus our attention on the constructions of bent functions with the form
+h(x) = f(x) + F ◦ φ(x),
+(1)
+where f is a bent function on F2n, F is a Boolean function on Fr
+2, and φ = (φ1, φ2, . . . , φr) is an
+(n, r)-function. In fact, the research on the bent-ness of h can be dated back to [3], where Carlet
+presented a sufficient condition for a particular case (called Carlet function) of h to be bent, that
+is, the case of h to be bent when r = 2, f = f1, φ1 = f1 + f2, φ2 = f1 + f3 and F(x1, x2) = x1x2
+in (1). That sufficient condition had been proved by Mesnager [13] to be necessary. Mesnager
+∗Corresponding author
+Y. Li is with Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics,
+Bengbu, Anhui 233030, China (Email: [email protected]).
+H. Kan is with Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fu-
+dan University, Shanghai 200433, China; Shanghai Engineering Research Center of Blockchain, Shanghai 200433,
+China; and Yiwu Research Institute of Fudan University, Yiwu City 322000, China (E-mail: [email protected]).
+J. Peng and C. Chen are with Mathematics and Science College of Shanghai Normal University, Guilin Road
+#100, Shanghai 200234, China (Emails: [email protected] and [email protected]).
+L. Zheng is with the School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001,
+China (Email: [email protected]).
+1
+arXiv:2301.04456v1  [cs.IT]  11 Jan 2023
+
+[13] also studied the bent-ness of two particular cases of Carlet function. The first particular
+case is to let f2(x) = f1(x) + Trn
+1(ax) and f3(x) = f1(x) + Trn
+1(bx) for some a, b ∈ F∗
+2n; and the
+second particular case is to let f3(x) = f1(x) + Trn
+1(ax) for some a ∈ F∗
+2n in Carlet function,
+from which Mesnager obtained a lot of bent functions and gave their duals. Thereafter, several
+papers, such as [10, 20, 22, 23, 24, 28, 30], were done for generalizing Carlet and Mesnager’s
+works. The main results of those papers are listed in Table 1.
+Table 1: The bent functions of the form h(x) = f(x) + F ◦ φ(x)
+r
+φ = (φ1, φ2, . . . , φr)
+F
+Condition
+Ref.
+2
+φ1 = f + f2, φ2 = f + f3
+F(x1, x2) = x1x2
+see Theorem 1 of
+this paper
+[3]
+2
+φ1(x) = Trn
+1(ax),
+φ2(x) = Trn
+1(bx)
+F(x1, x2) = x1x2
+f is bent,
+DaDbf ∗ = 0
+[13]
+2
+φ1 = f + f2, φ2(x) = Trn
+1(ax)
+F(x1, x2) = x1x2
+f is bent,
+Da(f + f2)∗ = 0
+[13]
+3
+φi(x) = Trn
+1(µix)
+F(x1, x2, x3) = x1x2x3
+see [24]
+[24]
+r
+φi(x) = Trn
+1(µix)
+F(x1, x2, . . . , xr) = Πr
+i=1xi
+see Theorem 1 of
+[22]
+[22]
+r
+φi(x) = Trn
+1(µix)
+any Boolean function on Fr
+2
+see Theorem 2 of
+this paper
+[20]
+r
+φi(x) = Trn
+1(µix)
+any Boolean function on Fr
+2
+f is bent,
+DµiDµjf ∗ = 0 for
+any i ̸= j
+[28, 30]
+r
+φi = f + gi
+any Boolean function on Fr
+2
+see Theorem 4 of
+this paper
+[10]
+r
+φ
+any Boolean function on Fr
+2
+f satisfies Pr with φ
+Thm. 5
+r
+φ1 = f + g,
+φi(x) = Trn
+1(µix), 2 ≤ i ≤ r
+any Boolean function on Fr
+2
+see Corollary 5 of
+this paper
+Cor. 5
+r
+φ1(x) = f(x) + f(x + α),
+φi(x) = Trn
+1(µix), 2 ≤ i ≤ r
+any Boolean function on Fr
+2
+α ∈
+�
+µ2, . . . , µr
+�⊥,
+DµiDµjf ∗ = 0 for
+any i ̸= j
+Cor. 6
+In this paper, we continue to study the bent-ness of h defined in (1). By analysing carefully
+many previous results on the constructions of bent functions in the form (1), we obtain a
+framework on the constructions of bent functions, which unifies many previous constructions of
+bent functions in [3, 10, 13, 20, 22, 24, 28]. This framework also enables us to find another efficient
+construction of bent functions. Based on that, we obtain a number of concrete bent functions
+and determine their duals. Consequently, we find that our results cover many previously known
+bent functions.
+2
+Preliminaries
+Throughout the paper, let n = 2m be an even positive integer. Let F2n be the finite field of order
+2n, F∗
+2n = F2n\{0}, and Fn
+2 be the n-dimensional vector space over F2. There is a one-to-one
+correspondence between F2n and Fn
+2, because every a ∈ F2n can be represented uniquely by
+a = a1α1 + a2α2 + · · · + anαn, where ai ∈ F2, {α1, α2, . . . , αn} is a basis of F2n over F2. So the
+finite field F2n is always identified to the n-dimensional vector space Fn
+2 in this paper.
+For a vector ω = (ω1, ω2, . . . , ωn) ∈ Fn
+2, the set suppt(ω) = {1 ≤ i ≤ n : ωi ̸= 0} is said to
+be the support of ω, whose cardinality is called the (Hamming) weight of ω, denoted by wt(ω).
+Namely, we have wt(ω) = |suppt(ω)|.
+A mapping φ from Fn
+2 to Fr
+2 is called an (n, r)-function.
+When n is divisible by r, the
+(n, r)-function
+Trn
+r (x) = x + x2r + x22r + · · · + x2n−r
+is called the trace function. The set of all (n, 1)-functions (namely, all Boolean functions) is
+denoted by Bn.
+2
+
+For a given Boolean function f on Fn
+2, the Walsh-Hadamard transform of f is a mapping
+from Fn
+2 to Z defined as
+Wf(µ) =
+�
+x∈Fn
+2
+(−1)f(x)+µ·x,
+µ ∈ Fn
+2,
+and its inverse transform is given by
+(−1)f(µ) = 2−n �
+x∈Fn
+2
+Wf(x)(−1)µ·x,
+µ ∈ Fn
+2,
+where µ · x denotes the canonical inner product of µ and x (in F2n, µ · x = Trn
+1(µx)). The first
+derivative of f in terms of µ ∈ Fn
+2 is defined as
+Dµf(x) = f(x) + f(x + µ).
+Definition 1. A Boolean function f over Fn
+2 is called bent if n is even and Wf(µ) = ±2
+n
+2 for
+all µ ∈ Fn
+2.
+Bent functions always appear in pairs, that is, for any bent function f on Fn
+2, there is always a
+unique bent function f∗ such that Wf(µ) = 2
+n
+2 (−1)f∗(µ) for all µ ∈ Fn
+2. Hence, in the literature,
+f∗ is called the dual of f.
+Two Boolean functions f and g are called EA-equivalent if there is an affine automorphism
+A and affine function ℓ such that f(x) = g ◦ A(x) + ℓ(x). The set of all functions which are
+EA-equivalent to f is called a complete class of f. To see whether a bent function is inside a
+complete class of another bent function is challenging in general.
+3
+Some known results and a framework of bent functions
+3.1
+Known results
+In this subsection, we review some known bent functions of the form
+h(x) = f(x) + F ◦ φ(x),
+(2)
+where f is a bent function on Fn
+2, φ = (φ1, φ2, . . . , φr) is an (n, r)-function and F is a Boolean
+function on Fr
+2. Firstly, when r = 2, f = f1, φ1 = f1 + f2, φ2 = f1 + f3 and F(x1, x2) = x1x2,
+Carlet gave the following result.
+Theorem 1. [3] Let f1, f2, f3 be three bent functions on F2n such that f1 + f2 + f3 is bent as
+well, and (f1 + f2 + f3)∗ = f∗
+1 + f∗
+2 + f∗
+3 . Then
+h(x) = f1(x)f2(x) + f1(x)f3(x) + f2(x)f3(x)
+is a bent function with dual
+h∗(x) = f∗
+1 (x)f∗
+2 (x) + f∗
+1 (x)f∗
+3 (x) + f∗
+2 (x)f∗
+3 (x).
+This theorem provides a general method to find new bent functions. By finding different
+bent functions f1, f2 and f3 satisfying the conditions of Theorem 1, several new bent functions
+have been constructed, see [7, 12, 13, 14, 16] for details.
+In 2014, Mesnager [13] revisited Theorem 1, and found that the conditions of Theorem 1 is
+also necessary. Then letting f1(x) = f(x), f2(x) = f(x) + Trn
+1(ax) and f3(x) = f(x) + Trn
+1(bx)
+for some distinct a, b ∈ F∗
+2n in Theorem 1, she obtained the following result.
+Corollary 1. [13] Let f be a bent function on F2n. Let a, b ∈ F∗
+2n. Then
+h(x) = f(x) + Trn
+1(ax)Trn
+1(bx)
+(3)
+is bent if and only if DaDbf∗ = 0. Moreover, the dual of h is given by h∗(x) = f∗(x)f∗(x + a) +
+f∗(x)f∗(x + b) + f∗(x + a)f∗(x + b).
+3
+
+Letting f3(x) = f1(x) + Trn
+1(ax) for some a ∈ F∗
+2n in Theorem 1, Mesnager also obtained the
+following result.
+Corollary 2. [13] Let a ∈ F∗
+2n. Let f1, f2 be two bent functions on F2n. Then
+h(x) = f1(x) + Trn
+1(ax)(f1(x) + f2(x))
+is bent if and only if Da(f��
+1 + f∗
+2 ) = 0. Moreover, the dual of h is that h∗(x) = f∗
+1 (x) + (f∗
+1 (x) +
+f∗
+2 (x))f∗
+1 (x + a).
+Using Corollaries 1 and 2, Mesnager found several infinite families of bent functions and
+derived their duals.
+After Mesnager’s work, many papers were dedicated to generalizing Corollary 1 for finding
+new infinite families of bent functions. For instance, the authors of [24] generalized the function
+h of Corollary 1 to the form
+h(x) = f(x) + Trn
+1(ax)Trn
+1(bx)Trn
+1(cx),
+(4)
+and studied its bent-ness, where f is a bent function on F2n and a, b, c ∈ F∗
+2n satisfy certain
+conditions. The authors of [22] generalized the function h of Corollary 1 to the form
+h(x) = f(x) +
+r
+�
+i=1
+Trn
+1(µix),
+(5)
+and studied its bent-ness, where f is a bent function on F2n and µ1, µ2, . . . , µr ∈ F∗
+2n satisfy
+certain conditions, see [22, Theorem 1].
+The authors of [20] generalized the function h of
+Corollary 1 to its extreme form. Their result is given as follows.
+Theorem 2. [20] Let f be a bent function over F2n. If there exist r elements µ1, µ2, . . . , µr in
+F∗
+2n and r Boolean functions g1, g2, . . . , gr on F2n such that
+f∗
+�
+x+
+r
+�
+i=1
+µiωi
+�
+=f∗(x)+
+r
+�
+i=1
+ωigi(x)
+for all x ∈ F2n and all (ω1, . . . , ωr) ∈ Fr
+2, then for any F ∈ Br, the Boolean function
+h(x) = f(x) + F
+�
+Trn
+1(µ1x), Trn
+1(µ2x), . . . , Trn
+1(µrx)
+�
+(6)
+is bent with dual
+h∗(x) = f∗(x) + F
+�
+g1(x), g2(x), . . . , gr(x)
+�
+.
+Obviously, the functions h given in (3), (4), (5) and (6) are special cases of the form (2).
+But the conditions of h (in (3), (4), (5) and (6)) to be bent become more and more complicated.
+Therefore, a nature question is to ask that whether there is an unified simple condition such that
+the function h in (2) is bent. To this end, the authors of [28] simplified Theorem 2 as follows.
+Theorem 3. [28] Let f be a bent function on F2n.
+Let µ1, µ2, . . . , µr ∈ F∗
+2n be such that
+DµiDµjf∗ = 0 for any 1 ≤ i < j ≤ r. Then for any F ∈ Br, the function h given by (6) is bent,
+whose dual is that
+h∗(x) = f∗(x) + F(ϕ1(x), ϕ2(x), . . . , ϕr(x)),
+where ϕi(x) = f∗(x) + f∗(x + µi) for each i ∈ {1, 2, . . . , r}.
+Theorem 3 is clearly more concise than Theorem 2. But it does not contain Theorem 1 and
+Corollary 2. In order to find a more general uniform, the authors of [10] presented the following
+result.
+4
+
+Theorem 4. [10, Theorem 3] For any 1 ≤ i ≤ r, let f, gi ∈ Bn, and let φ = (φ1, φ2, . . . , φr) be
+the (n, r)-function with φi = f + gi. If the sum of any odd number of functions in f, g1, . . . , gr
+is a bent function, and its dual is equal to the sum of the duals of corresponding bent functions.
+Then for any Boolean function F on Fr
+2, the function h given by (2) is bent. Moreover, the dual
+of h is given by
+h∗(x) = f∗(x) + F ◦ ϕ(x),
+where ϕ = (ϕ1, ϕ2, . . . , ϕr) is the (n, r)-function with ϕi(x) = f∗(x) + g∗
+i (x) for any 1 ≤ i ≤ r.
+Theorem 4 reduces to Theorem 1 when r = 2 and F(x1, x2) = x1x2; and reduces to Theorems
+2 and 3 when gi(x) = f(x) + Trn
+1(µix) for each 1 ≤ i ≤ r. So in this sense, Theorem 4 is very
+general, and it seems difficult to be generalized any more.
+3.2
+A framework of bent functions
+In this subsection, we try to generalize Theorem 4. By analysing carefully the conditions of h to
+be bent in Theorems 1, 2, 3, and 4, respectively, we find that all conditions can be summarized
+by the following property.
+Definition 2 (Pr). Let f be a Boolean function over F2n. If there is an (n, r)-function φ =
+(φ1, φ2, . . . , φr) such that the following two conditions are satisfied:
+(i) f(x) + ω · φ(x) = f(x) + �r
+i=1 ωiφi is bent for any ω = (ω1, ω2, . . . , ωr) ∈ Fr
+2;
+(ii) there is an (n, r)-function ϕ = (ϕ1, ϕ2, . . . , ϕr) such that
+�
+f(x)+ω·φ(x)
+�∗ = f∗(x)+ω·ϕ(x)
+for any ω ∈ Fr
+2,
+then we say that f satisfies Pr with respect to the (n, r)-function φ.
+According to this property, we give the following framework of bent functions, which is main
+result of this subsection.
+Theorem 5. Let n = 2m. Let φ be an (n, r)-function, and let f be a Boolean function on F2n
+satisfying Pr with respect to φ. Then for any Boolean function F on Fr
+2, the function h given
+by (2) is bent, and the dual of h is
+h∗(x) = f∗(x) + F ◦ ϕ(x).
+Proof. By the definition of the inverse Walsh-Hadamard transform, it holds that
+(−1)F◦φ(x) = 2−r �
+ω∈Fr
+2
+WF (ω)(−1)ω·φ(x),
+∀ x ∈ Fn
+2.
+Hence, the Walsh-Hadamard transform of h at β ∈ F2n is that
+Wh(β) =
+�
+x∈F2n
+(−1)f(x)+Trn
+1 (βx)(−1)F◦φ(x)
+=2−r �
+x∈F2n
+(−1)f(x)+Trn
+1 (βx) �
+ω∈Fr
+2
+WF (ω)(−1)ω·φ(x)
+=2−r �
+ω∈Fr
+2
+WF (ω)
+�
+x∈F2n
+(−1)f(x)+Trn
+1 (βx)+ω·φ(x)
+=2−r �
+ω∈Fr
+2
+WF (ω)Wg(β),
+where g(x) = f(x) + ω · φ(x). Recall that f satisfies Pr with respect to φ, that is, g is bent and
+g∗(x) = f∗(x) + ω · ϕ(x) for any ω ∈ Fr
+2. Hence, we have
+Wh(β) = 2m−r �
+ω∈Fr
+2
+WF (ω)(−1)f∗(β)+ω·ϕ(β) = 2m(−1)f∗(β)+F◦ϕ(β).
+The proof is completed.
+5
+
+According to Theorem 5, we can deduce the following corollaries.
+Corollary 3. Theorem 5 reduces to that of Theorem 3 when φ = (φ1, φ2, . . . , φr) is an (n, r)-
+function with φi(x) = Trn
+1(µix), where µi ∈ F∗
+2n for each 1 ≤ i ≤ r.
+Proof. To prove this result, by Theorem 5, it suffices to show that f satisfies Pr with respect to
+φ if and only if f is bent and DµiDµjf∗ = 0 for any 1 ≤ i < j ≤ r. In fact, this fact has been
+presented in [28, Lemma 3.3]. Here we provide a sketchy proof for the readers convenience. Note
+that Item (i) of Pr is satisfied if and only if f is bent when φi(x) = Trn
+1(µix) for each 1 ≤ i ≤ r.
+Now assume that Item (ii) of Pr is satisfied, then it is easily seen that ϕi(x) = f∗(x)+f∗(x+µi)
+for each 1 ≤ i ≤ r when wt(ω) = 1, and DµiDµjf∗ = 0 for any 1 ≤ i < j ≤ r when wt(ω) = 2.
+Conversely, by induction on wt(ω), one can check easily that Item (ii) of Pr is also satisfied.
+Corollary 4. Theorem 5 reduces to that of Theorem 4 when φ = (φ1, φ2, . . . , φr) is an (n, r)-
+function with φi = f + gi, where f and gi are any Boolean functions on F2n for 1 ≤ i ≤ r.
+Proof. Suppose that φ = (φ1, φ2, . . . , φr) with φi = f + gi for any 1 ≤ i ≤ r. Then for any
+ω = (ω1, ω2, . . . , ωr) ∈ Fr
+2, we have
+f(x) + ω · φ(x) = f(x) +
+r
+�
+i=1
+ωi(f(x) + gi(x)) =
+�
+Gω(x),
+if wt(ω) is odd,
+f(x) + Gω(x),
+if wt(ω) is even,
+where Gω(x) = ω1g1(x)+ω2g2(x)+· · ·+ωrgr(x). Therefore, Item (i) of Pr holds if and only if the
+sum of any odd number of functions in f, g1, g2, . . . , gr is bent. Note that when suppt(ω) = {i},
+f(x) + ω · φ(x) = gi(x) and f∗(x) + ω · ϕ(x) = f∗(x) + ϕi(x).
+So Item (ii) of Pr holds only if ϕi(x) = f∗(x) + g∗
+i (x) for any 1 ≤ i ≤ r. In this case,
+f∗(x) + ω · ϕ(x) = f∗(x) +
+r
+�
+i=1
+ωi(f∗(x) + g∗
+i (x)) =
+�
+G∗
+ω(x),
+if wt(ω) is odd,
+f∗(x) + G∗
+ω(x),
+if wt(ω) is even,
+where G∗
+ω(x) = ω1g∗
+1(x) + ω2g∗
+2(x) + · · · + ωrg∗
+r(x). Hence, Item (ii) of Pr holds if and only if
+(Gω)∗ = G∗
+ω when wt(ω) is odd, and (f + Gω)∗ = f∗ + G∗
+ω when wt(ω) is even. Equivalently,
+the dual of the sum of any odd number of functions in f, g1, g2, . . . , gr is equal to the sum of the
+duals of corresponding bent functions. This completes the proof.
+From the proof of Corollary 4, it is easily seen that for a given Boolean function f on F2n,
+and an (n, r)-function φ = (φ1, φ2, . . . , φr), Pr holds if and only if the sum of any odd number
+of functions in f, f + φ1, f + φ2, . . . , f + φr is bent, and its dual is equal to the sum of the duals
+of corresponding bent functions. Namely, Theorem 4 is the same as Theorem 5. So in this
+sense, Theorem 4 indeed cannot be generalized any more. Note that Theorem 4 was proved by
+induction in [10]. Here we provide a more simple alternative proof from another perspective.
+Theorem 5 also allows us to deduce the following result.
+Corollary 5. Let n = 2m. Let f and g be two bent functions on F2n. Let µ2, µ3, . . . , µr ∈ F∗
+2n.
+If the following two conditions are satisfied:
+(A) DµiDµjf∗ = 0 for any 2 ≤ i < j ≤ r;
+(B) for any ω′ = (ω2, ω3, . . . , ωr) ∈ Fr−1
+2
+, it holds that
+g∗(x +
+r
+�
+i=2
+ωiµi) =
+�
+g∗(x) + f∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is odd,
+g∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is even,
+(7)
+6
+
+then for any Boolean function F on Fr
+2, the function h given by
+h(x) = f(x) + F(f(x) + g(x), Trn
+1(µ2x), Trn
+1(µ3x), . . . , Trn
+1(µrx))
+is bent. Moreover, the dual of h is
+h∗(x) = f∗(x) + F(ϕ1, ϕ2, . . . , ϕr),
+where ϕ1(x) = f∗(x) + g∗(x) and ϕi(x) = f∗(x) + f∗(x + µi) for any 2 ≤ i ≤ r.
