Dynamical quantum ergodicity from energy level statistics
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Vikram, Amit (ORCID:0000000272321271)↗
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Computational Response Theory for Dynamics
Quantifying the sensitivity - how a quantity of interest (QoI) varies with respect to a parameter – and response – the representation of a QoI as a function of a parameter - of a computer model of a parametric dynamical system is an important and challenging problem. Traditional methods fail in this context since sensitive dependence on initial conditions implies that the sensitivity and response of a QoI may be ill-conditioned or not well-defined. If a chaotic model has an ergodic attractor, then ergodic averages of QoIs are well-defined quantities and their sensitivity can be used to characterize model sensitivity. The response theorem gives sufficient conditions such that the local forward sensitivity – the derivative with respect to a given parameter - of an ergodic average of a QoI is well-defined. We describe a method based on ergodic and response theory for computing the sensitivity and response of a given QoI with respect to a given parameter in a chaotic model with an ergodic and hyperbolic attractor. This method does not require computation of ensembles of the model with perturbed parameter values. The method is demonstrated and some of the computations are validated on the Lorenz 63 and Lorenz 96 models.
97 MATHEMATICS AND COMPUTING↗
Chaotic roots of the modular multiplication dynamical system in Shor's algorithm
Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the “maximally random” Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an “integrable” quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized 𝐴-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other “chaotic” quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity, and chaos in and via quantum algorithms.
Dynamical systems↗
From Anderson localization on random regular graphs to many-body localization
Highlights: • Anderson transition from ergodicity to localization on random regular graphs (RRG). • Analytical, pool method, and exact-diagonalization study of correlations on RRG. • Many-body localization (MBL) to ergodicity transition: quantum dots and spin chains. • Dynamical eigenstate correlation functions in RRG and MBL problems. • Anderson localization on RRG as a toy model for MBL. The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in delocalized and localized phases, as well at criticality, are discussed. In the many-body part, models with short-range and power-law interactions are considered, as well as the quantum-dot model representing the limit of the “most long-range” interaction. Central themes – which are common to the RRG and MBL problems – include ergodicity of the delocalized phase, localized character of the critical point, strong finite-size effects, and fractal scaling of eigenstate correlations in the localized phase.
36 MATERIALS SCIENCE↗
Time domain probabilistic seismic risk analysis using ground motion prediction equations of Fourier amplitude spectra
Modeling of Fourier amplitude spectra (FAS) of seismic motions has gained much attention in engineering seismology. In the past few years, several ground motion prediction equations (GMPEs) and inter-frequency correlation structure of FAS have been established. Due to many preferable characteristics of FAS, probabilistic seismic hazard/risk analysis is rapidly changing from ergodic, spectrum acceleration Sa(T 0 )-based approach to non-ergodic, site-specific, FAS-based approach. This paper presents time domain intrusive framework for probabilistic seismic risk analysis using GMPE of FAS. Herein, methodology for time domain stochastic ground motion modeling based on GMPEs of FAS is presented in some detail. The simulated uncertain motions are modeled as a random process and represented by polynomial chaos Karhunen-Loève expansion. The random process excitations are further propagated into the uncertain structural system using Galerkin stochastic finite element method (SFEM). Probabilistic evolution of structural response is solved, and such solution is used to develop seismic risk for any damage state. The presented framework is illustrated through seismic risk analysis of a four-story building subjected to possible earthquakes from two strike slip faults. The influences of the epistemic uncertainties in source stress drop Δσ and site attenuation κ0 on seismic risk are investigated. The need for non-ergodic seismic risk analysis with source-specific and site specific characterizations is emphasized.
58 GEOSCIENCES↗
Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma
Abstract For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback–Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein’s lemma for quantum hypothesis testing beyond independent and identically distributed situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations.
Physics↗
Approximate CFTs and random tensor models
Abstract A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS 3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.