+Proof. Let φ = (φ1, φ2, . . . , φr) be the (n, r)-function with φ1(x) = f(x) + g(x) and φi(x) =
+Trn
+1(µix) for each 2 ≤ i ≤ r. Then for any ω = (ω1, ω2, . . . , ωr) ∈ Fr
+2, it is easily seen that
+f(x) + ω · φ(x) =
+�
+f(x) + Trn
+1((ω2µ2 + ω3µ3 + · · · + ωrµr)x),
+if ω1 = 0,
+g(x) + Trn
+1((ω2µ2 + ω3µ3 + · · · + ωrµr)x),
+if ω1 = 1.
+This implies that Item (i) of Pr is satisfied when f and g are bent. So we have that
+�
+f(x) + ω · φ(x)
+�∗ =
+�
+f∗(x + ω2µ2 + ω3µ3 + · · · + ωrµr),
+if ω1 = 0,
+g∗(x + ω2µ2 + ω3µ3 + · · · + ωrµr),
+if ω1 = 1.
+Note that when suppt(ω) = {i}, we have
+f(x) + ω · φ(x) =
+�
+g(x),
+if i = 1,
+f(x) + Trn
+1(µix)
+otherwise,
+and f∗(x) + ω · ϕ(x) = f∗(x) + ϕi(x).
+So Item (ii) of Pr holds only if ϕ1(x) = f∗(x) + g∗(x) and ϕi(x) = f∗(x) + f∗(x + µi) for any
+2 ≤ i ≤ r. In this case,
+f∗(x) + ω · ϕ(x) =
+�
+f∗(x) + �r
+i=2 ωi(f∗(x) + f∗(x + µi)),
+if ω1 = 0,
+g∗(x) + �r
+i=2 ωi(f∗(x) + f∗(x + µi)),
+if ω1 = 1.
+Hence, Item (ii) of Pr holds if and only if the following two relations hold:
+f∗(x + ω2µ2 + · · · + ωrµr) = f∗(x) +
+r
+�
+i=2
+ωi(f∗(x) + f∗(x + µi))
+=
+�
+f∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is even,
+�r
+i=2 ωif∗(x + µi),
+if wt(ω′) is odd,
+(8)
+and
+g∗(x + ω2µ2 + · · · + ωrµr) = g∗(x) +
+r
+�
+i=2
+ωi(f∗(x) + f∗(x + µi))
+=
+�
+g∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is even,
+g∗(x) + f∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is odd,
+(9)
+where ω′ = (ω2, ω3, . . . , ωr). By Corollary 3, we know that Relation (8) holds if and only if
+DµiDµjf∗ = 0 for any 2 ≤ i < j ≤ r. Then the result follows from Theorem 5 immediately.
+Remark 1. In Corollary 5, let φi(x) = f(x) + g(x) + Trn
+1(µix) for some 1 ≤ i ≤ r, where
+µi ∈ F2n. Then one can obtain a similar result as that of Corollary 5.
+Remark 2. Corollary 5 is a generalization of Corollary 2, since Corollary 5 reduces to that of
+Corollary 2 when r = 2 and F(x1, x2) = x1x2.
+Note that Condition (B) of Corollary 5 is elusive when r > 2. In the following corollary, we
+give a reduced form by applying Corollary 5 to g(x) = f(x + α) for some α ∈ F∗
+2n.
+7
+
+Corollary 6. Let f be a bent function on F2n. Let α ∈ F2n and µ2, µ3, . . . , µr ∈ F∗
+2n be such
+that α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and DµiDµjf∗ = 0 for any 2 ≤ i < j ≤ r. Then for any Boolean
+function F on Fr
+2, the function
+h(x) = f(x) + F(f(x) + f(x + α), Trn
+1(µ2x), Trn
+1(µ3x), . . . , Trn
+1(µrx))
+is bent. Moreover, the dual of h is
+h∗(x) = f∗(x) + F(ϕ1, ϕ2, . . . , ϕr),
+where ϕ1(x) = Trn
+1(αx) and ϕi(x) = f∗(x) + f∗(x + µi) for any 2 ≤ i ≤ r.
+Proof. Let g(x) = f(x+α). Then it easily seen that g∗(x) = f∗(x)+Trn
+1(αx), and then Relation
+(7) becomes that
+f∗(x +
+r
+�
+i=2
+ωiµi) =
+��r
+i=2 ωif∗(x + µi),
+if wt(ω′) is odd,
+f∗(x) + �r
+i=2 ωif∗(x + µi),
+if wt(ω′) is even,
+since α ∈
+�
+µ2, µ3, . . . , µr
+�⊥.
+Hence, Condition (B) of Corollary 5 is satisfied if and only if
+DµiDµjf∗ = 0 for any 2 ≤ i < j ≤ r by Corollary 3, and the result follows from Corollary 5
+directly.
+Remark 3. Note that though the conditions of h to be bent in Corollary 6 are similar as that
+of Theorem 3 (in fact, Corollary 6 reduces to Theorem 3 when α = 0), the corresponding bent
+functions in Corollary 6 and Theorem 3 can be EA-inequivalent. For instance, let n = 6 and
+f(x) = (x1, x2, x3) · (x4, x5, x6).
+Let µ2 = (1, 0, 0, 0, 0, 0), µ3 = (0, 1, 1, 0, 0, 0). Then it is easy to check that Dµ2Dµ3f∗ = 0.
+Hence, by Theorem 3, we have that
+h(x) = f(x) + F(µ2 · x, µ3 · x) = f(x) + F(x1, x2 + x3)
+is bent for any Boolean function F on F2
+2; and by Corollary 6, we have that
+ˆh(x) = f(x) + ˆF
+�
+f(x) + f(x + α), µ2 · x, µ3 · x
+�
+= f(x) + ˆF
+�
+f(x) + f(x + α), x1, x2 + x3
+�
+is bent for any α ∈
+�
+µ2, µ3
+�⊥ and any Boolean function ˆF on F3
+2. These two bent functions can
+be clearly EA-inequivalent, since the algebraic degree of h is 2, while the algebraic degree of ˆh is
+3 when α = µ3 and ˆF(x1, x2, x3) = x1x2x3.
+Corollary 6 is efficient in producing new bent functions, since it is only required to find
+some α ∈ F2n and µ2, µ3, . . . , µr ∈ F∗
+2n such that DµiDµjf∗ = 0 for any 2 ≤ i < j ≤ r and
+α ∈
+�
+µ2, µ3, . . . , µr
+�⊥. In the next section, we will use Corollary 6 to construct a number of
+concrete bent functions and compute their duals.
+4
+Several concrete bent functions and their duals
+The authors of [10] have found two kinds of f and φ satisfying the conditions of Theorem 4
+(that is, Pr by the previous section) for constructing new bent functions. The first kind is to
+let f be a bent function and φ be a linear (n, r)-function; and the second kind is to let f and
+f + φi be some self-dual bent functions for each 1 ≤ i ≤ r. They also invited the readers to find
+more kinds of f and φ for obtaining more classes of bent functions in Conclusion of [10]. In the
+previous section, we have shown that Theorem 5 is the same as that of Theorem 4. In addition,
+we have found another new kind of f and φ satisfying Pr by Theorem 5 (see Corollary 6). In
+this section, we find a number of concrete bent functions by using Corollary 6.
+8
+
+4.1
+New bent functions from Gold functions
+In this subsection, we construct some concrete bent functions by applying Corollary 6 to Gold
+function g(x) = Trn
+1(λx2t+1), where t is a positive integer and λ ∈ F∗
+2n. We first recall the
+following result.
+Lemma 1. [6][8] Let n = 2m and d = gcd(t, n). Let g(x) = Trn
+1(λx2t+1) for some λ ∈ F∗
+2n.
+Then g is bent on F2n if and only if n
+d is even and λ /∈ S, where S = {x2t+1 : x ∈ F2n}.
+Moreover, the dual of g is that g∗(x) = Trn
+1(λx2t+1
+0
+) + ( m
+d mod 2), where x0 ∈ F2n satisfies that
+λx0 + λ2tx22i
+0
+= x2t.
+(10)
+Note that g is explicit, but g∗ is not explicit in Lemma 1. To find some bent functions
+by Corollary 6, we need to determine µ2, µ3, . . . , µr ∈ F∗
+2n such that DµiDµjf∗ = 0 for any
+2 ≤ i < j ≤ r. So we take f = g∗ and present the following theorem, which is the main result
+of this subsection.
+Theorem 6. Take the same notations as in Lemma 1. Let λ ∈ F2n\S and µ2, µ3, . . . , µr ∈ F∗
+2n
+be such that Trn
+1(λ(µ2t
+i µj+µiµ2t
+j )) = 0 for any 2 ≤ i < j ≤ r. Then for any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥
+and any Boolean function F on Fr
+2, the function h∗ given by
+h∗(x) = Trn
+1(λx2t+1) + F(Trn
+1(αx), ϕ2(x), ϕ3(x), . . . , ϕr(x)),
+(11)
+is bent, where ϕi(x) = Trn
+1
+�
+λ(µix2t + µ2t
+i x + µ2t+1
+i
+)
+�
+for each 2 ≤ i ≤ r.
+Proof. Let f(x) = g∗(x) = Trn
+1(λx2t+1
+0
+) + ( m
+d mod 2), where x0 ∈ F2n satisfies (10). Then from
+Lemma 1, we know that f is bent and its dual is f∗(x) = g(x) = Trn
+1(λx2t+1). Hence, we have
+f∗(x) + f∗(x + µi) = Trn
+1
+�
+λ(x2t+1 + (x + µi)2t+1)
+�
+= Trn
+1
+�
+λ(µix2t + µ2t
+i x + µ2t+1
+i
+)
+�
+, ∀ µi ∈ F2n,
+and
+DµiDµjf∗(x) =Trn
+1
+�
+λ(µix2t + µ2t
+i x + µ2t+1
+i
+)
+�
++ Trn
+1
+�
+λ(µi(x + µj)2t + µ2t
+i (x + µj) + µ2t+1
+i
+)
+�
+=Trn
+1
+�
+λ(µiµ2t
+j + µ2t
+i µj)
+�
+,
+∀ µi, µj ∈ F2n.
+This means that DµiDµjf∗ = 0 if Trn
+1
+�
+λ(µiµ2t
+j + µ2t
+i µj)
+�
+= 0. Then by Corollary 6, we obtain
+that
+h(x) = f(x) + F(f(x) + f(x + α), Trn
+1(µ2x), Trn
+1(µ3x), . . . , Trn
+1(µrx))
+(12)
+is bent for any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and any Boolean function F on Fr
+2, and the dual of h is
+exactly that of (11). This completes the proof.
+Remark 4. When α = 0, Theorem 6 is exactly that of [28, Theorem 4.1].
+Applying Theorem 6 to t = m, we can deduce the following corollary.
+Corollary 7. Let n = 2m. Let θ ∈ F∗
+2m and µ2, µ3, . . . , µr ∈ F∗
+2n be such that Trn
+1(θ−1µiµ2m
+j ) = 0
+for any 2 ≤ i < j ≤ r. Then for any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and any F ∈ Br, the function
+h(x) = Trm
+1 (θx2m+1) + F
+�
+Trn
+1(θα2mx) + Trm
+1 (θα2m+1), Trn
+1(µ2x), . . . , Trn
+1(µrx)
+�
++ 1
+is bent, whose dual is that
+h∗(x) = Trm
+1 (θ−1x2m+1) + F(Trn
+1(αx), ϕ2(x), ϕ3(x), . . . , ϕr(x)),
+where ϕi(x) = Trn
+1(θ−1µ2m
+i
+x) + Trm
+1 (θ−1µ2m+1
+i
+) for each 2 ≤ i ≤ r.
+9
+
+Proof. Let t = m, λ ∈ F2n\F2m and θ−1 = λ+λ2m. Then it is easily checked that Trn
+1(λ(µ2t
+i µj +
+µiµ2t
+j )) = Trn
+1(θ−1µiµ2m
+j ). Let x0 = θx2m = (λ+λ2m)−1x2m, that is, x0 satisfies (10). Then from
+Lemma 1, we obtain that f(x) = Trn
+1(λx2m+1
+0
+) + 1 = Trm
+1 (θx2m+1) + 1 is bent (since S = F2m
+when t = m), and the dual of f is that f∗(x) = Trn
+1(λx2m+1) = Trm
+1 (θ−1x2m+1). The result
+follows then from Theorem 6 and the calculations for (11) and (12).
+Remark 5. When α = 0, Corollary 7 reduces to Theorem 12 of [20], which contains Theorem
+2 of [22] (where F(x1, x2, . . . , xr) = x1x2 · · · xr), the part of bent functions in Theorem 1 of
+[24] (where r = 3 and F(x1, x2, x3) = x1x2x3), and Theorem 9 of [13] (where r = 2 and
+F(x1, x2) = x1x2) as special cases.
+When n = 2m = 4t, the authors of [10] have given the explicit form of g∗(x) = Trn
+1(λx2t+1
+0
+)+
+( m
+d mod 2) by solving (10), see [10, Lemma 3], which is g∗(x) = Trn
+1(P(λ)x2t+1), where P(λ) =
+λ2m+1+1+λ2t+2m+23t
+Trm
+t (Nn
+m(λ2))
+and Nn
+m(λ) = λ2m+1. They have also pointed out in Remark 16 of [10] that
+g∗ is self-dual if λ ∈ F2m with λ + λ2t = 1. This result enables us to give the following corollary.
+Corollary 8. Let n = 2m = 4t.
+Let λ ∈ F2n\S and µ2, µ3, . . . , µr ∈ F∗
+2n be such that
+Trn
+1(λ(µ2t
+i µj + µiµ2t
+j )) = 0 for any 2 ≤ i < j ≤ r, where S = {x2t+1 : x ∈ F2n}. Then for
+any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and any F ∈ Br, the function
+h(x) = Trn
+1(P(λ)x2t+1) + F
+�
+Trn
+1(P(λ)(α2tx + αx2t + α2t+1)), Trn
+1(µ2x), . . . , Trn
+1(µrx)
+�
+(13)
+is bent, whose dual is that
+h∗(x) = Trn
+1(λx2t+1) + F(Trn
+1(αx), ϕ2(x), ϕ3(x), . . . , ϕr(x)),
+where ϕi(x) = Trn
+1
+�
+λ(µix2t + µ2t
+i x + µ2t+1
+i
+)
+�
+for each 2 ≤ i ≤ r. In particular, for any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and any Boolean function F on Fr
+2, h is bent if λ ∈ F2m with λ + λ2t = 1.
+Proof. Let f(x) = Trn
+1(P(λ)x2t+1). Then for any α ∈ F2n, it is easily seen that f(x)+f(x+α) =
+Trn
+1
+�
+P(λ)(αx2t +α2tx+α2t+1)
+�
+, and hence (12) becomes (13). Then result follows from Theorem
+6 immediately.
+Remark 6. When α = 0 and λ ∈ F2m with λ + λ2t = 1 (i.e., P(λ) = λ), Corollary 8 reduces to
+Theorem 23 of [20], which contains Theorem 3 of [22] (where F(x1, x2, . . . , xr) = x1x2 · · · xr),
+and the part of bent functions in Theorems 3 and 4 of [24] (where r = 3 and F(x1, x2x3) =
+x1x2x3) as special cases.
+4.2
+New bent functions from a class of bent functions inside the completed
+Maiorana-MacFarland class
+The authors of [10] have shown that the following function
+f(x) = Trn
+1(λx2tπ(x + x2m)) + g(x + x2m)
+(14)
+is bent if and only if λ ∈ F2n\F2m, where t is a non-negative integer, n = 2m, π is a permutation of
+F2m, and g is a Boolean function on F2m. This bent function is inside the completed Maiorana-
+MacFarland class, and it is a generalization of [18, Theorem 4], [29, Theorem 4.6], and [17,
+Theorem 9]. In this subsection, we intend to find more bent functions by using this bent function
+and Corollary 6, for which we need first to determine the dual of f. We use the technique used
+in [10, Proposition 1] to complete this task.
+Lemma 2. The dual of the bent function f in (14) is that
+f∗(x) = Trn
+1
+�
+ωxπ−1(Λ−1(x + x2m)2t)
+�
++ G
+�
+π−1(Λ−1(x + x2m)2t)
+�
+,
+(15)
+where Λ = λ + λ2m and G(z) = Trn
+1
+�
+λ(ωz)2tπ(z)
+�
++ g(z).
+10
+
+Proof. Let ω ∈ F2n with ω + ω2m = 1. Then F2n can be decomposed as F2n = F2m + ωF2m,
+that is, for any x ∈ F2n, there are unique y, z ∈ F2m such that x = y + ωz. This expression also
+means that z = x + x2m and y = ω2mx + ωx2m. Then f can be represented by
+f(x) =f(y + ωz) = Trn
+1
+�
+λ(y + ωz)2tπ(z)
+�
++ g(z) = Trm
+1
+�
+Λy2tπ(z)
+�
++ G(z),
+where Λ = λ + λ2m and G(z) = Trn
+1
+�
+λ(ωz)2tπ(z)
+�
++ g(z). Then for any θ = a + ωb, where
+a, b ∈ F2m, we have
+Wf(θ) =
+�
+x∈F2n
+(−1)f(x)+Trn
+1 (θx)
+=
+�
+y,z∈F2m
+(−1)f(y+ωz)+Trn
+1 ((a+ωb)(y+ωz))
+=
+�
+z∈F2m
+(−1)G(z)+Trm
+1
+�
+(a+b)z
+� �
+y∈F2m
+(−1)Trm
+1
+�
+(Λπ(z)+b2t)y2t�
+.
+This implies that f is bent if and only if |{z ∈ F2m : Λπ(z) + b2t = 0}| = 1 for any b ∈ F2m.
+Recall that π is a permutation of F2m. Thus, f is bent if and only if Λ = λ + λ2m ̸= 0, that is,
+λ /∈ F2m. In this case, z = π−1(Λ−1b2t), and
+Wf(θ) = Wf(a + bω) = 2m(−1)G(z)+Trm
+1
+�
+(a+b)z
+�
+.
+This implies that
+f∗(a + bω) = G(z) + Trm
+1
+�
+(a + b)z
+�
+= G
+�
+π−1(Λ−1b2t)
+�
++ Trm
+1
+�
+(a + b)π−1(Λ−1b2t)
+�
+.
+Hence, the dual of f satisfies that
+f∗(x) = f∗(y + zω) = G
+�
+π−1(Λ−1z2t)
+�
++ Trm
+1
+�
+(y + z)π−1(Λ−1z2t)
+�
+.
+Recall that y = ω2mx + ωx2m and z = x + x2m. Then we have
+f∗(x) =G
+�
+π−1(Λ−1(x + x2m)2t)
+�
++ Trm
+1
+�
+(ωx + (ωx)2m)π−1(Λ−1(x + x2m)2t)
+�
+=G
+�
+π−1(Λ−1(x + x2m)2t)
+�
++ Trn
+1
+�
+ωxπ−1(Λ−1(x + x2m)2t)
+�
+.
+This completes the proof.
+From Lemma 2 and Corollary 6, we can deduce the following result.
+Theorem 7. Take the same notations as in Lemma 2. Let f be the bent function given in (14).
+Then for any µ2, µ3, . . . , µr ∈ F∗
+2m, any α ∈
+�
+µ2, µ3, . . . , µr
+�⊥, and any Boolean function F on
+Fr
+2, the function
+h(x) = f(x) + F(f(x) + f(x + α), Trn
+1(µ2x), Trn
+1(µ3x), . . . , Trn
+1(µrx))
+is bent, and the dual of h is
+h∗(x) = f∗(x) + F(Trn
+1(αx), ϕ2(x), ϕ3(x), . . . , ϕr(x)),
+where f∗ is given by (15) and ϕi(x) = Trn
+1
+�
+ωµiπ−1(Λ−1(x + x2m)2t)
+�
+for each 2 ≤ i ≤ r.
+Proof. Let T(x) = x + x2m. Then for any µi, µj ∈ F∗
+2m, it is easily seen that T(x) = T(x + µi),
+which implies that f∗(x) + f∗(x + µi) = Trn
+1
+�
+ωµiπ−1(Λ−1(x + x2m)2t)
+�
+and DµiDµjf∗ = 0. The
+result follows then from Corollary 6 immediately.