Physics↗
Translational diffusion in supercooled water at and near the glass transition temperature—136 K
The properties of amorphous solid water at and near the calorimetric glass transition temperature, T g , of 136 K have been debated for years. One hypothesis is that water turns into a “true” liquid at T g (i.e., it becomes ergodic) and exhibits all the characteristics of an ergodic liquid, including translational diffusion. A competing hypothesis is that only rotational motion becomes active at T g , while the “real” glass transition in water is at a considerably higher temperature. To address this dispute, we have investigated the diffusive mixing in nanoscale water films, with thicknesses up to ∼100 nm, using infrared (IR) spectroscopy. The experiments used films that were composed of at least 90% H 2 O with D 2 O making up the balance and were conducted under conditions where H/D exchange was essentially eliminated. Because the IR spectra of multilayer D 2 O films (e.g., thicknesses of ∼3–6 nm) embedded within thick H 2 O films are distinct from the spectrum of isolated D 2 O molecules within H 2 O, the diffusive mixing of (initially) isotopically layered water films could be followed as a function of annealing time and temperature. The results show that water films with total thicknesses ranging from ∼20 to 100 nm diffusively mixed prior to crystallization for temperatures between 120 and 144 K. The translational diffusion had an Arrhenius temperature dependence with an activation energy of 40.8 ± 3.5 kJ/mol, which indicates that water at and near T g is a strong liquid. The measured diffusion coefficient at 136 K is 6.25 ± 1.4 × 10 −21 m 2 /s.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Circuit complexity and functionality: A statistical thermodynamics perspective
Circuit complexity, defined as the minimum circuit size required for implementing a particular Boolean computation, is a foundational concept in computer science. Determining circuit complexity is believed to be a hard computational problem. Recently, in the context of black holes, circuit complexity has been promoted to a physical property, wherein the growth of complexity is reflected in the time evolution of the Einstein-Rosen bridge (“wormhole”) connecting the two sides of an anti-de Sitter “eternal” black hole. Here, we are motivated by an independent set of considerations and explore links between complexity and thermodynamics for functionally equivalent circuits, making the physics-inspired approach relevant to real computational problems, for which functionality is the key element of interest. In particular, our thermodynamic framework provides an alternative perspective on the obfuscation of programs of arbitrary length—an important problem in cryptography—as thermalization through recursive mixing of neighboring sections of a circuit, which can be viewed as the mixing of two containers with “gases of gates.” This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits of same size and functionality that cannot be connected via a polynomial number of local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of computational complexity theory to its first level.
Science & Technology - Other Topics↗
Stabilizer Scars
Quantum many-body scars are eigenstates in nonintegrable isolated quantum systems that defy typical thermalization paradigms, violating the eigenstate thermalization hypothesis and quantum ergodicity. Here, we identify exact analytic scar solutions in a 2+1 dimensional lattice gauge theory in a quasi-1D limit as zero-magic resource stabilizer states. Our results also highlight the importance of magic resources for gauge theory thermalization, revealing a connection between computational complexity and quantum ergodicity.
eigenstate thermalization↗
Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems
The resource theory of thermal operations, an established model for small-scale thermodynamics, provides an extension of equilibrium thermodynamics to nonequilibrium situations. On a lattice of any dimension with any translation-invariant local Hamiltonian, we identify a large set of translation-invariant states that can be reversibly converted to and from the thermal state with thermal operations and a small amount of coherence. These are the spatially ergodic states, i.e., states that have sharp statistics for any translation-invariant observable, and mixtures of such states with the same thermodynamic potential. As an intermediate result, we show for a general state that if the gap between the min- and the max-relative entropies to the thermal state is small, then the state can be approximately reversibly converted to and from the thermal state with thermal operations and a small source of coherence. Our proof provides a quantum version of the Shannon-McMillan-Breiman theorem for the relative entropy and a quantum Stein’s lemma for ergodic states and local Gibbs states. In this work, our results provide a strong link between the abstract resource theory of thermodynamics and more realistic physical systems as we achieve a robust and operational characterization of the emergence of a thermodynamic potential in translation-invariant lattice systems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity
We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector q exhibit almost no relaxation until times O ( exp ( 1 / q ) ) , at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions. Published by the American Physical Society 2024
Physics↗
Topological kinetic crossover in a nanomagnet array