+11
+
+Similarly as that of Theorems 6 and 7, by applying Corollary 6 to the following two monomial
+bent functions
+f1(x) = Tr6k
+1 (λx22k+2k+1) and f2(x) = Tr4k
+1 (λx22k+2k+1+1)
+given by [2] and [8], respectively; and to the following bent functions with Niho exponents
+f3(x) = Trm
+1 (x2m+1) + Trn
+1
+� 2k−1−1
+�
+i=1
+x(2m−1) i
+2k +1
+�
+given by [9], we can also obtain certain concrete bent functions, since the duals of f1, f2, f3
+have been determined in [10], [11] and [1], respectively, and hence by Corollary 6, we only need
+to find some elements α ∈ F2n and µ2, µ3, . . . , µr ∈ F∗
+2n such that α ∈
+�
+µ2, µ3, . . . , µr
+�⊥ and
+DµiDµjf∗
+e = 0 for any 2 ≤ i < j ≤ r and 1 ≤ e ≤ 3. Here, the concrete results are not unfolded
+in details.
+5
+Conclusion
+In this paper, we gave another characterization for the generic construction of bent functions
+given in [10], which enabled us to obtain another efficient construction of bent functions. Based
+on this construction, we found several infinite families of bent functions and confirmed their
+duals. Consequently, our results cover a lot of known bent functions. It remains to verify the
+EA-equivalence of the bent functions obtained in this paper to known families.
+Acknowledgments
+This work was supported in part by the National Key Research and Development Program
+of China under Grant 2019YFB2101703; in part by the National Natural Science Founda-
+tion of China under Grants 61972258, 62272107 and U19A2066; in part by the Innovation
+Action Plan of Shanghai Science and Technology under Grants 20511102200 and 21511102200;
+in part by the Key Research and Development Program of Guangdong Province under Grant
+2020B0101090001, in part by Scientific Research Fund of Hunan Provincial Education Depart-
+ment under Grant 19B485, and in part by Open Reseach Program of Shanghai Key Lab of
+Intelligent Information Processing under Grant IIPL201902.
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+functions with maximal number of bent components, Des. Codes Cryptogr. 88 (2020) 2171-
+2186. 10
+[30] L. Zheng, J. Peng, H. Kan, D. Tang, Constructing vectorial bent functions via second-order
+derivatives, Discret. Math. 344 (8) (2021) 112473. 2
+14
+

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+Wave correlations and quantum noise in cosmology
+Ulf Leonhardt
+Department of Physics of Complex Systems,
+Weizmann Institute of Science,
+Rehovot 7610001, Israel
+January 11, 2023
+Abstract
+Wave noise is correlated. While it may look random in space, correlations ap-
+pear in space–time, because the noise is carried by wave propagation. These corre-
+lations of wave noise give rise to fluctuation forces such as the Casimir force, they
+are responsible for the particle creation in the dynamical Casimir effect and in the
+expanding universe. This paper considers the noise correlations for light waves in
+non-exponentially expanding flat space. The paper determines the high-frequency
+asymptotics of the correlation spectrum in the conformal vacuum. These noise cor-
+relations give rise to a nontrivial vacuum energy that may appear as the cosmological
+constant.
+1
+arXiv:2301.03795v1  [gr-qc]  10 Jan 2023
+
+1
+Introduction
+Explorers have mapped every corner of the Earth, but the time of exploration has only just
+began: 95% of the current content of the universe is completely unknown. The uncharted
+95% are called the “dark sector” with 25% belonging to dark matter and 70% to dark
+energy [1]. While there are many ideas from particle physics on the nature of dark matter,
+and several experimental programmes for detecting dark–matter particles [2] dark energy
+has been an enigma [3, 4]. However, it might actually be the other way round: dark energy
+could be the easier problem to solve, but not as a problem of high–energy physics. Rather,
+it might belong to an area of low–energy physics, extrapolated to cosmological scales. In
+this paper I will follow up on the hypothesis [5, 6, 7] that dark energy, this arcane force
+that drives the universe apart, is a form of much more mundane forces, the van der Waals
+and Casimir forces, that cause ordinary things to stick. These are forces of the quantum
+vacuum [8, 9].
+This is not a new idea. In 1968 Zel’dovich [10] suggested that vacuum fluctuations
+create Einstein’s cosmological constant Λ [11]. Einstein’s Λ is what was later called
+dark energy [12]. However, Zel’dovich’s and similar suggestions [13] disagree with the
+measured value of Λ by some 120 orders of magnitude. The idea that Λ comes from
+the quantum vacuum is not new — and seem to have failed spectacularly. What is new
+is a better theory of the quantum vacuum, inspired by precision measurements and ma-
+nipulations of Casimir forces [14, 15, 16], by the analogy between dielectric media and
+space–time geometries [17, 18, 19, 20, 21, 22, 23, 24] tried and tested in transformation
+optics [23, 24, 25, 26] and in optical analogues of black holes [27, 28, 29, 30, 31, 32, 33],
+and inspired by the person to whom this volume is dedicated: Michael Berry. Not only
+did he encourage me to pursue unconventional ideas, these ideas resonate with his work
+on the infinite intricacies of light [34].
+The theory [5, 6, 7] is still mostly a hypothesis, but it appears to agree with astronom-
+ical data [7] and seems to resolve [7] a major inconsistency in the conventional interpreta-
+tion of that data [35]: the 5σ tension between the directly measured Hubble constant [36]
+and the Hubble constant inferred from the Cosmic Microwave Background [1]. There are
+some 102 theories to explain the Hubble tension [37]. All of them require modifications
+of known physics — changes to the standard model of particle physics, general relativ-
+ity or the cosmological principle; all make some experimentally untested modifications,
+with one exception. The theory advocated here is the only one in the field rooted on
+experiments and relying on “new things in old things” — to quote a phrase of Michael
+Berry.
+These results are encouraging, but much more work needs to be done to prove or
+disprove the theory on astronomical data [7], to test its physical mechanism in laboratory
+analogues [38] and also to improve the theory itself. Let me explain. The renormalized
+vacuum expectation value εvac of the electromagnetic energy density can be expressed
+such that [5]
+4πG
+3c2 εvac = −αΛ∆
+(1)
+in terms of the gravitational constant G, the speed of light in vacuum c and the dimen-
+sionless coupling parameter αΛ. The parameter αΛ depends on the inverse squared of
+the cutoff length ℓΛ with [5] αΛ = (9π)−1 if ℓΛ is the Planck length ℓp =
+�
+ℏG/c3 (ℏ
+2
+
+being the reduced Planck constant). The energy density εvac does two things: it gravitates
+and it generates a trace anomaly [5, 38, 39] with energy density εΛ that appears as the
+cosmological term Λ, but is no longer constant. The total vacuum energy εΛ + εvac grows
+with −4εvac times the Hubble parameter [5]. The cosmological term εΛ thus accumulates
+εvac during the cosmic evolution, it grows with negative εvac and falls with positive εvac.
+The cosmological constant still appears in the theory, yet not as a fundamental constant
+of nature but only as an integration constant [7] that depends on the initial conditions and
+presumably was zero at the beginning of time.
+The quantity ∆ in the vacuum energy density (1) carries the physical units of a fre-
+quency squared and depends on the nature of the quantum vacuum. In the first version [5]
+of the theory ∆ was found to be
+∆ = ∂3
+t
+1
+H + H∂2
+t
+1
+H
+(2)
+where H denotes the Hubble parameter [40]. One sees from a scale analysis that εvac
+carries the correct order of magnitude of the cosmological constant1. In the second incar-
+nation [7] of the theory2 the expression
+∆ = ∂3
+t
+1
+H
+(3)
+was published and used to compare theory with data [7] assuming εvac as a perturbation
+of the cosmic dynamics [7]. While Eqs. (2) and (3) agree on the leading term, they
+differ in the subdominant term. The data ruled out Eq. (2) whereas Eq. (3) agrees with
+the astronomical data with the precision of that data for exactly the Planck–scale value
+αΛ = (9π)−1. However, this is only true within first–order perturbation theory; the full
+solution of the cosmic dynamics contains oscillatory modulations, suggesting that some
+vital ingredient was missing that dampens these oscillations. In this paper I hope to have
+identified the missing component and to have finally deduced the correct vacuum energy.
+The paper also clarifies the role the quantum vacuum plays in cosmology and it offers
+an explanation why quantum electromagnetism, and quantum electromagnetism alone, is
+responsible for what appears as dark energy in the current era. The heart of the problem
+of explaining dark energy from vacuum fluctuations is the physics of wave noise.
+Wave noise is organized. In space, it may look completely random, but in space–
+time patterns of correlations are clearly visible (Fig. 1). There we see the characteristic
+diagonal features of wave propagation. Waves are traveling to the left or the right with the
+wave velocity c/n, and the noise they carry travels with them. If n varies the noise pattern
+varies as well. The most dramatic of such modifications are reflections, for example at
+obstacles where n is discontinuous. Reflected wave noise gives rise to fluctuation forces
+[8, 9] such as the Casimir forces [42]. If n varies in time, waves may be reflected in time
+as well [43, 44]. A reflection in space is the change of sign in the wave number, in time it
+is a sign change in frequency. In the dynamical Casimir effect [45, 46, 47, 48, 49] these
+negative–frequency components correspond to newly–created particles, simply because if
+1The argument [5] goes as follows. According to the Friedman equation [40, 41] expression (1) gives
+1
+2H2 for the realistic case of zero spatial curvature [1]. As H varies on the scale of H the energy density
+εvac goes like H2 and thus plays a role in the cosmic dynamics.
+2Actually, this was the result of my first, unpublished version of the theory.
+3
+
+Figure 1:
+Wave noise. Space–time diagram of waves with Gaussian noise. Although the wave
+field looks random in space {x} features appear in space–time {ct, x} following the causal cones
+of wave propagation (with speed c). For this picture 128 normalized left–moving and 128 right–
+moving plane waves [Eqs. (5) and (6)] with periodic boundary conditions and of random Gaussian
+complex coefficients were summed up. Increasing the number of waves produces finer and finer
+structures, but ultimately the noise field diverges, illustrating the divergence of the bare vacuum
+noise.
+part of a wave of positive frequency ω is converted to −ω the energy ℏω of the remaining
+positive–frequency component must grow, particles are created. Here we focus less on the
+particle aspects, but rather on the amplitude correlations of wave noise. We begin with a
+brief review on a familiar example, the noise seen by accelerated observers [50, 51, 52].
+Then we show how this is related to the noise perceived by an observer at rest in an
+exponentially expanding universe [53] before turning to the discussion of vacuum modes
+in a universe of arbitrary expansion [54]. We confirm the extension [54] of Gibbons’ and
+Hawking’s formula for the radiation temperature [53] and find a new feature not present
+in exponential expansion: the Hawking partners appear as red–shifted thermal radiation.
+The multiple interference of all Hawking processes in the expanding universe gives the
+effective vacuum energy; to calculate it we use the Wigner function of wave noise.
+2
+Uniform acceleration
+Wave noise is organized, because waves can be organized in terms of modes, and the
+noise appears solely in the amplitudes and phases of the mode coefficients. Consider a
+simple 1+1 dimensional example: a scalar wave field ˆA in empty Minkowski space given
+4
+
+by the mode decomposition
+�A =
+� +∞
+−∞
+�
+�akAk + �a†
+kA∗
+k
+�
+dk
+(4)
+where the Ak are the mode functions Ak(x, t) describing how the modes propagate in
+space x and time t. The �ak are the mode coefficients, and only they are subject to statistical
+or quantum ���uctuations. The mode functions should be normalized such that each mode
+accounts for the field of exactly one particle. This is conveniently done with the help of
+the scalar product [55]
+(A1, A2) = i
+ℏ
+� +∞
+−∞
+(A∗
+1 ∂tA2 − A2 ∂tA∗
+1) dx
+(5)
+requiring
+(A1, A2) = δ(k1 − k2) ,
+(A∗
+1, A2) = 0 .
+(6)
+For example, if the modes are plane waves Ak = A exp(ikx − iωt) with ω = c|k| we
+must require A2 = ℏ/(4πω). From the canonical commutation relations between field
+and momentum density then follow [55] — for Bosonic fields like the electromagnetic
+field — the standard Bose commutation relations:
+[�ak1,�a†
+k2] = δ(k1 − k2) ,
+[�ak1,�ak2] = 0 .
+(7)
+The Minkowski vacuum |0⟩ is the quantum state annihilated by all the plane–wave oper-
+ators:
+�ak|0⟩ = 0 .
+(8)
+The Minkowski vacuum is the vacuum with respect to an observer at rest in Minkowski
+space. It also appears as the vacuum to observers in uniform motion, because they per-
+ceive the modes Ak as plane waves as well, Doppler–shifted of course. But this is no
+longer true for accelerated observers [50, 51, 52].
+Uniform acceleration is described by the transformation to Rindler coordinates [56]
+as follows. Suppose we write the Cartesian space–time coordinates in terms of hyperbolic
+polar coordinates:
+x = ξ cosh η ,
+ct = ξ sinh η .
+(9)
+The Rindler coordinates {ξ, η} cover the two wedges with x ≥ |η| for ξ ≥ 0 on the right
+and −x ≥ |η| for ξ ≤ 0 on the left of the space–time diagram (Fig. 2). In analogy to the
+regular polar coordinates {r, φ} with spatial metric dr2 + r2dφ2 we get for the hyperbolic
+space–time metric
+ds2 = c2dt2 − dx2 = ξ2dη2 − dξ2 .
+(10)
+A space–time metric measures the proper time τ with increment dτ = ds/c. In particular,
+as ds = ξdη for dξ = 0, the proper time along a trajectory with fixed ξ is (ξ/c)η. We can
+draw another conclusion from the analogy of the Rindler coordinates with polar coordi-
+nates. In space a rotation corresponds to a shift in the angle. In Minkowski space–time, a
+hyperbolic rotation corresponds to a Lorentz transformation to a frame moving with ve-
+locity u. An infinitesimal Lorentz boost shifts the hyperbolic angle by du/c. A sequence
+5
+
+R
+L
++ξ
+-ξ
+x
+ct
+η
+η
+Figure 2:
+Accelerated observers. Space–time diagram of accelerated observers (black curves) in
+Minkowski space with Cartesian coordinates x and t. The observers follow the Rindler trajectories
+of Eq. (9) with fixed ξ and variable parameter η. The acceleration is given by c2/ξ while (ξ/c)η
+gives the proper time of each observer. For negative ξ the parameter η needs to run backwards
+(reversed arrow) as proper time always runs forwards. The observer on the right (R) is separated
+from the observer on the left (L) by horizons (red). Neither left– nor right–moving light from R
+can reach the shaded region in L.
+of infinitesimal boosts thus draws an entire Rindler coordinate line along varying η for
+ξ = const. Now, uniform acceleration is just such a sequence of infinitesimal Lorentz
+transformations. We thus conclude that the Rindler line is the world line of a uniformly
+accelerated observer with acceleration du/dτ = c2/ξ.
+Consider such a uniformly accelerated observer. Suppose the observer is equipped
+with a spectrometer. A spectrometer consists of a spectral element to decompose the field
+�A into frequencies, and a detector to measure the spectral components. It is not important
+what the detector is. It may be a particle detector [52] or an amplitude detector [57],
+the physically important feature of the spectrometer is the ability to perform a frequency
+analysis, and there the important aspect is the fact that the spectrometer responds to its
+proper time τ and not to the coordinate time t. As τ = (ξ/c)η we may describe the effect
+of the spectrometer as a Fourier transformation with respect to η. Note, however, that for
+ξ < 0 (on the left side L of the Rindler diagram of Fig. 2) η needs to run backwards, since
+proper time always runs forwards.
+Imagine now a pair of accelerated observers — one with positive ξ on R and one
+with the exact opposite −ξ on L. Figure 2 reveals that the two observers are separated by
+horizons. The entire world line of observer L lies in the shadow of left– or right–moving
+waves that touch observer R. But it turns out the two observers can and must communicate
+by sharing the same noise field. To work this out, consider the spectral components they
+6
+
+Figure 3:
+Plane wave. The accelerated observer (Fig. 2) samples noise made of plane waves
+with random amplitudes and phases. Each plane wave is sampled along the Rindler trajectory of
+Eq. (9) with proper time (ξ/c)η. The panel shows the real and imaginary part of the wave sampled
+along the path with parameter η. Fourier analysis reveals that the positive–frequency components
+for η contain negative–frequency components for t enhancing the quantum noise perceived by one
+observer at +ξ by correlations with its partner at −ξ (Fig. 2).
+measure:
+�AR = 1
+2π
+� +∞
+−∞
+�A
+���
+R eiνη dη ,
+�AL = 1
+2π
+� +∞
+−∞
+�A
+���
+L e−iνη dη
+(11)
+in terms of the dimensionless Fourier components ν. Here the R and L indicate the space–
+time trajectories of the two observers. They sample the plane–wave Minkowski modes
+(Fig. 3) as oscillations with phases
+ϕR = k(x ∓ ct)|R = kξ e∓η ,
+ϕL = k(x ∓ ct)|L = −kξ e∓η .
+(12)
+Now, components with positive Rindler frequencies ν may also sample negative Minkow-
+ski frequencies, i.e. the complex–conjugated modes A∗
+k. In fact, moving the contour of the
+Fourier integral by +iπ on R and by −iπ on L changes the sign in the phases (12) while
+preserving the convergence of the Fourier integrals (11). We thus see that the Fourier
+transform of the conjugate A∗
+k is exactly e−πν times the Fourier transform of Ak, on both
+sides of the Rindler wedge.
+Accelerated observers sample negative Minkowski frequencies. To see how this af-
+fects the wave noise perceived by the accelerate observers, we introduce a set of modes
+7
+
+Figure 4:
+Rindler modes. The figure shows examples of modes that are monochromatic for
+the two accelerated observers (white hyperbolas, see also Fig. 2). For a monochromatic mode the
+phase increases linearly with time, but for the observers this is proper time, not coordinate time.
+Each accelerated observer comes in with asymptotically the speed of light and leaves asymptot-
+ically with the speed of light. For such velocities proper time ticks exponentially slowly, and so
+the phase grows only logarithmically. Near the horizon (Fig. 2) the phase diverges logarithmically
+[Eq. (13)]. An exponentially small part of the wave crosses to the other side if this wave is made
+of a superposition of positive–norm plane waves, describing the quantum vacuum.
+that are monochromatic with respect to those observers (Fig. 4). Any mode in Minkowski
+space must be a superposition of left– or right–moving waves. The left–moving waves are
+functions of x− = x + ct while the right–moving modes depend on x+ = x − ct. From
+x± = ξe∓η follows that the phases of monochromatic Rindler modes must be logarithmic
+in x±, which means that the Rindler modes are purely imaginary powers of x±. There we
+have two possibilities: x± or −x± to an imaginary power. In the first case the wave is
+predominately localized on the right side of the space–time diagram (Fig. 2), in the sec-
+ond case on the left side. On R we should give the Rindler wave a positive η–frequency
+ν, i.e. the power ±ν of x±, while on L it should oscillate with −ν as η runs backwards
+for forward–running proper time, which also corresponds to the power ±ν but this time
+of −x±. We thus define
+Aν = A
+�
+(x±)±iν
+: ν > 0
+(−x±)±iν
+: ν < 0
+with
+x± = x ∓ ct
+(13)
+and represent the field as
+�A =
+�
+±
+� +∞
+−∞
+�
+�aνAν + �a†
+νA∗
+ν
+�
+dν .
+(14)
+8
+
+ct
+XIt only remains to determine the normalization factor A from Eqs. (6). We substitute the
+modes (13) into the scalar product (5) with the understanding that (A1, A2) differs from
+zero only when ν1 ∼ ν2. We define δ = ±(ν2 − ν1) and obtain for ν > 0:
+(A1, A2) = 2cν
+ℏ A2
+� +∞
+−∞
+(x ∓ ct)iδ−1 dx = 2cν
+ℏ A2 �
+1 − e−2πν� � ∞
+0
+ξiδ dξ
+ξ .
+(15)
+Writing ξ as an exponential gives 2π times the standard Fourier representation of the delta
+function. Defining the parameter ζ by
+tanh ζ = e−πν
+(16)
+with cosh ζ = (1 − e−2πν)−1/2 we thus get
+A = B cosh ζ ,
+B2 =
+ℏ
+4πcν .
+(17)
+This concludes the normalization of the Rindler modes and hence the Rindler representa-
+tion of the field. Only one important, subtle point remains to be discussed.
+The Rindler modes (13) are understood to be analytic on the upper half complex plane
+for x+ and on the lower half plane for x− such that the left side is suppressed for ν >
+0 and the right side for ν < 0. In either case, the Aν are then analytic on the lower
+complex plane for the time t. From this follows that we can always close the contour
+of a Fourier transformation with respect to Minkowski time t for negative frequencies
+ω and get zero. In other words, the Rindler modes (13) have only positive Minkowski
+frequencies. Therefore, they are superpositions of positive–norm Minkowski waves, and
+so their associated annihilation operators �aν are also just superpositions of the Minkowski
+�ak, which implies that both share the same vacuum state |0⟩.
+Having established the vacuum in the Rindler representation, it is elementary to work
+out the spectral components seen by the two accelerated observers. We obtain from
+Eqs. (11) and (14) for the modes (13) with norm (17) and x± = ξe∓η the expressions
+�AR = B
+�
+�aν cosh ζ + �a†
+−ν sinh ζ
+�
+,
+�AL = B
+�
+�a−ν cosh ζ + �a†
+ν sinh ζ
+�
+.
+(18)
+We see here again that the observers sample negative–frequency components �a† with rel-
+ative weight tanh ζ = e−πν. Representing the mode operators in terms of their real
+and imaginary parts (Hermitian and anti–Hermitian parts) we see that the sampled field
+amplitudes are connected — the real parts are correlated and the imaginary parts anti–
+correlated. This means that the wave noise perceived by the observer on R is correlated
+with the noise perceived by observer L. Observer R is influenced by some extra random-
+ness that comes from this connection to observer L and vice versa. That excess noise
+appears in the intensity as an additional contribution to the standard vacuum noise:
+⟨ �A†
+R �AR⟩ = ⟨ �A†
+L �AL⟩ = B2
+�1
+2 +
+1
+e2πν − 1
+�
+.
+(19)
+As the dimensionless η is related to the proper time by the factor c/ξ, the frequencies mea-
+sured in the spectrometers of the accelerated observers are related to the dimensionless ν
+9
+
+by the same factor. We may read the (e2πν − 1)−1 in Eq. (19) as the Planck distribution
+(eℏω/kBT − 1)−1 with Unruh temperature [52]
+kBT = ℏc
+2πξ
+(20)
+where kB denotes Boltzmann’s constant. Each one of the two observers perceives the
+vacuum as thermal radiation with temperature (20). Each one receives this extra noise,
+because the noise is correlated. These correlations do appear when the field amplitudes
+are Fourier–transformed: they are spectral correlations. In terms of particles, they appear
+as entangled Einstein–Podolski–Rosen pairs [55]. When the spectrometer of observer R
+detects a particle at frequency ω so does the spectrometer of observer L (provided they
+are perfectly efficient). But here we are primarily concerned with amplitude noise and its
+cosmological implications.
+3
+Exponential expansion
+Turn now from accelerated observers in static Minkowski space to an observer at rest in
+the expanding universe. Consider first the conceptually simplest case: pure exponential
+expansion (de Sitter space [58]). This is the phase of the cosmic evolution we are entering
+at the present time and, presumably, it was the phase of inflation [59] just after the Big
+Bang (although with a much higher expansion rate then in the current era). Assume
+in agreement with astronomical observations [60] that the universe is homogeneous and
+isotropic, and spatially flat [1]. In this case, the space–time geometry is given by the
+flat–space Friedmann–Lemaitre–Robertson–Walker metric [40]:
+ds2 = c2dt2 − a2dr2
+(21)
+with time–dependent scale factor a(t). The scale factor describes how spatial distances
+expand, as the physical distance between two points at the same time t is given by a times
+the coordinate difference r. The spatial coordinates r are called comoving coordinates,
+because they do not move relative to the universe. The coordinate time t is called cosmo-
+logical time and, physically, it is the proper time of an observer at rest with the universe
+(dr = 0). We may introduce a new time τ called conformal time, defined as
+τ =
+� dt
+a
+(22)
+such that the metric becomes conformally flat:
+ds2 = a2 �
+c2dτ 2 − dr2�
+.
+(23)
+For light rays (ds = 0) the conformal factor a2 is irrelevant, and so light rays travel
+in conformal time and comoving space like in empty Minkowski space. As Maxwell’s
+equations are conformally invariant [24] this remains true for full electromagnetic fields
+and their quantum fluctuations. We assume that the quantum vacuum is carried by plane
+waves in conformal time. The notation is the exact opposite as in the case of uniform
+10
+
+acceleration: there t is the time the vacuum propagates with and τ denotes the proper
+time of the accelerated observer, whereas in the expanding universe the vacuum waves
+propagate with τ while t is the proper time of the observer at rest with the universe.
+Note that the gravitational field of the universe (the space–time geometry) does distin-
+guish a global frame — only in this frame the metric is homogeneous and isotropic. We
+can of course move this frame to any point (as the universe is homogeneous) and rotate it
+(as it is isotropic) but the metric is different for an observer in uniform motion. Note also
+that although the universe is spatially flat, it is curved in space–time. One obtains for the
+curvature scalar [41]
+R = − 6
+c2
+�
+∂tH + 2H2�
+(24)
+in terms of the Hubble parameter
+H = ∂ta
+a .
+(25)
+In the case of exponential expansion the Hubble parameter is a constant H0 such that
+a = a0 eH0t .
+(26)
+In this case, the space–time curvature is negative and constant3 as we also see from R =
+−12H2
+0/c2.
+Figure 5:
+Exponential expansion. An observer at rest samples a plane wave in the exponentially
+expanding universe. The wave oscillates with conformal time [Eq. (27)] that differs exponentially
+from the proper time of the observer (the cosmological time t) in perfect analogy to the Minkowski
+wave sampled by the accelerated observer (Fig. 3).
+Suppose the observer at rest with the universe samples the plane waves of the quantum
+vacuum (Fig. 5). They oscillate with frequencies Ω in the conformal time τ of Eq. (22).
+3The space–time of exponential expansion (de Sitter space) is a maximally symmetric space with con-
+stant Riemann tensor Rαβ
+µν = −(H0/c)2 (δα
+µδβ
+ν − δα
+ν δβ
+µ). The negative prefactor indicates the negative
+curvature.
+11
+
+de Sitter
+extension
+r
+τ = 0
+τ
+∞
+t
+t
+Figure 6:
+Extended de Sitter space. Radial space–time diagram {cτ, r} in conformal time τ
+and comoving radius r = |r|. Cosmological time t runs according to the arrows indicated and
+ends (t = +∞) at the horizontal line (τ = 0).. Light travels along diagonal lines in the conformal
+diagram and may cross over to the next world, the extension, for τ > 0. Light beyond the horizon
+(red line) cannot reach the observer (black vertical line up until t = +∞) before this world ends
+(τ = 0). Light coming in within the white area — within the horizon — leaves in the shaded area,
+but cannot reach the double–shaded region in the extended world, in perfect analogy to the Rindler
+horizon of uniform acceleration (Fig. 2).
+We obtain for the case of exponential expansion:
+τ = − 1
+aH0
+.
+(27)
+Note that conformal time is negative and ends at τ = 0 in the infinite future (t = +∞).
+The observer samples the phase
+ϕ = Ωτ = Ω
+a0
+e−H0t .
+(28)
+This is the same phase as the one of a right–moving wave sampled by Rindler observer R
+(Fig. 4). We see from Eq. (12) that Ω/a0 corresponds to kξ and H0t to the dimensionless
+Rindler time η.
+The observer at rest with the exponentially expanding universe thus perceives waves
+in the same way as the uniformly accelerated observer in Minkowski space, including the
+waves of the quantum vacuum. Like in uniform acceleration, the observer is surrounded
+12
+
+by a horizon (Fig. 6). Seen in conformal time and comoving space, incoming rays out-
+side of the radius rH = −cτ will never arrive at the observer before the world ends in
+conformal time (τ = 0). From Eq. (27) we get
+rH =
+c
+aH .
+(29)
+Unlike the accelerated observer, there is no partner L to the observer R, at least in this uni-
+verse. We may construct an artificial partner by extending de Sitter space to τ > 0 (simi-
+lar to the Kruskal extension of the black hole [56]). For this we imagine another universe
+with infinite cosmological time related to positive conformal time by τ = H−1
+0 e−H0t. In
+this netherworld time runs backwards from +∞ to −∞ such that conformal time and
+light smoothly passes from one world into the other (Fig. 6). The partner observer in the
+netherworld is then shrouded behind a horizon (Fig. 6) from the observer in this world,
+in perfect analogy to uniform acceleration. In particular, we may conclude that the de
+Sitter observer perceives the vacuum as thermal radiation as well [53]. From the corre-
+spondence to the case of the accelerated observer with Unruh temperature (20) we obtain
+the Gibbons–Hawking temperature [53]
+kBT = ℏH0
+2π .
+(30)
+Exponential expansion is a clear, simple, perfectly understood case of quantum noise in
+cosmology, but it is largely an academic case. In reality, the universe does not expand
+exponentially yet nor did it in the past. Very few papers have tackled the problem beyond
+the case of de Sitter space [54, 61, 62], because it is a difficult problem of — appar-
+ently — hardly any relevance, as the Gibbons–Hawking temperature of the real universe
+is astronomically small (T lies in the order of 10−29K for 1/H0 of 10Gy). But if the
+quantum noise of general cosmological horizons is indeed the key to understanding the
+cosmological constant [5], understand it we must.
+4
+Expanding flat space
+Apart from exponential expansion, there is no other case when an expanding flat space
+establishes a genuine event horizon [54, 63] (Fig. 7a). One sees this as follows. The cos-
+mological horizon [64] is the spherical surface around a given point where the expansion
+velocity reaches the speed of light. The expansion velocity u is the derivative of the proper
+length ℓ = ar with respect to cosmological time t. Differentiating ℓ gives Hubble’s law,
+u = Hℓ, in terms of the Hubble parameter H defined in Eq. (25). We see that u reaches
+c at rH of Eq. (29). For the cosmological horizon to be an event horizon it needs to be
+light–like, parallel to light rays in the {cτ, r} space–time diagram, because otherwise light
+may cross it. Since
+τ =
+�
+da
+a2H
+(31)
+the conformal time τ does only agree with −1/(aH) for H = const, i.e. exponential
+expansion, which proves that cosmological horizons are not event horizons, except in
+the exponential case. In fact, the light of distant galaxies and the Cosmic Microwave
+13
+
+Figure 7:
+Cosmological horizon. Space–time diagrams of the horizon (red curve) based on
+actual cosmological data [1, 40] (plotted in units c/H0 with Hubble constant H0). a: in co–
+moving spatial coordinates r and conformal time �� light (black and white lines) propagates like in
+Minkowski space. The region outside the horizon is shaded in grey. Light may cross the horizon,
+except when, in the final stage of cosmic evolution, the horizon becomes light–like and hence a
+genuine event horizon. b: vacuum modes in analogy to the Rindler modes (Fig. 4). The modes are
+defined with respect to a specific time, here τ = 0 (the present time). The figure shows the phase
+pattern of the incident light only, not the outgoing light; Eq. (36) describes both.
+Background reaches us from beyond our horizon [40, 63]. Therefore, it is not clear from
+the outset how to generalize the Gibbons–Hawking formula (30) to the case of expanding
+flat space in general.4
+Consider light in a universe with metric (21). Space shall be expanding, H > 0. For
+conceptual simplicity we do not start from Maxwell’s equations, but rather describe each
+polarization component by a conformally–coupled scalar field with modes satisfying the
+wave equation [24, 65]:
+1
+√−g ∂α
+√−g gαβ∂βA − R
+6 A = 0
+(32)
+in terms of the metric tensor gαβ, its determinant g and matrix–inverse gαβ, and the cur-
+vature scalar R of Eq. (24). Einstein’s summation convention over repeated indices is
+adopted. The modes shall be normalized according to Eq. (6) with the scalar product
+4This section closely follows Ref. [54] but corrects an error in the conformal factor. Despite this error,
+the ideas and results of the paper [54] are correct, as we show here and in Sec. 5.
+14
+
+co-moving r
+a
+0
+conformal
+-2
+-3
+0
+2r
+b
+0
+T
+-2
+-3
+0
+1
+2[65]:
+(A1, A2) = ic
+ℏ
+� �
+A∗
+1 ∂0A2 − A2 ∂0A∗
+1
+� √−g d3x ,
+∂0 = g0α∂α .
+(33)
+One sees from the wave equation that the scalar product (33) is a conserved quantity for
+arbitrary wave packets satisfying Eq. (32). Writing A as A0/a reduces the wave equation
+(32) to the free wave equation for A0 with respect to the conformal time τ of Eq. (22),
+which shows that light waves propagate in the expanding universe like in free Minkowski
+space {cτ, r} (not just light rays). We may use the plane waves
+A = (A/a) eik·r−iωτ
+with
+ω = c|k| ,
+A2 =
+ℏ
+16π3ω
+(34)
+as normalized modes. We assume that the cosmological quantum vacuum is in the vac-
+uum state (8) with respect to these conformal plane waves. This cosmological vacuum
+is called the conformal vacuum [65]. However, as we know from the case of exponen-
+tial expansion, an observer at rest may not perceive the conformal vacuum as vacuum
+fluctuations.
+Imagine a point–like observer at rest with the expanding universe. We use spherical
+coordinates with the origin attached to the point of the observer. Only radial waves will
+matter, because all waves with higher orbital angular momentum vanish at the origin.
+Write the radial modes as
+A =
+1
+√
+4π arAν(r, τ) .
+(35)
+From the wave equation (32) follows that the Aν satisfy one–dimensional wave propaga-
+tion, which means that Aν consists of a superposition of incoming and outgoing waves
+f(r±cτ). As A must not diverge for r → 0 we need to require Aν = f(r+cτ)−f(r−cτ),
+the outgoing wave is the ingoing wave reflected at the focus. Inspired by the cases of uni-
+form acceleration and exponential expansion, we wish to define modes in close analogy
+to the Rindler modes of Eq. (13). These modes can only capture the cosmological hori-
+zon at a given moment in time, i.e. for a given scale factor a0 and corresponding Hubble
+parameter H0. We define [54] (Fig. 7b) in analogy to the Rindler modes [Eq. (13), Fig. 4]:
+Aν = A
+�
+(η − ρ)iν − (η + ρ)iν
+: ν > 0
+(ρ − η)−iν − (−η − ρ)−iν
+: ν < 0
+(36)
+where η (not to be confused with the Rindler η) and ρ are defined as (Fig. 8)
+η = 1 + a0H0(τ0 − τ) ,
+ρ = a0H0
+c
+r .
+(37)
+Like in the case of the Rindler modes, the modes (36) are analytic on the lower half τ
+plane. Consequently, they consist entirely of positive–frequency plane–wave modes (34)
+and share the conformal vacuum. Let us call them vacuum modes. The phase of each of
+the vacuum–mode components, incoming or outgoing, is logarithmic:
+ϕ = ν ln [1 + a0H0(τ0 − τ ∓ r/c)] .
+(38)
+15
+
+ρ = 1
+0
+1
+2
+η = 1
+η = 0
+co-moving r
+conformal τ
+Figure 8:
+Characteristic events. Space–time diagram showing a part of the actual cosmolog-
+ical horizon (Fig. 7a). Vacuum modes (Fig. 7b) are established in analogy to the Rindler modes
+(Fig. 4). The vacuum modes are characterized by the time parameter η and the space parameter
+ρ defined in Eq. (37). The η parameter runs backwards from η = 1 when the vacuum mode is
+defined (t = t0) to η = 0 when the Hawking partners arrive at the origin. At the time t0 (η = 1)
+the spatial parameter reaches unity at the horizon.
+Like the Rindler modes (Fig. 4) the vacuum modes (36) are not monochromatic (Fig. 7b);
+the frequency ω = −∂tϕ varies in space and time. At the defining time of the modes t0
+we have
+ω|t=t0 =
+ω0
+1 ∓ u/c ,
+u = H0ℓ ,
+ℓ = a0r
+(39)
+where ω0 denotes the frequency at the origin and at t = t0. This frequency is related to
+the dimensionless parameter ν by
+ω0 = νH0 .
+(40)
+Equation (39) shows that the vacuum modes are Doppler–shifted in the expanding uni-
+verse. Incoming waves propagate against the Hubble flow u and are blue–shifted, outgo-
+ing waves are red–shifted. Note that the Doppler profile (39) was originally used to define
+the modes (36). Here we have derived them from the analogy to the case of uniform ac-
+celeration.
+It remains to normalize the radial vacuum modes. For this we express the scalar
+product (33) in conformal time τ and spherical coordinates {r, θ, φ} with metric tensor
+gαβ = a2 diag(1, −1, −r2, −r2 sin2 θ). We obtain for the radial waves (35):
+(A1, A2) = i
+ℏ
+� ∞
+0
+�
+A∗
+ν1 ∂τAν2 − Aν2 ∂τA∗
+ν1
+�
+dr .
+(41)
+For the vacuum modes (36) with definitions (37) we have ∂τ = −a0H0 ∂η and a0H0 dr =
+16
+
+c dρ and get
+(A1, A2) = −ic
+ℏ
+� ∞
+0
+�
+A∗
+ν1 ∂ηAν2 − Aν2 ∂ηA∗
+ν1
+�
+dρ .
+(42)
+We may normalize the vacuum modes at a convenient moment (η = 0) as the scalar
+product remains constant at any time. We find exactly the same norm as for the Rindler
+waves, Eqs. (16) and (17).
+Finally, consider the mode overlap between the vacuum modes defined at different
+times. The most relevant case is the overlap between the vacuum modes at one horizon,
+say at t2, with the modes at the previous horizon at t1. By this we mean that t2 is the
+time when the Hawking partners generated at t1 arrive. The overlap tells how the modes
+at one instant of creating Gibbons–Hawking radiation are related to the modes at the next
+stage of creation. In particular, the phases between the modes are important, as the acts
+of creation will interfere with each other. This is because particle creation works like
+parametric amplification [55] where the phase of the incident light determines whether
+particles are created or annihilated. We calculate the scalar product (A1, A2) at time t2
+where η1 = 0 (arrival of the partners) and η2 = 1 (primary Hawking radiation). We
+denote the scale factors and Hubble parameters as a1, H1 and a2, H2, and use ρ = ρ2 as
+integration variable with ρ1 = ρ2(a1H1)/(a2H2) from Eq. (37). In this way we get
+(A1, A2) = c
+ℏ (ν1 + ν2) A1A2
+�a2H2
+a1H1
+�iν1
+cosh2 ζ I12
+(43)
+with definition (16) and the remaining overlap integral
+I12
+=
+� ∞
+0
+ρ−iν1 (1 + ρ)iν2 dρ
+ρ −
+� 1
+0
+ρ−iν1 (1 − ρ)iν2 dρ
+ρ
+(44)
+=
+Γ(−iν1)
+Γ(−iν2) Γ(iν2 − iν1) −
+Γ(1 + iν2)
+Γ(1 − iν1 + iν2) Γ(−iν1)
+(45)
+in terms of Gamma functions. Note that we gave the ν an appropriate small imaginary
+part such that the integrals (44) converge. The dominant contribution to the mode overlap
+appears for ν1 → ν2 where Γ(iν2−iν1) ∼ 1/(iν1−iν2). In the mode expansion
+�
+(�aνAν +
+�a†
+νA∗
+ν)dν the overlap (A1, �A) picks out a single mode with ν1 = ν2 = ν by Cauchy’s
+theorem. Taking into account the normalization (17) we arrive at the simple result:
+�a2 ∼
+�a2H2
+a1H1
+�iν
+�a1 .
+(46)
+Therefore, to a good approximation, the coefficients of the vacuum modes at time t2 are
+given by the mode coefficients at time t1 multiplied by the characteristic logarithmic phase
+factor ν(ln a2H2 − ln a1H1) of the cosmological horizons. This concludes our discussion
+of the vacuum modes in the expanding universe.
+5
+Radiating horizons
+Consider now the noise the observer perceives. The observer, at rest with the expanding
+universe at r = 0, samples the field with respect to cosmological time t, but the field
+17
+
+oscillates with conformal time τ. In the radial vacuum modes we have organized all the
+superpositions of conformal plane waves the observer perceives, such that
+�A
+���
+r=0 =
+� +∞
+−∞
+�
+�aνA0,ν + �a†
+νA∗
+0,ν
+�
+dν
+(47)
+where according to Eq. (35) the A0,ν are given by
+A0,ν =
+1
+√
+4π ar Aν
+����
+r=0
+.
+(48)
+We obtain from expressions (36) and (37) for the modes:
+lim
+r→0
+Aν
+r = ∓2iνA (±η)±iν−1 a0H0
+c
+.
+(49)
+Consider the radiation field around two times in the cosmic evolution: near the time t0
+when particles are produced in the Gibbons–Hawking effect for the Hubble parameter
+H0 and then around the time when the corresponding Hawking partners arrive, given by
+the condition η = 0 (Fig. 8). The time t0 is arbitrary, but for each t0 a new system of
+modes needs to be constructed according to Eqs. (36) and (37). Since any such system
+is a superposition of positive–frequency plane waves, Eq. (34), the vacuum state with
+respect to the mode operators �aν is the cosmic vacuum, regardless of t0.