Ergodic kinetics, which are critical to equilibrium thermodynamics, can be constrained by a system’s topology. Here, we studied a model nanomagnetic array in which such constraints visibly affect the behavior of the magnetic moments. In this system, magnetic excitations connect into thermally active one-dimensional strings whose motion can be imaged in real time. At high temperatures, our data showed the merging, breaking, and reconnecting of strings, resulting in the system transitioning between topologically distinct configurations. Below a crossover temperature, the string motion is dominated by simple changes in length and shape. In this low-temperature regime, the system is energetically stable because of its inability to explore all possible topological configurations. This kinetic crossover suggests a generalizable conception of topologically broken ergodicity and limited equilibration.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Hierarchical Gaussian Random Field Sampling for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and the Karhunen–Loève Decomposition
This work introduces structure preserving hierarchical decompositions for sampling Gaussian random fields (GRFs) within the context of multilevel Bayesian inference in high-dimensional space. Existing scalable hierarchical sampling methods, such as those based on stochastic partial differential equations (SPDEs), often reduce the dimensionality of the sample space at the cost of accuracy of inference. Other approaches, such that those based on Karhunen-Loève (KL) expansions, offer sample space dimensionality reduction but sacrifice GRF representation accuracy and ergodicity of the Markov chain Monte Carlo (MCMC) sampler and are computationally expensive for high-dimensional problems. The proposed method integrates the dimensionality reduction capabilities of KL expansions with the scalability of SPDE-based sampling, thereby providing a robust, unified framework for high-dimensional uncertainty quantification (UQ) that is scalable and accurate, preserves ergodicity, and offers dimensionality reduction of the sample space. The hierarchy in our multilevel algorithm is derived from the geometric multigrid hierarchy. By constructing a hierarchical decomposition that maintains the covariance structure across the levels in the hierarchy, the approach enables efficient coarse-to-fine sampling while ensuring that all samples are drawn from the desired distribution. The effectiveness of the proposed method is demonstrated on a benchmark subsurface flow problem, demonstrating its effectiveness in improving computational efficiency and statistical accuracy. Furthermore, our proposed technique is more efficient and accurate and displays better convergence properties than existing methods for high-dimensional Bayesian inference problems.
Gaussian random fields↗
Regional-Scale 3D Ground-Motion Simulations of M w 7 Earthquakes on the Hayward Fault, Northern California Resolving Frequencies 0–10 Hz and Including Site-Response Corrections
Large earthquake ground-motion simulations in 3D Earth models provide constraints on site-specific shaking intensities but have suffered from limited frequency resolution and ignored site response in soft soils. In this work we report new regional-scale 3D simulations for moment magnitude 7.0 scenario earthquakes on the Hayward Fault, northern California with SW4. Simulations resolved significantly broader band frequencies (0–10 Hz) than previous studies and represent the highest resolution simulations for any such earthquake to date. Seismic waves were excited by a kinematic rupture following Graves and Pitarka (2016) and obeyed wave propagation in a 3D Earth model with topography from the U.S. Geological Survey (USGS) assuming a minimum shear wavespeed, VSmin, of 500 m/s. We corrected motions for linear and nonlinear site response for the shear wavespeed, VS, from the USGS 3D model, using a recently developed ground-motion model (GMM) for Fourier amplitude spectra . At soft soil locations subjected to strong shaking, the site-corrected intensities reflect the competing effects of linear amplification by low V S material, reduction of stiffness during nonlinear deformation, and damping of high frequencies. Sites with near-surface V S of 500 m/s or greater require no linear site correction but can experience amplitude reduction due to nonlinear response. Averaged over all sites, we obtained reasonable agreement with empirical ergodic median GMMs currently used for seismic hazard and design ground motions (epsilon less than 1), with marked improvement at soft sedimentary sites. At specific locations, the simulated shaking intensities show systematic differences from the GMMs that reveal path and site effects not captured in these ergodic models. Results suggest how next generation regional-scale earthquake simulations can provide higher spatial and frequency resolution while including effects of soft soils that are commonly ignored in scenario earthquake ground-motion simulations.
58 GEOSCIENCES↗
Spectral anomalies and broken symmetries in maximally chaotic quantum maps
Spectral statistics such as the level spacing statistics and spectral form factor (SFF) are widely expected to accurately identify “ergodicity,” including the presence of underlying macroscopic symmetries, in generic quantum systems ranging from quantized chaotic maps to interacting many-body systems. By studying various quantizations of maximally chaotic maps that break a discrete classical symmetry upon quantization, we demonstrate that this approach can be misleading and fail to detect macroscopic symmetries. Notably, the same classical map can exhibit signatures of different random matrix symmetry classes in short-range spectral statistics depending on the quantization. While the long-range spectral statistics encoded in the early time ramp of the SFF are more robust and correctly identify macroscopic symmetries in several common quantizations, we also demonstrate analytically and numerically that the presence of Berry-like phases in the quantization leads to spectral anomalies, which break this correspondence. Finally, we provide numerical evidence that long-range spectral rigidity remains directly correlated with ergodicity in the quantum dynamical sense of visiting a complete orthonormal basis.
Shou, Laura [Univ. of Maryland, College Park, MD (↗
Local Poincaré algebra from quantum chaos
The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.
79 ASTRONOMY AND ASTROPHYSICS↗