+As in the cases of uniform acceleration and exponential expansion, imagine the ob-
+server as equipped with a spectrometer measuring the Fourier transformation of the field
+with respect to the proper time of the observer, cosmological time. Consider the Fourier
+transform near the time t0. We write for each vacuum mode
+�A0,ν =
+� +∞
+−∞
+A0,ν eiωt dt
+(50)
+with the understanding that the integration is performed near t0. There we get from
+Eqs. (37) and (22):
+dη = −a0H0
+a
+dt ,
+t ∼ − 1
+H0
+ln η ,
+(51)
+and hence from Eqs. (48) and (49):
+�A0,ν =
+√
+4π iνcA δ(ν − ν0) ,
+ν0 = ω
+H0
+.
+(52)
+For positive frequencies ω the Fourier transform �A0,−ν of the negative–index modes van-
+ishes. However, like in the case of the accelerated observer, the Fourier transform of the
+complex conjugate negative–index modes A∗
+0,−ν does not disappear:
+�
+A∗0,−ν = e−πν �A0,ν .
+(53)
+From relation (16) and the normalization (17) of the vacuum modes we obtain the compact
+result:
+� +∞
+−∞
+�A eiωt dt
+����
+r=0
+=
+√
+ℏν
+c
+i
+�
+�aν cosh ζ + �a†
+−ν sinh ζ
+�
+,
+ν = ω
+H0
+.
+(54)
+18
+
+The result shows that the observer, sampling the vacuum noise with respect to cosmologi-
+cal time around t0, experiences the creation of Hawking particles [54], even in the case of
+non–exponential expansion when the cosmological horizon is not an event horizon [63].
+Now turn to the time t1 when the Hawking partners are expected to arrive, i.e. when
+η ∼ 0 (Fig. 8). It follows from Eqs. (22) and (37):
+η ∼ −a0H0
+a1
+t
+(55)
+where a1 denotes the scale factor at t1. Defining now ν1 = (a1/a0)(ω/H0) we thus obtain
+�A0,−ν = iν
+√π
+A
+c
+� +∞
+−∞
+(−η)−iν−1 e−iν1η dη ∼ i
+√
+2iν A
+c (ν1/ν)iν eiν
+(56)
+in the saddle–point approximation for ν ≫ 1. Similarly, for the negative–frequency
+Fourier–transform of the complex conjugate modes with positive index ν we get
+�
+A∗0,+ν = e−πν �A0,−ν .
+(57)
+Substituting these results in the mode expansion (47) we calculate the integral over the
+mode index in the saddle–point approximation as well. The phase of the integrand, ϕ =
+ν ln(ν1/ν) + ν, is stationary (∂νϕ = 0) for ν = ν1. We obtain in perfect analogy to
+Eq. (54):
+� +∞
+−∞
+�A eiωt dt
+����
+r=0
+=
+√
+ℏν
+c
+i
+�
+�a−ν cosh ζ + �a†
+ν sinh ζ
+�
+,
+ν =
+a1
+a0H0
+ω .
+(58)
+Like the accelerated observer [Eq. (18)] the observer at rest with the expanding universe
+measures spectral correlations expressed in the Bogoliubov transformations (54) and (58).
+These correlations appear as extra noise with Planck spectrum (19).
+6
+Cosmic cascade
+We have thus derived the thermal radiation of cosmological horizons in expanding flat
+space from the physical picture of wave noise (Fig. 1). This picture reproduces the gen-
+eralization [54] of Gibbons’ and Hawking’s [53] result, Eq. (30), to arbitrary expansion.
+In this general case H0 in Eq. (30) refers to the Hubble parameter (25) at any given time
+t0, not required to be constant as in Gibbons’ and Hawking’s case [53] of exponential
+expansion, de Sitter space (Fig. 6). In addition, we also derived a new aspect of Gibbons–
+Hawking radiation not seen in de Sitter space. There the Hawking partners never arrive
+before the world ends in conformal time (Fig. 6) whereas in reality they do (Fig. 8). The
+light of distant galaxies and the Cosmic Microwave Background easily cross the cosmo-
+logical horizon [40, 63] and so do the Hawking partners. We have found that the partners
+are correlated with the primary particles, Eqs. (54) and (58), for the same dimensionless
+frequency ν. For the primary particles, ν is given by the frequency ω divided by the
+Hubble parameter H0, which gives in the Planck spectrum (19) the Gibbons–Hawking
+temperature (30). For the Hawking partners, ν is given by ω divided by (a0/a1)H0 where
+19
+
+0
+1
+2
+-3
+-2
+-1
+0
+co-moving r
+conformal τ
+Figure 9:
+Cascade of horizons. In the actual universe (Fig. 7) depicted in conformal time τ
+and comoving radius r, the Gibbons–Hawking radiation at present (τ = 0) depends on a cascade
+(zigzag line) of radiation generated by past cosmological horizons (red) curve. Depending on
+the relative phase, radiation is created or annihilated. The multiple interference of all creation
+processes gives rise to the effective Gibbons–Hawking temperature and vacuum energy density.
+a1 denotes the scale factor at their time of arrival. This means that the Hawking partners
+also arrive as thermal radiation, but with the red–shifted temperature
+kBT1 = a0
+a1
+ℏH0
+2π .
+(59)
+These results are simple and intuitive, but they are still incomplete. If the present Hawk-
+ing partners arrive in the future as thermal radiation, so should the Hawking partners of
+the past arrive in the present. Call the scale factor and Hubble parameter of the past cos-
+mological horizon a−1 and H−1. The present radiation of Hawking partners should then
+have the temperature
+kBT−1 = a−1
+a0
+ℏH−1
+2π
+.
+(60)
+20
+
+But neither this nor the primary temperature (30) is the effective temperature Teff of
+the radiation in total, because the Hawking particles interfere with their partners. We
+have worked out that they have the logarithmic phase difference (46). Like in parametric
+amplification [55] the phase of the incident radiation determines whether it gets ampli-
+fied or de–amplified, whether particles are created or annihilated. The Hawking partners
+from the previous horizon may very well annihilate some of the Gibbons–Hawking radi-
+ation at the present, depending on the relative phase. Furthermore, the horizon before the
+previous horizon interferes with the particle production as well, and so does the whole
+cascade of past cosmological horizons (Fig. 9). Each horizon establishes the Bogoliubov
+transformation
+�b±ν = �a±ν cosh ζ + �a†
+∓ν sinh ζ
+with
+tanh ζ = e−πν .
+(61)
+Between horizons, the modes are phase shifted according to Eq. (46). As the frequencies
+relevant to the vacuum energy much exceed the Hubble parameter, we are in the regime
+of ν ≫ 1 where we get for the final �bν in terms of the initial vacuum mode operators �a±ν:
+�bν ∼ �aν + �a†
+−ν S e−πν
+(62)
+with S summing up the phase factors of the m–th previous horizons relative to the present
+one:
+S =
+∞
+�
+m=1
+�a−mH−m
+a0H0
+�2iν
+.
+(63)
+This sum is highly oscillatory, but we are interested in the net effect of the interfering
+horizons, i.e. in the average. When averaged over δν ∼ 1 only an exponentially small
+contribution will remain that, together with the primary e−πν, turns the Bogoliubov trans-
+formation (62) into
+�bν ∼ �aν + �a†
+−ν eiΦ−πω/Heff
+(64)
+with some phase Φ that does not affect vacuum correlations. The exact expression for
+Heff we shall derive in the next section, but here we can already draw some qualitative
+conclusions. Since Heff depends on the history of cosmic evolution, it will introduce
+a memory effect in the cosmologically relevant vacuum energy. This memory of the
+past should remove the oscillations that would otherwise plague the cosmic dynamics.
+Destructive interference from past cosmological horizons may also explain why first–
+order perturbation theory with the primary H instead of the full Heff agrees so remarkably
+well with astronomical data [7].
+It is also interesting to note that the cosmic vacuum energy vanishes within one cosmic
+era and thrives in the transition periods between different eras. By era we mean a period
+in the cosmic evolution dominated by one type of fluid with a characteristic equation of
+state. In the radiation–dominated era [40] the Hubble parameter H goes with a−2, in the
+matter–dominated era [40] H ∝ a−3/2 and during vacuum domination H would become
+constant. Apart from the exponential expansion in the vacuum era, all other eras are
+characterized by a power law:
+H = H0 a−γ
+(65)
+21
+
+with constant H0 and γ > 1 (where H0 denotes H at a = 1 here). The partner radiation
+arriving at time t with scale factor a and Hubble parameter H originates from the past
+cosmological horizon the conformal time interval τ earlier, with
+τ =
+�
+da
+a2H =
+1
+γ − 1
+� 1
+aH −
+1
+a−1H−1
+�
+=
+1
+a−1H−1
+,
+(66)
+which gives
+aH
+a−1H−1
+= 1
+γ .
+(67)
+This recurrence relation remains true for all the phases in the sum (63) such that the sum
+forms a perfect harmonic series with vanishing zero–frequency component. The cycle
+average of such a series vanishes: the number of particles produced is exactly zero. For
+a power–law expansion, creation and annihilation thus cancels out exactly as the result of
+multiple interference between past horizons (Fig. 9).
+7
+Wigner function
+The interferences in the cosmic cascade of creation and annihilation at horizons (Fig. 9)
+are captured in the sum (63). Yet this sum is difficult to evaluate and mathematically ill–
+defined. Let us therefore try to deduce a better formula for the effective Gibbons–Hawking
+temperature. The principal problem of our previous approach (Sec. 5) is the Fourier trans-
+formation. We wish to deduce the radiation spectrum as it evolves in time, and there we
+are interested in spectral features ∼ exp(−2π/H) that would require an integration time
+in the order of 1/H for their accurate resolution. However, the universe also evolves on
+a time scale of 1/H and with it the Gibbons–Hawking spectrum. The two aspects, spec-
+tral accuracy and temporal resolution, appear to be mutually exclusive. Frequency and
+time are as mutually exclusive as position and momentum in quantum mechanics (being
+Fourier transforms of each other). Fortunately, there are good compromises. To give a
+simple example, music sheets describe tones – frequencies — in time; to give a sophis-
+ticated example, quantum quasiprobability distributions [55, 66, 67, 68] describe both
+position and momentum. Probably the best compromise is the Wigner function [66]. In
+quantum mechanics, the Wigner function is a partial Fourier transformation of the density
+matrix [55]. The density matrix is a correlation function of two variables, for example
+two positions. The Wigner function performs a Fourier transformation with respect to
+the position difference as a function of the position average. In this way the Wigner
+function captures the momentum spectrum as a function of position. The marginal dis-
+tributions (reduced probability distributions) all give the correct probability distributions
+of either position or momentum, or of any linear combination of the two, with perfect
+accuracy. This property defines the Wigner function uniquely [69] and explains why the
+Wigner function describes conjugate variables (position and momentum, time and fre-
+quency) with the highest possible precision. Here we employ the time–frequency Wigner
+function:
+W = 1
+2π
+� +∞
+−∞
+K(t + θ/2, t − θ/2) eiωθ dθ
+(68)
+22
+
+of the two–time field correlation function K defined as the vacuum expectation value
+K = ⟨ �A1 �A2 + �A2 �A1⟩
+(69)
+with the indices indicating the two times and positions {t1, r1} and {t2, r2}. In the con-
+formal vacuum, the electromagnetic field fluctuations propagate like in empty Minkowski
+space of the conformal times τ and comoving positions r. We may thus use the well–
+known expression of the Minkowski vacuum correlations [8]:
+K =
+1
+(2π)2s2 ,
+s2 = a1a2
+�
+c2(τ2 − τ1)2 − (r2 − r1)2�
+(70)
+in terms of the Minkowski metric s with reciprocal conformal factor a1a2. Here we are
+interested in the spectrum measured with respect to the cosmological times t1 and t2 at a
+given point comoving with the universe:
+r2 = r1 ,
+t2 = t + θ/2 ,
+t1 = t − θ/2 .
+(71)
+There are several ways to derive Eq. (70) — we may expand the field in terms of the
+plane–wave modes (34) and integrate, or we may use the fact that K is the real part of
+the analytic function ⟨ �A1 �A2⟩ with imaginary part given by the difference between retarded
+and advanced Green function, and derive K in one line from the Kramers–Kronig relation
+[5].5 Note that these vacuum correlations exist outside of the causal cone (s < 0) as
+has been recently measured in quantum optics [70]. The correlations peak at s = 0,
+because electromagnetic waves propagate along light cones, including electromagnetic
+noise. Space–time points on the light cone (s = 0) are thus strongly correlated. Wave
+noise is organized (Fig. 1).
+Cosmology adds one subtle complication to the definition (68) of the Wigner function:
+there was a beginning of time (say at t = 0). For a given cosmological time t the Fourier
+time θ runs only from −2t to +2t in the real world. Close to the beginning, the expansion
+factor a develops a branch point [41] such that a becomes complex in the time before,
+which explains [41] why there was nothing real before the beginning of reality. The
+conformal time τ, being defined as the integral of the inverse of a, inherits the branch
+point and ceases to be real for |θ| > 2t as well. The branch points of τ are harmless in the
+integrand (70), but a goes to zero with some fractional power [41]. We might be inclined
+to run the integral in the definition (68) of the Wigner function from −2t to +2t, but
+the branch points of a1 or a2 at ±2t in the integrand (70) would then create oscillations
+with period π/t in the spectrum. The spectral oscillations average out for frequencies ω
+much larger than the inverse cosmic age, but like the oscillations in the cosmic cascade
+(Sec. 6) they obscure the subtle thermal spectrum of Gibbons–Hawking radiation. It is
+therefore wise to analytically continue a around the beginning of time. If we lead the
+integral (68) slightly above the branch points ±2t for ω > 0 and slightly below for ω < 0
+the oscillations are gone, because if we approximate a(t) by some root for t ∼ 0 we could
+close the integration contour on the upper half plane for ω > 0 and on the lower half plane
+for ω < 0 (due to the Fourier factor eiωt) and get zero.
+5There is a sign error in Eq. (50) of Ref. [5] and subsequent expressions, because the wrong half plane
+was taken in closing the integration contour. Fortunately — thanks to another sign error — the result (80)
+carries the correct sign.
+23
+
+Having cleared the way we are now ready to calculate the Wigner function. It is wise
+not to use the explicit expression (70) of the correlation function, but rather our experience
+with Rindler modes in expanding flat space (Sec. 4). We expand [Eq. (47)] the radiation
+field �A in terms of the vacuum modes (36) defined at some arbitrary time t0 ≥ t. The
+value of t0 is not important, as the modes capture the conformal vacuum for all times.
+In the vacuum expectation value (69) of K the ⟨�a† and �a⟩ vanish while the ⟨�aν and �a†
+ν′⟩
+produce delta functions δ(ν − ν′). For the modes at r = |r2 − r1| = 0 we apply Eqs. (48)
+and (49) with the normalization (16) and (17), note that the negative–ν modes are reduced
+by the factor e−πν, and obtain the expression
+K =
+a2
+0H2
+0
+(2π)2c2a1η1a2η2
+� ∞
+0
+2ν
+�1
+2 +
+1
+e2πν − 1
+�
+cos
+�
+ν ln η2
+η1
+�
+dν .
+(72)
+Let us check that this formula agrees with the standard result (70) for K. Formula (72)
+contains the typical Planck term ν(e2πν − 1)−1 plus the contribution ν/2 of the vacuum
+energy. We express these terms in a geometrical series:
+ν
+2 + ν
+�1
+2 +
+1
+e2πν − 1
+�
+= ν
+∞
+�
+m=0
+′
+e−2πmν
+(73)
+where the prime should indicate that the zeroth term is meant to be divided by 2. As
+1
+4 sinh2(z/2) =
++∞
+�
+m=−∞
+1
+(z − 2πmi)2 = −∂z
++∞
+�
+m=−∞
+1
+z − 2πmi
+(74)
+we see that the term (73) is the Fourier transform of [4 sinh2(z/2)]−1 for ν > 0 where
+we can close the integration contour on the upper half plane. Running through the pole at
+zero (instead of surrounding it) produces the factor 1/2 in the vacuum term. For ν < 0
+we close the contour on the lower half plane and get the same expression with ν replaced
+by |ν|. From the inverse Fourier transformation then follows
+� ∞
+0
+ν
+�1
+2 +
+1
+e2πν − 1
+�
+cos νz dν =
+1
+4 sinh2(z/2)
+(75)
+and from this — and definition (37) for η — we obtain Eq. (70). We have thus reproduced
+the known vacuum correlation, but only for times less than t0. In the Wigner function (68)
+we must integrate from −∞ to +∞. Therefore we should move t0 to +∞. In the infinite
+future the expansion a0 goes to infinity and H0 to a finite value, and so [Eq. (37)] the ratio
+η2/η1 goes to (τ∞ − τ2)/(τ∞ − τ1) while the factors (a0H0)/(aη) go to 1/(τ∞ − τ). In
+Eq. (72) we may thus replace η by
+η = τ∞ − τ =
+� ∞
+a
+da
+a2H
+(76)
+and remove the a0H0 altogether. We thus obtain for the thermal part of the Wigner func-
+tion
+Wth =
+1
+(2π)2c2
+� ∞
+0
+ν
+e2πν − 1 D(ν, ω) dν
+(77)
+24
+
+with the kernel
+D = 1
+π e−ωσ
+� +∞
+−∞
+1
+a1η1a2η2
+cos
+�
+ν ln η2
+η1
+�
+eiωϑ dϑ
+(78)
+where we have lifted the integration line in expression (68) by the constant positive imag-
+inary time σ, assuming positive frequencies where, as we know, we should lead the inte-
+gration contour above the branch points at the origin of physical time:
+θ = ϑ + iσ
+with
+σ > 0
+for
+ω > 0 .
+(79)
+Consider de–Sitter space as a test case of our formula. In this case, a grows exponentially
+with constant H0, τ = −(H0a)−1 with τ∞ = 0, and ln(τ2/τ1) = −H0(ϑ + iσ). We get
+D = H2
+0δ(ω − H0ν)
+(80)
+and hence a perfect Planck spectrum with Gibbons–Hawking temperature (30). Formula
+(78) thus reproduces Gibbons’ and Hawking’s classic result [53]. Consider now the realis-
+tic case of cosmic evolution, which deviates from pure exponential expansion. The kernel
+D is of course independent of the integration contour (unless singularities or branch cuts
+are crossed) but for any given real time t there will be only one imaginary time σ when
+D does approach the defining integral of a delta function in the asymptotic limit of large
+frequencies ω (whereas for de–Sitter space all σ do). In the following we work out the
+condition when this is the case.
+But first we need to consider some realistic cosmology in order to estimate the validity
+of the approximation we are going to make. In the spatially flat, isotropic and homoge-
+neous universe the square of the Hubble parameter is proportional to the energy density
+(by Friedman’s equations [40, 41]). For radiation (photons and neutrinos) the energy
+density goes with the inverse fourth power of the expansion factor a, because the energy
+falls with the inverse wavelength and hence with a−1 and the density falls with a−3. For
+matter (baryonic and dark) the energy density is essentially the rest–mass mass density
+multiplied by c2 and a−3. Dark energy Λ — being the cosmological constant — remains
+constant. This gives the Λ Cold Dark Matter (ΛCDM) model:
+H2 = H2
+0
+�ΩR
+a4 + ΩM
+a3 + ΩΛ
+�
+(81)
+where H0 denotes the Hubble constant at the present time (a = 1) and the Ωm describe
+the weights of the various contributions to the energy density with all Ωm summing up to
+unity. The cosmic parameters are retrieved from the fluctuations of the Cosmic Microwave
+Background [1] and are listed in Ref. [40]. For a ≫ ΩR/ΩM ≈ 0.3 × 10−3 we can
+ignore the radiation contribution and enter a stage of cosmic evolution entirely dominated
+by matter and Λ. For describing this matter–vacuum era in the simplest possible way we
+change the scale of a and the units of time replacing (ΩM/ΩΛ)1/3a → a and H0
+√ΩΛt → t
+such that
+H2 = a−3 + 1 .
+(82)
+From t being the integral of 1/(aH) with respect to a we obtain
+a =
+�
+sinh 3t
+2
+�2/3
+and
+H = coth 3t
+2 .
+(83)
+25
+
+We get the conformal time
+τ =
+� a
+0
+da
+a2H = 2√a 2F1
+�
+1/6, 1/2, 7/6, −a3�
+,
+τ∞ = Γ(1/3) Γ(7/6)
+Γ(3/2)
+(84)
+in terms of the hypergeometric function 2F1 and the Gamma function Γ, and from the
+relationship (e.6) [71] of the hypergeometric function:
+η = τ∞ − τ = a−1
+2F1
+�
+1/3, 1/2, 4/3, −a−3�
+.
+(85)
+Consider now the curves in the complex a–plane where the Hubble parameter is real.
+For the Λ–matter model (82) we get three curves where H2 is real: straight lines going
+through the origin with angles {0, π/3, −π/3}. The Hubble parameter itself is real for
+∞ > H2 > 0. So the curves come in from ∞ and end at the points where H = ∞
+or H = 0, which is {0, eiπ/3, e−iπ/3} for the Λ–matter stage (82). The positive real axis
+corresponds to the real world with real time t, the π/3–line in the upper half plane corre-
+sponds to the line with positive imaginary part π/3 in the complex plane of cosmological
+time. In terms of the time t + θ/2 in the Wigner function (68) it draws a line (79) parallel
+to the real axis with σ = 2π/3. This is the line we are going to need in our integral
+(78). The ΛCDM model (81) has four roots of H2 = 0 we can calculate from Ferrari’s
+formula for the roots of quartic equations, two are real and negative, the other two com-
+plex conjugate to each other; we take the root a+ on the upper half plane, for which
+|a+| = 0.775 and arg a+ = π/3 − 1.09 × 10−4. Calculating η according to Eq. (76) we
+find arg η+ = −π/3 + 1.34 × 10−4. We see that a+η+ is real to an accuracy in the order
+of 10−5.
+This has consequences if we calculate the integral (78) in the saddle–point approxima-
+tion for large ω, because we get for the first and second derivatives of the phase ln(η2/η1)
+in the cosine:
+∂θ ln η2
+η1
+����
+iσ
+= −Re 1
+aη
+����
+iσ
+,
+∂2
+θ ln η2
+η1
+����
+iσ
+= 1
+2 Im
+� H
+aη −
+1
+(aη)2
+�����
+iσ
+(86)
+and so the second derivative vanishes: the integral (78) gives a delta function. In fact,
+for large ω only ϑ ∼ 0 matters where we may approximate ln(η2/η1) ∼ −i�σ − �Hϑ and
+(a1η1a2η2)−1 ∼ �H2 with the definitions
+�H = 1
+aη
+����
+iσ
+,
+�σ = 2 arg a|iσ .
+(87)
+We thus obtain from Eq. (78):
+D = e−(ωσ−ν�σ) �H2δ(ω − �Hν) .
+(88)
+For the matter–vacuum universe in our scaled units we have in particular
+�H = 2F1
+�
+1/3, 1/2, 4/3, sech2(3t/2)
+�
+,
+�σ = σ = 2π/3 .
+(89)
+From Eq. (88) follows that, in the full ΛCDM model, the thermal part (77) of the Wigner
+function (68) approaches the high–frequency asymptotics:
+Wth ∼
+ω
+(2π)2c2 e−2πω/Heff
+for
+ω ≫ �H
+(90)
+26
+
+expressed in terms of the effective Hubble parameter
+Heff =
+�H
+1 +
+1
+2π(σ �H − �σ)
+.
+(91)
+The problem is solved.
+8
+Summary and outlook
+We have derived the Gibbons–Hawking temperature for the standard cosmological model
+— the Λ Cold Dark Matter model — from the physical picture of wave noise (Fig. 1).
+The resulting temperature,
+kBT = ℏHeff
+2π
+,
+(92)
+depends on the effective Hubble parameter Heff of Eq. (91) with
+1
+Heff
+= 1
+�H
++ σ
+2π −
+�σ
+2π �H
+.
+(93)
+The effective Hubble parameter sums up the multiple interferences in the cascade of cre-
+ation and annihilation at cosmological horizons (Fig. 9). It does it by analytic continuation
+of the cosmic dynamics to complex times. The parameter �H is given by
+�H =
+1
+a(τ∞ − τ)
+����
+t+iσ/2
+(94)
+in terms of the scale factor a and the conformal time τ evaluated at infinity and at a
+certain complex time t + iσ/2 on the upper half plane. The real part of this complex time
+is the cosmological time at which Gibbons–Hawking radiation is acting at the moment,
+the imaginary part σ/2 needs to be determined from the requirement
+Im �H
+���
+t+iσ/2 = 0 .
+(95)
+The parameter �σ is given by twice the argument of a at the complex time:
+�σ = 2 arg a|t+iσ/2 .
+(96)
+Expression (94) generalizes the Gibbons–Hawking formula (30) for de–Sitter space [53].
+In de–Sitter space [58] the scale factor a grows exponentially as a = eH0t while the
+conformal time τ falls as −H−1
+0 e−H0t approaching τ∞ = 0 in the infinite future. The
+product a(τ∞ − τ) = H−1
+0
+clearly is constant and real for all imaginary times. In a
+realistic cosmological model σ needs to be calculated. For example, in the most relevant
+case, the matter–vacuum dominated period of cosmic evolution, we get σ = 2π/3 for all
+times t (in appropriate units6).
+6Here time is measured in the inverse units of √ΩΛH0 where ΩΛH2
+0 describes the contribution of the
+cosmological constant Λ to the square of the Hubble parameter at the present time (a = 1).
+27
+
+The temperature (92) lies in the order of 10−29K (at the present cosmological time)
+and so the particles of Gibbons–Hawking radiation are completely negligible, but the
+amplitude fluctuations are not — according to Lifshitz theory [5]. They are predicted to
+produce the contribution (1) to the renormalized vacuum energy proportional to
+∆ = ∂3
+t
+1
+Heff
+.
+(97)
+This contribution drives the cosmological term εΛ [5, 6, 7] (but is not proportional to
+εΛ itself). Expression (97) with effective Hubble parameter (93) hopefully is the final
+formula in a series of attempts [5, 7] to determine the correct vacuum energy of expanding
+flat space. For the matter–vacuum dominated period we obtain in our units
+∆ = 1
+�H4
+�
+4 − 8H �H −
+�
+4 − 26
+3 H2
+�
+�H2 +
+�
+6 − 20
+3 H2
+�
+H �H3
+�
+(98)
+in terms of expression (94) at the complex time t + iπ/3 where �H is real — with �H
+given by Eq. (89) — and the Hubble parameter [Eq. (83)] that is real as well — with
+H = tanh(3t/2). Figure 10 compares this result with the previous attempts for the
+vacuum energy, Eqs. (2) and (3).
+t
+0.0
+0.5
+1.0
+1.5
+2.0
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Figure 10: Comparison. Black curve: Heff for the matter–vacuum dominated period of cosmic
+evolution in scaled units. We see that Heff gently falls from 1.24 to unity for t → ∞ (de Sitter
+space in the far future). Red curves: ∆ (proportional to −εvac) in scaled units. Solid curve: result
+of this paper, Eq. (98). Dashed curve: 1
+6∆ obtained from Eq. (3) and used, in perturbation theory,
+in the comparison [7] with astronomical data. Dashed–and–dotted curve: result of Eq. (2), ruled
+out by the data [7]. The factor 1
+6 was chosen such that the curves have the same asymptotics for
+t → ∞. The curves are similar, but with a different prefactor that would correspond to a different
+cutoff [5]. The cutoff is a parameter of the theory, because it is not precisely known (only in its
+order of magnitude). It remains to be seen how the solid curve compares with astronomical data.
+Formula (93) depends on the history of cosmic expansion — being determined by
+analytic continuation of the entire expansion. As the vacuum energy acts back on the
+28
+
+cosmic evolution due to its gravity, it has the tendency of developing oscillations in the
+Hubble parameter if ∆ depends on just the local values of a. It is hoped that the memory
+effect in the vacuum energy derived here will eliminate such artefacts. The multiple
+interference of cosmic creation and annihilation (Fig. 9) summed up in Heff may also
+explain why first–order perturbation theory is remarkably good at fitting the cosmological
+data [7] while the full theory with the previous expressions would fail.
+We obtained our result (93) assuming that the electromagnetic vacuum noise consists
+of modes oscillating with conformal time (22) whereas an observer at rest with the uni-
+verse counts time as cosmological time. Furthermore we assumed, inspired by optical
+analogues of gravity [17, 18, 19, 20, 21, 22, 23, 24], that the “medium” of space behaves
+like a medium, comoving with the universe like the observer at rest. We therefore re-
+quired that the spectrum of vacuum fluctuations perceived by space is the spectrum with
+respect to cosmological time. As the two times differ the spectrum becomes nontrivial
+and, as it turned out, thermal. We used the physical picture of wave noise (Fig. 1) and the
+asymptology [34] of Wigner functions [68] to work out the temperature.
+We can draw another conclusion from the picture of wave noise (Fig. 1). Our analy-
+sis has been entirely local: we picked an arbitrary point in the spatially flat universe and
+considered the vacuum fluctuations at this point evolving in time. Nevertheless, the quan-
+tum vacuum is arriving from long distances away, in particular the noise of the Hawking
+partners. For sustaining the correlations responsible for the Gibbons–Hawking effect and
+hence the vacuum energy εvac, perfect vacuum modes need to be formed according to
+Eq. (36). These are superpositions of perfect, non–dispersive plane waves (34) sustaining
+correlations across vast distances in space. Such long–range correlations cannot exist in
+massive fields, even for energies at the Planck scale where mass is almost irrelevant. Let
+us estimate the requirement for maintaining correlations. A field with particles of mass m
+obeys the dispersion relation
+ℏ2ω2 = ℏ2c2k2 + m2c4 .
+(99)
+Assuming λ = 2πc/ω = ℓp with Planck length ℓp we get for the deviation of the phase
+from the cosmological horizon to the point of observation:
+δϕ = rHδk ∼ −πrH
+ℓP
+λ2
+C
+,
+λC = 2πℏ
+mc
+(100)
+where λC denotes the Compton wavelength. We obtain that for rH ∼ 1010ly the mass m
+must not exceed 10−2eV for not ruining the noise correlations. There is only one field
+with particles of such low mass, the electromagnetic field. Gluons are massless like the
+electromagnetic photons, but they are short–range due to interactions with themselves.
+Neutrinos have masses ≲ 0.8eV [72] and are therefore probably too heavy as well. More-
+over, neutrinos are fermions, and it seems questionable whether fermions can create vac-
+uum forces. The standard vacuum fluctuations acting in the Casimir or van der Waals
+forces [42] are not fluctuations of particles and antiparticles, but field fluctuations. What
+are the physically relevant field fluctuations of fermions in the vacuum state? The need
+for massless bosonic fields to sustain wave correlation across cosmological distances may
+thus explain why only the electromagnetic field seems to contribute to the cosmological
+vacuum energy, as the comparison with astronomical data suggests [7].
+29
+
+Finally, we found that for cosmological eras dominated by only one type of matter
+or energy the effective Gibbons–Hawking temperature is strictly zero or constant. These
+eras are the radiation–dominated era at the youth of the universe (a ≪ 10−3), the matter–
+dominated era in its middle age, and the vacuum–dominated era for the eternity to follow
+(a ≫ 1). We found this by summing up the cosmic interferences, but we also see it in one
+glance from our analytic theory. Pure eras are described by power laws with H = H0 a−γ,
+γ > 1 or exponential expansion with γ = 0. For a power law the conformal time τ
+grows with aγ−1 and hence is analytic. We may close the integration contour of the
+Wigner function (68) of the vacuum correlations (70) at infinity, get the vacuum term by
+integrating through the double pole at τ2 = τ1 but zero thermal contribution. Formula (94)
+also indicates that power laws in H generate zero Gibbons–Hawking temperature, because
+τ tends to infinity for a → ∞, but the formula requires a finite τ∞. For exponential
+expansion, the Gibbons–Hawking temperature (92) is constant, and so its contribution
+(97) to the dynamical vacuum energy density (1) vanishes as well. As the cosmological
+term is driven by the dynamical vacuum energy it remains constant. The vacuum energy
+acts only in transitions.
+This is a typical feature of the Casimir effect. In dielectrics [8, 9], the Casimir energy
+thrives on differences in the dielectric properties of a medium causing forces at interfaces
+and boundaries. In Casimir cosmology [40], the vacuum energy arises in the transitions
+between cosmic eras, changing the cosmological constant there [5, 6, 7]. The current
+era is such a transition period — the transition from matter to vacuum domination —
+and so the cosmological constant varies, which affects the Hubble constant (the Hubble
+parameter at the present time). The predicted variation of the Hubble constant [7] appears
+to agree with the astronomical data [36], giving some empirical support to the theory
+presented here. Wave noise (Fig. 1) may thus not only explain the mundane, the stickiness
+of the microworld, but perhaps also the arcane, the force of the macroworld that drives
+the universe apart.
+Acknowledgements
+Two and a half decades ago Michael Berry’s work on the optical Aharonov Bohm effect
+inspired me to look for connections between quantum optics and general relativity, and
+he has been an inspiration ever since. I am most grateful to him and wish him a happy
+anniversary. I would also like to thank Dror Berechya, David Bermudez, Nikolay Ebel,
+Jonathan Kogman, Amaury Micheli, and Scott Robertson for discussions and comments
+on this paper. The paper has been supported by the Israel Science Foundation and the
+Murray B. Koffler Professorial Chair.
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+Tuning a two-impurity Kondo system by a moir´e superstructure
+Sergey Trishin,1 Christian Lotze,1 Friedemann Lohss,1 Giada Franceschi,1
+Leonid I. Glazman,2 Felix von Oppen,3 and Katharina J. Franke1
+1Fachbereich Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany
+2Department of Physics, Yale University, New Haven, Connecticut 06520, USA
+3Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany
+Two-impurity Kondo models are paradigmatic for correlated spin-fermion systems. Working with
+Mn atoms on Au(111) covered by a monolayer of MoS2, we tune the inter-adatom exchange via the
+adatom distance and the adatom-substrate exchange via the location relative to a moir´e structure of
+the substrate. Differential-conductance measurements on isolated adatoms exhibit Kondo peaks with
+heights depending on the adatom location relative to the moir´e structure. Mn dimers spaced by a few
+atomic lattice sites exhibit split Kondo resonances. In contrast, adatoms in closely spaced dimers
+couple antiferromagnetically, resulting in a molecular-singlet ground state.
+Exciting the singlet-
+triplet transition by tunneling electrons, we find that the singlet-triplet splitting is surprisingly
+sensitive to the moir´e structure. We interpret our results theoretically by relating the variations in
+the singlet-triplet splitting to the heights of the Kondo peaks of single adatoms, finding evidence
+for coupling of the adatom spin to multiple conduction electron channels.
+Exchange interactions between magnetic adatoms and
+itinerant electrons of a substrate can induce correla-
+tion effects. For strong exchange coupling, the adatom
+spin becomes Kondo screened [1, 2].
+For intermedi-
+ate coupling, where the Kondo temperature is compa-
+rable to temperature or other competing couplings, the
+Kondo renormalizations remain in the perturbative do-
+main [3].
+When the exchange coupling is weak, com-
+peting couplings such as single-ion anisotropy can dom-
+inate, in which case Kondo screening can be neglected
+and spin excitations can be probed [4]. The panorama
+becomes yet broader when exchange coupling the adatom
+to a second magnetic atom in its vicinity.
+The na-
+ture of this coupling depends on the interatomic spacing.
+In close proximity, direct exchange tends to dominate,
+while larger separations favor substrate-mediated cou-
+plings such as the oscillatory Rudermann-Kittel-Kasuya-
+Yosida (RKKY) [5–7] and Dzyaloshinskii-Moriya (DM)
+[8, 9] interactions. The resulting ground states may be
+ferromagnetic [10, 11], antiferromagnetic [10, 11], or non-
+collinear [12].
+The competition between inter-adatom and adatom-
+substrate exchange leads to a rich phase diagram with
+multiple correlated ground states. Theoretically, the two-
+impurity Kondo problem has been treated extensively
+[13, 14], and motivated numerous experiments [10, 15–
+18]. The parameter space can be most directly explored
+by scanning-tunneling-microscope (STM) experiments.
+Atom manipulation with the STM tip admits manoev-
+ering the atoms into lattice sites at various distances
+and thus investigating different interatomic interaction
+strengths [19]. Tuning of the exchange coupling to the
+surface is somewhat less straightforward. An early ap-
+proach used strain-induced changes in the band gap of
+a decoupling interlayer [20].
+A more controlled strat-
+egy would exploit well-defined superstructures.
+Prime
+candidates to impose a spatially periodic modulation of
+the atom–substrate interaction strength are interlayers
+which form moir´e structures with the underlying metal
+substrate [21–23]. Most notably, monolayers of MoS2 on
+Au(111) have been successfully employed for tuning the
+exchange coupling of single magnetic Fe atoms from es-
+sentially uncoupled to strongly Kondo screened [23].
+Here,
+we
+exploit
+the
+moir´e
+pattern
+formed
+by
+monolayer-MoS2 on Au(111) to tune the exchange cou-
+pling of Mn dimers with the substrate, thereby probing
+the competition between interatomic and atom-substrate
+exchange. We find that the direct exchange coupling be-
+tween closely-spaced Mn atoms leads to a singlet ground
+state and study the remarkably strong variations of the
+singlet-triplet splitting across the moir´e pattern. We in-
+troduce a new experimental signature of multi-channel
+Kondo coupling by exploiting a theoretical relation be-
+tween the singlet-triplet excitation energy of the dimer
+and the Kondo renormalizations of individual adatoms
+and find evidence that the Mn adatoms are coupled to
+several conduction-electron channels.
+We use the previously established moir´e-patterned de-
+coupling layer MoS2 on Au(111) [23], grown by deposit-
+ing Mo atoms and subsequent annealing to 800 K in H2S
+gas at a pressure p = 10−5 mbar [24, 25]. A moir´e struc-
+ture forms as a result of the lattice mismatch between
+adlayer and substrate, easily seen in the STM images as
+a modulation of the apparent height with a periodicity
+of ≈ 3.3 nm (Fig. 1a) [24–26]. Deposition of Mn atoms
+at low temperatures (< 10 K) leads to isolated atoms ob-
+servable as round protrusions with an apparent height of
+≈ 300 pm. Some round protrusions with smaller appar-
+ent height are attributed to Mn atoms attached to defects
+and excluded from further analysis. We also find some
+oval protrusions. As discussed in more detail below, we
+attribute these to Mn dimers.
+We start by characterizing individual Mn atoms.
+These exhibit a narrow zero-bias resonance in differential-
+arXiv:2301.01517v1  [cond-mat.mes-hall]  4 Jan 2023
+
+2
+conductance (dI/dV ) spectra as shown for two examples
+in Fig. 1b. At our experimental temperature of 1.1 K, the
+lineshape is well reproduced by a temperature-broadened
+logarithmic peak. The peak splits when applying a mag-
+netic field (Fig. 1c). At 3 T, the Zeeman split amounts
+to 600 µV. This behavior is reminiscent of a weakly cou-
+pled Kondo impurity, with the experimental temperature
+larger than or of the order of the Kondo temperature
+[3]. Mn atoms at different positions with respect to the
+moir´e lattice exhibit lineshapes with small variations in
+intensity, but the same broadening (see Fig. 1b for two
+extremal cases). The intensity modulations can be un-
+derstood as modulations of Jν0, where J is the strength
+of the exchange coupling to the conduction electrons and
+ν0 the density of states (DoS) at the Fermi level as dis-
+cussed in more detail below. The observed variations are
+consistent with the DoS modulations due to the adatoms’
+position on the moir´e structure (Fig. 1d).
+Next, we characterize dimer structures formed by two
+adatoms in close proximity to each other.
+Density-
+functional calculations suggest that isolated atoms sit in
+hollow sites of the terminating S layer [28]. Starting with
+this assumption, we can tentatively assign model struc-
+tures to the most commonly found dimer arrangements
+on the surface by evaluating the separation and orienta-
+tion in the STM images. Figure 2a,b shows an arrange-
+ment, where two Mn atoms are separated by three lattice
+sites of the MoS2 substrate. At this separation, the atoms
+show a Kondo resonance as previously described for in-
+dividual adatoms (Fig. 2c), indicating that interatomic
+interactions are negligible. At a distance of two atomic
+lattice sites (Fig. 2d,e), the Kondo resonance develops a
+dip at the Fermi level (Fig. 2f). The spectrum is reminis-
+cent of a Zeeman-split Kondo resonance, indicating mag-
+netic interactions between the atoms, presumably result-
+ing from substrate-mediated RKKY interactions. When
+the atoms are in even closer proximity, their shapes are
+no longer individually resolved in the STM image (Fig.
+2g,h). While a definite assignment of the adsorption sites
+is thus difficult, the oval shape and its orientation with
+respect to the underlying lattice suggest that the atoms
+lie in nearest-neighbor hollow sites (for details and STM
+manipulations, see section S2 in Supplementary Material
+(SM) [29]).
+Differential-conductance spectra measured
+on this type of dimer are radically different from those
+of individual atoms or weakly interacting dimers. The
+Kondo resonance is now replaced by pronounced inelas-
+tic steps at ±10 mV (Fig. 2i).
+It is rather surprising to detect inelastic excitations
+of a relatively large energy, considering that individ-
+ual atoms do not show a noticeable magnetocrystalline
+anisotropy.
+Instead of a change in magnetocrystalline
+anisotropy energy, we suggest that the threshold energy
+is associated with a spin-changing transition of the dimer.
+Such excitations have been observed for Mn dimers on
+CuN [30].
+The close proximity of the atoms may al-
+3 nm
+4.2
+3.8
+3.4
+dI/dV (G0) x 10-3
+dI/dV (G0) x 10-3
+-10
+-5
+0
+5
+10
+bias voltage (mV)
+a)
+d)
+c)
+b)
+0.20
+0.15
+0.10
+0.05
+Jν0
+1.6
+1.2
+0.8
+0.4
+distance (nm)
+moiré min.
+moiré max..
+0 T
+3 T
+3.6
+3.2
+2.8
+2.4
+-15 -10 -5
+0
+5
+10
+15
+bias voltage (mV)
+moiré maximum
+moiré minimum
+Figure 1. Variation of the Kondo coupling across the moir´e
+structure.
+a) STM topography of Mn atoms on the moir´e
+structure of MoS2 on Au(111) (recorded at 100 mV, 20 pA). b)
+dI/dV spectra taken on Mn atoms adsorbed close to a moir´e
+maximum (black) and a moir´e minimum (red). The dashed
+lines show fits using a code based on Ref. [27]. The fits yield
+a (dimensionless) adatom-substrate exchange Jν0 of -0.080
+(red) and -0.049 (black). c) dI/dV spectra on a Mn atom at 0
+T (black) and 3 T (orange). The zero-bias resonance splits at
+3 T (fit: dashed line). [Spectra were recorded at a setpoint of
+15 mV, 3 nA (panel b) and 10 mV, 3 nA (panel c)]. d) Values
+of Jν0 obtained from fitting dI/dV spectra (keeping T 2
+0 =
+0.000415 constant, as obtained from a best fit with B-field)
+on atoms at various positions within the moir´e superstructure
+(distance to the moir´e maximum). Symbols indicate different
+measurement sets. The black dashed line is a linear guide to
+the eye through the data points. The red dashed lines are
+corresponding lines obtained from fits using different tunnel
+couplings T 2
+0 . The upper line corresponds T 2
+0 = 2.635×10−4
+and the lower line to T 2
+0 = 5.6×10−4. These boundary values
+have been determined from error margins of fits at 3 T.
+low for direct exchange as a result of finite overlap of
+the atomic d orbitals. Mn atoms are indeed likely cou-
+pled antiferromagnetically when interacting via direct ex-
+change [31]. This would lead to a singlet ground state
+|Stot = 0, M = 0⟩.
+Magnetic excitations must then in-
+volve a spin-changing transition such as the singlet-triplet
+transition and the excitation energy directly reflects the
+exchange coupling JD (for details, see below). To further
+corroborate the antiferromagnetic nature of the exchange
+coupling, we apply an external magnetic field of 3 T to
+the dimer. The inelastic steps become slightly broader
+(Fig. 2k). This is consistent with a singlet-triplet transi-
+
+3
+2a
+5.5Å
+1a
+i)3.0
+2.8
+2.6
+2.4
+2.2
+-15 -10 -5
+0
+5
+10
+15
+bias voltage (mV)
+a)
+b)
+c)
+d)
+e)
+f)
+g)
+h)
+0T
+3T
+S=0
+S=1
+Energy
+B field
+gμBB
+~J D
+|S,M>
+|1,0>
+|1,+1>
+|1,-1>
+|0,0>
+j)
+k)
+4.0
+3.5
+3.0
+-10
+-5
+0
+5
+10
+bias voltage (mV)
+5.1Å
+3a
+5.6Å
+4.2
+4.0
+3.8
+3.6
+-10
+-5
+0
+5
+10
+bias voltage (mV)
+4.0
+3.0
+2.0
+20
+-10
+0
+10
+20
+bias voltage (mV)
+5.5Å
+dI/dV (G ) x 10
+0
+-3
+dI/dV (G ) x 10
+0
+-3
+dI/dV (G ) x 10
+0
+-3
+dI/dV (G ) x 10
+0
+-3
+Figure 2. Various dimer structures. a), d), g) Structure mod-
+els and b), e), f) corresponding STM topographies of Mn
+dimers on MoS2with various interatom spacings. Yellow, gray,
+and purple spheres represent S, Mo, and Mn, respectively.
+The Mn atoms, sitting in MoS2 hollow sites, are separated
+by three lattice sites (panels a,b), two lattice sites (panels
+d,e), and one lattice site (panels g,h). c), f), i) dI/dV spectra
+recorded at the locations indicated by the black crosses in
+(b,e,h). The spectra drastically depend on the dimer separa-
+tion, exhibiting a Kondo resonance (panel c), a split Kondo
+resonance (panel f), and a step-like increase in the differen-
+tial conductance (panel i). j) Energy-level diagram of the ob-
+served spin excitation in (i). The degeneracy of the M = 0, ±1
+sublevels of the excited state is lifted by a magnetic field. k)
+dI/dV spectra of a dimer with Mn in nearest-neighbor sites
+with and without magnetic field and respective fits with sym-
+metric step functions (dashed). For our measurement condi-
+tions at 1.1 K, a magnetic field of 3 T is not sufficient to fully
+resolve the splitting, but the excitation appears broadened by
+110 µV. STM topographies were recorded at 100 mV, 20 pA,
+the setpoint of the dI/dV spectra was 10 mV, 3 nA (c,f), 15
+mV, 3 nA (i), and 20 mV, 3 nA (k).
+tion to |Stot = 1, M⟩, where the excited state is Zeeman
+split in the magnetic field, but the sublevels are not in-
+dividually resolved at the experimental temperature of
+1.1 K (Fig. 2j).
+Importantly, we do not observe addi-
+tional excitations around zero bias, which would indicate
+a higher-spin ground state as favored by ferromagnetic
+coupling of the atoms.
+As discussed above, the moir´e superstructure weakly
+affects the height of the Kondo resonance of individual
+atoms reflecting the modulation of the dimensionless ex-
+change coupling Jν0. As the dimer is in a singlet ground
+state, one may naively expect that the moir´e structure
+does not influence the inelastic excitations. Remarkably,
+we observe strong variations of the singlet–triplet tran-
+sition by several meV as the dimer’s adsorption site is
+varied with respect to the moir´e lattice (Fig. 3). Dimers
+located on maxima of the moir´e structure (Fig. 3a,e)
+exhibit the smallest excitation energy (7.5 meV), while
+those on minima (Fig. 3d,e) show the largest excitation
+energy (10 meV).
+To understand these variations, we compute the shift
+of the singlet-triplet splitting ∆ due to the hybridiza-
+tion of the adatom d orbitals with the substrate and
+relate it to the exchange coupling between the adatom
+and conduction-electron spins. As we do not observe in-
+elastic excitations on single adatoms indicating negligible
+single-ion anisotropy, we assume that the Mn atoms are
+only weakly perturbed by the surrounding and retain the
+half-filled d-shell when placed on the substrate. Accord-
+ing to Hund’s rule, this implies a high-spin configuration
+with S = 5/2 and suggests that spin-orbit coupling will
+be weak, so that the inter-adatom exchange can be mod-
+eled by isotropic Heisenberg exchange, Hex = JDSA ·SB.
+Here, SA,B denotes the spins of adatom A,B.
+In the absence of hybridization with the substrate,
+states with magnitude Stot of the total spin Stot =
+SA + SB will have direct exchange energy
+Eex(SA, SB; Stot) = JD
+2 [Stot(Stot +1)−
+�
+j∈{A,B}
+Sj(Sj +1)].
+(1)
+Evaluating the singlet-triplet splitting for SA, SB =
+5
+2,
+we find
+∆ = Eex(5
+2, 5
+2; 1) − Eex(5
+2, 5
+2; 0) = JD.
+(2)
+This splitting is reduced by the hybridization of the
+adatom d orbitals with the conduction electrons. In gen-
+eral, the d orbitals hybridize with 2S = 5 (symmetry-
+adapted) conduction-electron channels [32].
+Since the
+substrate breaks rotational symmetry, the strength of
+hybridization Vm depends on the channel m.
+The en-
+ergies of the singlet and triplet states are then shifted by
+virtual excitation processes, in which a d electron hops
+into the substrate or a substrate electron hops into the
+d shell. Physically, these processes reduce the effective
+adatom spin, which results in a smaller direct exchange.
+A detailed calculation in second-order perturbation the-
+ory (see section S1 in SM for details [29]) gives a renor-
+malized singlet-triplet splitting
+∆ = JD
+�
+1 − 2
+5
+�
+m
+ν0|Vm|2
+� 1
+|ϵd| +
+1
+ϵd + U
+��
+.
+(3)
+Here, −ϵd > 0 is the energy to remove an electron from
+the filled d-shell and ϵd+U the energy to add an electron.
+The factor 2 in front of the sum over channels accounts
+
+84
+1nm
+1nm
+a)
+b)
+c)
+d)
+e)
++
++
++
++
+1.1
+1.0
+0.9
+0.8
+0.7
+0.6
+normalized dI/dV
+-15
+-10
+-5
+0
+5
+10
+15
+bias voltage (mV)
+1nm
+1nm
+Figure 3.
+Antiferromagnetically coupled Mn dimers (oval
+structures), which are identical apart from their location rela-
+tive to the moir´e structure. a-d) STM topographies of dimers
+(a) on the maximum, (b) close to the maximum, (c) further
+from the maximum, and (d) at the minimum of the moir´e
+structure. e) dI/dV spectra acquired on the dimers shown in
+(a-d), with colors matched to the crosses in (a-d). Topogra-
+phies recorded at 100 mV, 20 pA, setpoints of the recorded
+spectra were 20 mV, 1 nA (b), 20 mV, 3 nA (a) and 15 mV,
+3 nA (c), (d). Spectra are normalized for clarity.
+for the fact that both adatoms can be excited. The factor
+1/5 results from angular-momentum coupling.
+The singlet-triplet spacing can be directly related to
+experimentally measurable quantities by noting that the
+exchange coupling between the conduction electrons and
+the spin-S adatom is given by [32]
+Jm = ν0|Vm|2
+2S
+� 1
+|ϵd| +
+1
+ϵd + U
+�
+,
+(4)
+so that we can express the singlet-triplet splitting of the
+S = 5
+2 Mn dimer as
+∆ = JD
+�
+1 − 2
+�
+m
+ν0Jm
+�
+.
+(5)
+For weak coupling (ν0Jm ≪ 1), the relative change
+in the singlet-triplet spacing between minimum (∆min)
+and maximum (∆max) is approximately equal to δ ≃
+(∆min − ∆max)/JD.
+Equation (5) relates this directly
+to the corresponding change in the sum of the dimen-
+sionless exchange couplings �
+m ν0Jm to the substrate.
+Since information on the exchange couplings ν0Jm can
+be extracted from the Kondo data on a single adatom,
+applying this relation to the data in Fig. 3e gives direct
+information on the number of conduction-electron chan-
+nels coupled to the adatom spins.
+For analyzing the number of participating channels, we
+first assume that the adatom spin is coupled to a single
+channel. With this assumption, we can extract the di-
+mensionless adatom-substrate exchange coupling ν0J of
+the single channel by fitting the Kondo peak of the iso-
+lated atoms using a program based on Ref. [27]. Showcas-
+ing the variation between extremal positions with respect
+to the moir´e pattern, we extracted a value of ν0J = 0.049
+for the adatom on the moir´e minimum and ν0J = 0.080
+for an atom on the maximum from fitting the Kondo
+data in Fig. 1b. Equation 5 (specified to a single chan-
+nel) then predicts a relative change δ of the singlet-triplet
+splitting from minimum to maximum by ≈ 6%. This is
+clearly smaller than the experimentally observed varia-
+tion of ≈ 25% (Fig. 3e). We have extracted ν0J for sev-
+eral dozen isolated atoms in various positions across the
+moir´e structure. In all cases, ν0J decreases with increas-
+ing distance from the maxima of the moir´e pattern (Fig.
+1d). The variations of ν0J for similar distances from the
+moir´e maxima partially derive from the lack of rotational
+symmetry, so that the distance to the moir´e maximum
+does not uniquely specify the adsorption site. Moreover,
+the fitting procedure contains some uncertainty, as the
+strength of tunneling T 2
+0 and ν0J both affect the peak
+height. We first determined T 2
+0 from a spectrum of an
+atom subject to a magnetic field (for which the uncer-
+tainty is reduced due to the additional magnetic-field in-
+duced structure).
+We then fitted all spectra with the
+extracted value of T 2
+0 . To indicate the error margins of
+the fits, we reran all fits taking extremal values of T 2
+0
+consistent with the B-field data with sufficient accuracy.
+The black dashed line in Fig. 1d shows a linear guide-
+to-the-eye for the best fit results, while the red dashed
+lines indicate the scalings obtained when using the ex-
+tremal values of T 2
+0 . We find that with the assumption
+of a single channel, only the largest variation in ν0J (up-
+per red dashed line) would explain the variation of the
+singlet-triplet splitting.
+We can also apply Eq. (5), when assuming that all five
+channels are equally coupled. Since each channel renor-
+malizes independently, the value of ν0Jm for any m is
+equal to that extracted with the single-channel assump-
+tion. (In the leading-logarithm approximation underly-
+ing the Kondo fits, the number of channels enters only
+as an overall prefactor, which can be absorbed into T 2
+0 .)
+With this assumption, Eq. (5) predicts a variation in the
+singlet-triplet spacing, which is larger by a factor of five
+than in the single-channel case. We then find that the
+observed variation in the singlet-triplet spacing across
+the moir´e structure (Fig. 3e) would only be consistent
+with the opposite extreme case (lower red dashed line
+in Fig. 1d). Thus, while the uncertainties of the fitting
+procedure preclude a fully quantitative analysis, our re-
+sults strongly suggest that the Mn atoms have substan-
+tial coupling to several conduction-electron channels in
+the Au(111) substrate.
+In conclusion, we varied the adatom-substrate ex-
+change of Mn monomers and dimers by exploiting the
+
+:5
+moir´e pattern of a MoS2 layer on Au(111). The moir´e
+structure imprints density-of-states modulations, which
+in turn affect the Kondo resonance of the monomer and
+the singlet-triplet splitting of antiferromagnetically cou-
+pled dimers. Relating these variations through a theo-
+retical analysis, we find evidence that the adatoms are
+coupled to multiple conduction-electron channels. This
+constrasts with the commonly made assumption that
+adatoms couple only to a single channel of a metallic
+substrate. Our results show that this assumption is vi-
+olated in the perturbative limit.
+For the fully devel-
+oped Kondo effect, relatively small differences in ν0Jm
+between channels result in large differences in the asso-
+ciated Kondo temperatures TK,m ∝ e−1/ν0Jm. Then, the
+single-channel approximation can still be adequate pro-
+vided that only the first stage of the resulting multistage
+Kondo screening is accessible in experiment.
+Interest-
+ingly, coupling to multiple conduction-electron channels
+has previously been invoked to explain the appearance of
+multiple Yu-Shiba-Rusinov states induced by magnetic
+adatoms on superconductors [33, 34]. Our results em-
+phasize that adatom dimers realize a rich two-impurity
+problem. While theoretical studies have focused on spin-
+1
+2 impurities, adatom dimers typically have higher spins
+and couple to multiple conduction-electron channels.
+We acknowledge financial support by the Deutsche
+Forschungsgemeinschaft (DFG, German Research Foun-
+dation) through project numbers 328545488 (CRC 227,
+project B05) and 277101999 (CRC 183, project C02 and
+a Mercator professorship), as well as by the National Sci-
+ence Foundation through grant NSF DMR-2002275.
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+
+7
+SUPPLEMENTARY MATERIAL
+I.
+THEORETICAL CONSIDERATIONS
+We provide details concerning the theoretical considerations in the main text. We assume that Mn retains its half-
+filled d shell in the presence of the weak coupling to the substrate. The uncoupled state of Mn is thus fully rotationally
+symmetric and coupled to five conduction-electron channels. As the rotational symmetry is broken by the coupling to
+the substrate, their hybridization Vm with the various conduction-electron channels will be different. In the following,
+we compute the singlet-triplet splitting perturbatively, focusing on one channel (m = 0 for definiteness). The general
+result is obtained by adding the independent corrections for all five channels.
+A.
+Spin states of monomer
+First consider the spin states of a single Mn adatom. We can generate the spin states | 5
+2, Sz⟩ by applying the spin
+lowering operator S− = �2
+m=−2 c†
+m,↓cm,↑ to
+|5
+2, 5
+2⟩ =
+�
+m
+c†
+m,↑|vac⟩.
+(1)
+Then, we have
+|5
+2, 5
+2⟩ = | ↑↑↑↑↑⟩
+|5
+2, 3
+2⟩ =
+�
+1
+5
+�
+|states with one flipped spin⟩
+|5
+2, 1
+2⟩ =
+�
+1
+10
+�
+|states with two flipped spins⟩
+|5
+2, −1
+2⟩ =
+�
+1
+10
+�
+|states with three flipped spins⟩
+|5
+2, −3
+2⟩ =
+�
+1
+5
+�
+|states with four flipped spins⟩
+|5
+2, −5
+2⟩ = | ↓↓↓↓↓⟩.
+(2)
+Similarly, we can derive the states with one less electron, say in the m = 0 state. One finds
+|2, 2⟩ = | ↑↑↑↑⟩
+|2, 1⟩ =
+�
+1
+4
+�
+|states with one flipped spin⟩
+|2, 0⟩ =
+�
+1
+6
+�
+|states with two flipped spins⟩
+|2, −1⟩ =
+�
+1
+4
+�
+|states with three flipped spins⟩
+|2, −2⟩ = | ↓↓↓↓⟩.
+(3)
+
+8
+Applying c0,↑ to the S = 5
+2 states, one finds
+c0,↑|5
+2, 5
+2⟩ = |2, 2⟩
+c0,↑|5
+2, 3
+2⟩ =
+�
+4
+5|2, 1⟩
+c0,↑|5
+2, 1
+2⟩ =
+�
+6
+10|2, 0⟩
+c0,↑|5
+2, −1
+2⟩ =
+�
+4
+10|2, −1⟩
+c0,↑|5
+2, −3
+2⟩ =
+�
+1
+5|2, −2⟩
+c0,↑|5
+2, −5
+2⟩ = 0.
+(4)
+Applying c0,↓ to the S = 5
+2 states, one finds
+c0,↓|5
+2, 5
+2⟩ = 0
+c0,↓|5
+2, 3
+2⟩ =
+�
+1
+5|2, 2⟩
+c0,↓|5
+2, 1
+2⟩ =
+�
+4
+10|2, 1⟩
+c0,↓|5
+2, −1
+2⟩ =
+�
+6
+10|2, 0⟩
+c0,↓|5
+2, −3
+2⟩ =
+�
+4
+5|2, −1⟩
+c0,↓|5
+2, −5
+2⟩ = |2, −2⟩.
+(5)
+B.
+Singlet state of dimer – tunneling out
+The spin state of the dimer can either be expanded in product states |S1, M1⟩ ⊗ |S2, M2⟩ of the two adatoms, or
+according to magnitude Stot and projection Mtot of the total angular momentum Stot = S1 +S2 as |S1, S2; Stot, Mtot⟩.
+First consider the singlet state of the dimer. Using Clebsch-Gordan coefficients, we can expand it into product states
+as
+|5
+2, 5
+2; 0, 0⟩ =
+�
+1
+6
+�
+|5
+2, 5
+2⟩ ⊗ |5
+2, −5
+2⟩ − |5
+2, 3
+2⟩ ⊗ |5
+2, −3
+2⟩ + |5
+2, 1
+2⟩ ⊗ |5
+2, −1
+2⟩
+−|5
+2, −1
+2⟩ ⊗ |5
+2, 1
+2⟩ + |5
+2, −3
+2⟩ ⊗ |5
+2, 3
+2⟩ − |5
+2, −5
+2⟩ ⊗ |5
+2, 5
+2⟩
+�
+.
+(6)
+Applying cL,0,↑ for the left adatom gives
+cL,0,↑|5
+2, 5
+2; 0, 0⟩ =
+�
+1
+6
+�
+|2, 2⟩ ⊗ |5
+2, −5
+2⟩ −
+�
+4
+5|2, 1⟩ ⊗ |5
+2, −3
+2⟩ +
+�
+6
+10|2, 0⟩ ⊗ |5
+2, −1
+2⟩
+−
+�
+4
+10|2, −1⟩ ⊗ |5
+2, 1
+2⟩ +
+�
+1
+5|2, −2⟩ ⊗ |5
+2, 3
+2⟩
+�
+(7)
+Similarly, we have
+cL,0,↓|5
+2, 5
+2; 0, 0⟩ =
+�
+1
+6
+�
+−
+�
+1
+5|2, 2⟩ ⊗ |5
+2, −3
+2⟩ +
+�
+4
+10|2, 1⟩ ⊗ |5
+2, −1
+2⟩ −
+�
+6
+10|2, 0⟩ ⊗ |5
+2, 1
+2⟩
++
+�
+4
+5|2, −1⟩ ⊗ |5
+2, 3
+2⟩ − |2, −2⟩ ⊗ |5
+2, 5
+2⟩
+�
+(8)
+
+9
+We can compare these states to
+|2, 5
+2; 1
+2, 1
+2⟩ =
+�
+1
+15|2, 2⟩ ⊗ |5
+2, −3
+2⟩ −
+�
+2
+15|2, 1⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+1
+5|2, 0⟩ ⊗ |5
+2, 1
+2⟩
+−
+�
+4
+15|2, −1⟩ ⊗ |5
+2, 3
+2⟩ +
+�
+1
+3|2, −2⟩ ⊗ |5
+2, 5
+2⟩
+(9)
+|2, 5
+2; 1
+2, −1
+2⟩ =
+�
+1
+3|2, 2⟩ ⊗ |5
+2, −5
+2⟩ −
+�
+4
+15|2, 1⟩ ⊗ |5
+2, −3
+2⟩ +
+�
+1
+5|2, 0⟩ ⊗ |5
+2, −1
+2⟩
+−
+�
+2
+15|2, −1⟩ ⊗ |5
+2, 1
+2⟩ +
+�
+1
+15|2, −2⟩ ⊗ |5
+2, 3
+2⟩,
+(10)
+so that we identify
+cL,0,↑|5
+2, 5
+2; 0, 0⟩ = −
+�
+1
+2|2, 5
+2; 1
+2, −1
+2⟩
+;
+cL,0,↓|5
+2, 5
+2; 0, 0⟩ =
+�
+1
+2|2, 5
+2; 1
+2, 1
+2⟩.
+(11)
+C.
+Singlet state of dimer – tunneling in
+Now consider tunneling in of an electron. We can follow the same steps. Now, the m = 0 state of one of the atoms
+will be doubly occupied rather than empty, but this is also a zero-spin state. Thus, all the Clebsch-Gordan coefficients
+remain the same and one finds
+c†
+L,0,↑|5
+2, 5
+2; 0, 0⟩ =
+�
+1
+2|2, 5
+2; 1
+2, 1
+2⟩
+;
+c†
+L,0,↓|5
+2, 5
+2; 0, 0⟩ = −
+�
+1
+2|2, 5
+2; 1
+2, −1
+2⟩.
+(12)
+D.
+Triplet state of dimer – tunneling out
+We expand the triplet state of the dimer into product states of the two monomers. Due to rotational invariance,
+we can consider the M = 1 state without loss of generality,
+|5
+2, 5
+2; 1, 1⟩ =
+�
+1
+7|5
+2, 5
+2⟩ ⊗ |5
+2, −3
+2⟩ −
+�
+8
+35|5
+2, 3
+2⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+9
+35|5
+2, 1
+2⟩ ⊗ |5
+2, 1
+2⟩
+−
+�
+8
+35|5
+2, −1
+2⟩ ⊗ |5
+2, 3
+2⟩ +
+�
+1
+7|5
+2, −3
+2⟩ ⊗ |5
+2, 5
+2⟩.
+(13)
+Applying cL,0,↑ for the left adatom gives
+cL,0,↑|5
+2, 5
+2; 1, 1⟩ =
+�
+1
+7|2, 2⟩ ⊗ |5
+2, −3
+2⟩ −
+�
+8
+35
+�
+4
+5|2, 1⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+9
+35
+�
+6
+10|2, 0⟩ ⊗ |5
+2, 1
+2⟩
+−
+�
+8
+35
+�
+4
+10|2, −1⟩ ⊗ |5
+2, 3
+2⟩ +
+�
+1
+7
+�
+1
+5|2, −2⟩ ⊗ |5
+2, 5
+2⟩.
+(14)
+Similarly,
+cL,0,↓|5
+2, 5
+2; 1, 1⟩ = −
+�
+8
+35
+�
+1
+5|2, 2⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+9
+35
+�
+4
+10|2, 1⟩ ⊗ |5
+2, 1
+2⟩ −
+�
+8
+35
+�
+6
+10|2, 0⟩ ⊗ |5
+2, 3
+2⟩
++
+�
+1
+7
+�
+4
+5|2, −1⟩ ⊗ |5
+2, 5
+2⟩.
+(15)
+
+10
+We can compare this to
+|2, 5
+2; 1
+2, 1
+2⟩ =
+�
+1
+15|2, 2⟩ ⊗ |5
+2, −3
+2⟩ −
+�
+2
+15|2, 1⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+1
+5|2, 0⟩ ⊗ |5
+2, 1
+2⟩
+−
+�
+4
+15|2, −1⟩ ⊗ |5
+2, 3
+2⟩ +
+�
+1
+3|2, −2⟩ ⊗ |5
+2, 5
+2⟩
+|2, 5
+2; 3
+2, 1
+2⟩ =
+�
+32
+105|2, 2⟩ ⊗ |5
+2, −3
+2⟩ −
+�
+5
+21|2, 1⟩ ⊗ |5
+2, −1
+2⟩ +
+�
+2
+35|2, 0⟩ ⊗ |5
+2, 1
+2⟩
++
+�
+2
+105|2, −1⟩ ⊗ |5
+2, 3
+2⟩ −
+�
+8
+21|2, −2⟩ ⊗ |5
+2, 5
+2⟩
+|2, 5
+2; 3
+2, 3
+2⟩ =
+�
+4
+35|2, 2⟩ ⊗ |5
+2, −1
+2⟩ −
+�
+9
+35|2, 1⟩ ⊗ |5
+2, 1
+2⟩ +
+�
+12
+35|2, 0⟩ ⊗ |5
+2, 3
+2⟩ −
+�
+2
+7|2, −1⟩ ⊗ |5
+2, 5
+2⟩,
+(16)
+so that we identify
+cL,0,↑|5
+2, 5
+2; 1, 1⟩ =
+�
+7
+15|2, 5
+2; 1
+2, 1
+2⟩ +
+�
+2
+15|2, 5
+2; 3
+2, 1
+2⟩
+;
+cL,0,↓|5
+2, 5
+2; 1, 1⟩ = −
+�
+2
+5|2, 5
+2; 3
+2, 3
+2⟩
+(17)
+E.
+Triplet state of dimer – tunneling in
+This follows again by analogy with the tunneling-out terms, so that
+c†
+L,0,↓|5
+2, 5
+2; 1, 1⟩ =
+�
+7
+15|2, 5
+2; 1
+2, 1
+2⟩ +
+�
+2
+15|2, 5
+2; 3
+2, 1
+2⟩
+;
+c†
+L,0,↑|5
+2, 5
+2; 1, 1⟩ = −
+�
+2
+5|2, 5
+2; 3
+2, 3
+2⟩
+(18)
+F.
+Singlet-triplet splitting
+In the absence of coupling to the substrate, the impurity spins S1 and S2 of the two Mn adatoms are subject to
+antiferromagnetic exchange coupling of the dimer, Hex = JDS1 · S2 with JD > 0. Depending on the total spin Stot,
+the coupling energy is
+Eex(S1, S2; Stot) = JD
+2 [Stot(Stot + 1) − S1(S1 + 1) − S2(S2 + 1)].
+(19)
+For Mn adatoms with S1 = S2 = 5
+2, the splitting between the triplet (S = 1) excited state and the singlet (S = 0)
+ground state is equal to ∆E(0)
+st = JD.
+The singlet-triplet splitting is renormalized due the coupling of the adatoms to the substrate electrons. Tunneling
+of electrons between adatom d orbitals and substrate couples the singlet to the intermediate states |2, 5
+2; 1
+2, ± 1
+2⟩. The
+singlet state has exchange energy
+Eex(5
+2, 5
+2; 0) = −35JD
+4
+,
+(20)
+while the intermediate states have exchange energy
+Eex(2, 5
+2; 1
+2) = −7JD.
+(21)
+In the absense of hybridization, we can then write the energy of of singlet state as
+E(0)
+s
+= 2EMn + EFS + Eex(5
+2, 5
+2; 0),
+(22)
+where EMn denotes the energy of the uncoupled Mn adatom and EFS the energy of the unperturbed Fermi sea.
+Similarly, the intermediate state has energy
+E(0)
+s,out = 2EMn + |ϵd| + EFS + ξk + Eex(2, 5
+2; 1
+2)
+(23)
+
+11
+for tunneling out and
+E(0)
+s,in = 2EMn + ϵd + U + EFS − ξk + Eex(2, 5
+2; 1
+2)
+(24)
+for tunneling in. Here, −ϵd > 0 is the energy to remove an electron from the filled d-shell and ϵd + U the energy to
+add an electron. We can then compute the perturbative shift of the singlet state as
+∆Es = 2|V0|2
+�
+�
+�
+�
+ξk>0
+1
+[2EMn + EFS + Eex( 5
+2, 5
+2; 0)] − [2EMn + |ϵd| + EFS + ξk + Eex(2, 5
+2; 1
+2)]
++
+�
+ξk<0
+1
+[2EMn + EFS + Eex( 5
+2, 5
+2; 0)] − [2EMn + ϵd + U + EFS − ξk + Eex(2, 5
+2; 1
+2)]
+�
+�
+� .
+(25)
+Note that the two intermediate states |2, 5
+2; 1
+2, ± 1
+2⟩ give the same contributions, each with a factor 1/2 due to the
+matrix elements. Note also that the overall factor of two appears, since electrons can tunnel from either Mn adatom
+of the dimer. We can then simplify
+∆Es = −2ν0|V0|2
+ˆ ∞
+0
+dξ
+�
+1
+|ϵd| + ξ + Eex(2, 5
+2; 1
+2) − Eex( 5
+2, 5
+2; 0) +
+1
+ϵd + U + ξ + Eex(2, 5
+2; 1
+2) − Eex( 5
+2, 5
+2; 0)
+�
+(26)
+or
+∆Es = −2ν0|V0|2
+ˆ ∞
+0
+dξ
+�
+1
+|ϵd| + ξ + 7
+4JD
++
+1
+ϵd + U + ξ + 7
+4JD
+�
+.
+(27)
+Here, we introduced the density of states ν0. Assuming the dimer coupling JD to be small compared to the atomic-
+physics scales |ϵd| and U, we find
+∆Es = const + 7JD
+4 2ν0|V0|2
+� 1
+|ϵd| +
+1
+ϵd + U
+�
+,
+(28)
+where the constant is a contribution that is independent of the exchange couplings and that cancels out in the
+singlet-triplet spacing against a similar contribution to the shift of the triplet state.
+Now consider the shift of the triplet state. There are intermediate states with different energies, which have to be
+incorporated with the appropriate matrix elements. This yields
+∆Et = 2|V0|2
+�
+�
+�
+�
+ξk>0
+7
+15
+[2EMn + EFS + Eex( 5
+2, 5
+2; 1)] − [2EMn + |ϵd| + EFS + ξk + Eex(2, 5
+2; 1
+2)]
++
+�
+ξk<0
+7
+15
+[2EMn + EFS + Eex( 5
+2, 5
+2; 1)] − [2EMn + ϵd + U + EFS − ξk + Eex(2, 5
+2; 1
+2)]
++
+�
+ξk>0
+8
+15
+[2EMn + EFS + Eex( 5
+2, 5
+2; 1)] − [2EMn + |ϵd| + EFS + ξk + Eex(2, 5
+2; 3
+2)]
++
+�
+ξk<0
+8
+15
+[2EMn + EFS + Eex( 5
+2, 5
+2; 1)] − [2EMn + ϵd + U + EFS − ξk + Eex(2, 5
+2; 3
+2)]
+�
+�
+� .
+(29)
+Using the energies
+Eex(5
+2, 5
+2; 1) = −31JD
+4
+(30)
+Eex(2, 5
+2; 1
+2) = −7JD
+(31)
+Eex(2, 5
+2; 3
+2) = −11JD
+2
+,
+(32)
+we find, by the same steps as for the singlet shift,
+∆Et = const +
+� 7
+15
+3JD
+4
++ 8
+15
+9JD
+4
+�
+2ν0|V0|2
+� 1
+|ϵd| +
+1
+ϵd + U
+�
+= const + 31JD
+20 2ν0|V0|2
+� 1
+|ϵd| +
+1
+ϵd + U
+�
+.
+(33)
+
+12
+Combining results, we obtain the singlet-triplet splitting
+∆ = JD + ∆Et − ∆Es = JD
+�
+1 − 1
+52ν0|V0|2
+� 1
+|ϵd| +
+1
+ϵd + U
+��
+.
+(34)
+Schrieffer [1] has derived the sd exchange coupling J between adatom spins (magnitude S) and conduction electrons
+and finds
+J = |V0|2
+2S
+� 1
+|ϵd| +
+1
+ϵd + U
+�
+(35)
+(assuming dominant coupling to a single channel). Thus, we can express the renormalized singlet-triplet splitting as
+∆ = JD + ∆Et − ∆Es = JD(1 − 2ν0J).
+(36)
+Accounting for the coupling of the adatom to all five conduction electron channels m, this result generalizes to
+∆ = JD(1 − 2
+�
+m
+ν0Jm).
+(37)
+This equation is quoted in the main text.
+II.
+ADDITIONAL EXPERIMENTAL DATA
+A.
+Adsorption structure of Mn atoms on MoS2
+Figure 1a shows an overview topography image of a monolayer-island of MoS2 decorated with a large number of
+Mn atoms. A close-up view confirms that the individual atoms appear as round protrusions throughout a bias voltage
+range of -1 to 1 V (Fig. 1b). Owing to the convolution with the tip shape, the atoms appear with a large width
+(∼ 0.9 nm), impeding the determination of the exact adsorption site on the atomic lattice constant of MoS2. The
+similarity of apparent heights and spectroscopic signatures suggests that all atoms adsorb in equivalent lattice sites.
+This is in agreement the observation of unique adsorption sites of Fe on MoS2 [2]. DFT calculations further suggest
+hollow sites to be the energetically most favorable positions [3, 4]. Occasionally, we find elongated protrusions (see
+also lineprofiles in Fig. 1c), which we ascribe to dimers.
+10 nm
+3 nm
+a)
+b)
+c)
+300
+200
+100
+0
+apparent height (pm)
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+distance (nm)
+Supplementary Figure 1. a) Large-scale STM image of a monolayer-island of MoS2 on Au(111) after adsorption of Mn atoms
+at low temperature. Recorded at 1 V and 100 pA. b) Close-up view showing individual atoms as round protrusions and some
+elongated structures most probably being Mn dimers. Some point defects can be observed in the MoS2 layer. Recorded at 100
+mV and 20 pA. c) Height profiles along the black and red lines shown in b.
+
+13
+B.
+Manipulation of Mn atoms
+We mainly investigated Mn dimers statistically distributed over the surface. In rare cases, we were able to manip-
+ulate the Mn atoms in a controlled manner. Fig. 2 shows an example of consecutive manipulation events and the
+dI/dV spectra recorded on the obtained structures. In Fig. 2a two Mn atoms are separated at sufficiently far distance
+such that they exhibit a Kondo resonance (spectrum shown in 2d). At closer distance (b), the Kondo resonance is
+split (Fig. 2e). When the atoms are pushed into adjacent lattice sites as in Fig. 2c, the singlet-triplet excitation is
+observed (Fig. 2f).
+2.80
+2.70
+2.60
+-15
+-10
+-5
+0
+5
+10
+15
+bias voltage (mV)
+dI/dV (G0) x 10
+-3
+2.8
+2.6
+2.4
+2.2
+-15
+-10
+-5
+0
+5
+10
+15
+bias voltage (mV)
+dI/dV (G0) x 10
+-3
+b)
++
+c)
++
+a)
+d)
++
+5.0
+4.0
+3.0
+2.0
+dI/dV (G0) x 10
+-3
+-15
+-10
+-5
+0
+5
+10
+15
+bias voltage (mV)
+e)
+f)
+5 Å
+5 Å
+5 Å
+Supplementary Figure 2. Manipulation of two Mn atoms into dimer structures. a-c) STM topographies of the same atoms
+before and after successive manipulation events. The atom at the bottom of figure (a) was pushed closer towards the other
+upper atom, as seen in (b). Here the atoms are still distinguishable. In (c) the lower atom was pushed even closer to the upper
+atom, resulting in a dimer. d-f) dI/dV spectra performed on the upper atom in (a), (b) and (c) respectively. The topographies
+were recorded at 100 mV and 20 pA, the setpoint of the recorded spectra was 15 mV and 3 nA (f) and 10 mV and 3 nA (g).
+Fig. 3a shows one dimer where two Mn atoms are two lattice sites apart. The Kondo resonance is split (red line in
+Fig. 3c). Removing one of the atoms leads to an unperturbed Kondo resonance (green line in Fig. 2c).
+An unambiguous assignment of the adsorption sites of the Mn atoms within the dimer structures is challenging
+as the Mn atoms appear very large and cannot be separately resolved. Analyzing the orientation of the dimers on
+the surface, we observed only three orientations, suggesting the registry with the threefold atomic lattice structure
+of MoS2. While attempting to remove one of the Mn atoms from the densely-packed dimer structures by a voltage
+pulse, we often observed effectively a rotation of the dimers. Also the resulting dimers follow the main axes (Fig. 4).
+
+14
++
++
+a)
+b)
+4.5
+4.0
+3.5
+3.0
+dI/dV (G0) x 10
+-3
+-10
+-5
+0
+5
+10
+bias voltage (mV)
+c)
+5 Å
+5 Å
+Supplementary Figure 3. Disassembly of a Mn dimer. a,b) STM topographies of a Mn dimer before and after the removal of
+one atom. Here the right atom in (a) was removed, leading to a single Mn atom as shown in (b). c) dI/dV spectra performed
+on the left atom in (a) and on the same (remaining) atom (b) respectively. The topographies were recorded at 100 mV and 20
+pA, the setpoint of the recorded spectra was 10 mV and 3 nA (g).
+1
+1
+2
+2
+3
+3
+a)
+b)
+1 nm
+1 nm
+Supplementary Figure 4.
+Rotation of Mn dimers. a), b) STM topographies of single Mn dimers before (a) and after (b)
+applying a high bias voltage. In (a) the dimers 1 and 3 show the same orientation, whereas dimer 2 is rotated by roughly 120◦
+with respect to 1 and 3. After a bias voltage of 1.5 V was applied to the dimers in (a), dimer 1 and 2 appear rotated by 120◦.
+The topographies were recorded at 100 mV and 20 pA.
+C.
+RKKY coupled dimers in different moir´e sites
+In the main text, we showed the variation of singlet-triplet excitations along the moir´e superstructure. To probe
+whether RKKY-coupled Mn dimers are equally affected by the moir´e structure, we investigate Mn dimers with a
+spacing of two substrate lattice sites (Fig. 5). As described in the main text, substrate-mediated interactions lead to
+small excitation gaps around the Fermi level on top of the Kondo resonance (red lines in Fig. 5c,f). Various dimers
+in different moir´e sites display similar gap sizes while the height of the Kondo resonance varies. The same height
+modulation of the Kondo resonance is found on the isolated atoms in the same adsorption sites. This is shown by
+spectra taken on the same atoms after the neighbor has been removed by STM manipulation (black lines in Fig. 5c,f).
+Hence, once Kondo correlations of the individual atoms dominate the spectra and the coupling enters through a small
+perturbation, we hardly observe any moir´e induced modulations in the coupling.
+[1] J. R. Schrieffer, J. Appl. Phys. 38, 1143 (1967).
+[2] S. Trishin, C. Lotze, N. Bogdanoff, F. von Oppen, and K. J. Franke, Phys. Rev. Lett. 127, 236801 (2021).
+[3] X. Chen, L. Zhong, X. Li, and J. Qi, Nanoscale 9, 2188 (2017).
+[4] Y. Wang, B. Wang, R. Huang, B. Gao, F. Kong, and Q. Zhang, Physica E: Low-dimensional Systems and Nanostructures
+63, 276 (2014).
+
+.
+.:
+.15
+4.5
+4.0
+3.5
+3.0
+dI/dV (G0) x 10
+-3
+-10
+-5
+0
+5
+10
+bias voltage (mV)
+2.8
+2.4
+2.0
+1.6
+dI/dV (G0) x 10
+-3
+-10
+-5
+0
+5
+10
+bias voltage (mV)
++
+a)
++
++
++
+b)
+d)
+e)
+c)
+f)
+5 Å
+5 Å
+5 Å
+5 Å
+Supplementary Figure 5. Moir´e effect on RKKY-coupled Mn dimers. a), d) STM topographies of Mn dimers. Whereas in
+(a) the dimer is adsorbed close to the moir´e maximum, in (d) the dimer is adsorbed in the moir´e valley. c), f) dI/dV spectra
+performed at the crosses in (a) and (d). b),e) show the same scan frame, after one atom has been removed from the dimer.
+The black spectra in (c) and (f) show the spectra of the respective monomer. The topographies were recorded at 100 mV and
+20 pA, the setpoint of the recorded spectra was 15 mV and 3 nA (c) and 10 mV and 3 nA (f).
+