scholarly journals Euler-scale dynamical correlations in integrable systems with fluid motion

2020 ◽  
Vol 3 (2) ◽  
Author(s):  
Frederik Skovbo Møller ◽  
Gabriele Perfetto ◽  
Benjamin Doyon ◽  
Jörg Schmiedmayer
Keyword(s):  
Iterative Scheme ◽  
Scaling Limit ◽  
Fluid Motion ◽  
Many Body ◽  
Fluctuation Dissipation ◽  
Long Time ◽  
Interacting Systems ◽  
Point Correlation ◽  
Many Body Systems ◽  
Scale Limit

We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.

scholarly journals Quantum echo dynamics in the Sherrington-Kirkpatrick model

2020 ◽  
Vol 9 (2) ◽  
Author(s):  
Silvia Pappalardi ◽  
Anatoli Polkovnikov ◽  
Alessandro Silva
Keyword(s):  
Quantum Physics ◽  
Transverse Field ◽  
Many Body ◽  
Initial State ◽  
Large N ◽  
Semiclassical Dynamics ◽  
Long Time ◽  
Kirkpatrick Model ◽  
Ehrenfest Time ◽  
Many Body Systems

Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-N) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins N. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on N and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.

A Middle Way

2021 ◽  
Author(s):  
Robert W. Batterman
Keyword(s):  
Methodological Approach ◽  
Many Body ◽  
Proper Understanding ◽  
Equilibrium Behavior ◽  
Hydrodynamic Approach ◽  
Middle Way ◽  
Fluctuation Dissipation ◽  
Many Body Systems ◽  
Continuum Theories ◽  
Bending Of Beams

This book focuses on a method for exploring, explaining, and understanding the behavior of large many-body systems. It describes an approach to non-equilibrium behavior that focuses on structures (represented by correlation functions) that characterize mesoscale properties of the systems. In other words, rather than a fully bottom-up approach, starting with the components at the atomic or molecular scale, the “hydrodynamic approach” aims to describe and account for continuum behaviors by largely ignoring details at the “fundamental” level. This methodological approach has its origins in Einstein’s work on Brownian motion. He gave what may be the first instance of “upscaling” to determine an effective (continuum) value for a material parameter—the viscosity. His method is of a kind with much work in the science of materials. This connection and the wide-ranging interdisciplinary nature of these methods are stressed. Einstein also provided the first expression of a fundamental theorem of statistical mechanics called the Fluctuation-Dissipation theorem. This theorem provides the primary justification for the hydrodynamic, mesoscale methodology. Philosophical consequences include an argument to the effect that mesoscale parameters can be the natural variables for characterizing many-body systems. Further, the book offers a new argument for why continuum theories (fluid mechanics and equations for the bending of beams) are still justified despite completely ignoring the fact that fluids and materials have lower scale structure. The book argues for a middle way between continuum theories and atomic theories. A proper understanding of those connections can be had when mesoscales are taken seriously.

The Right Variables and Natural Kinds

2021 ◽  
pp. 120-136
Author(s):  
Robert W. Batterman
Keyword(s):  
Correlation Function ◽  
Natural Kinds ◽  
Scientific Methodology ◽  
Many Body ◽  
Material Parameters ◽  
Fluctuation Dissipation ◽  
Hydrodynamic Correlation ◽  
Many Body Systems ◽  
The Right

This chapter argues that mesoscale parameters (order parameters and material parameters) are the natural variables by which we can characterize and understand lawful behaviors of many-body systems. It engages in a debate about whether the determination of natural kinds flows from metaphysical considerations about fundamentality and carving nature at its joins or from goal oriented aims of ones scientific methodology. The chapter argues for a scientific determination of the natural variables (at least in the case of many-body systems) based on the previous discussions of the hydrodynamic, correlation function approach. The Fluctuation-Dissipation theorem provides a justification for taking the mesoscale variables and parameters to be natural kinds.

scholarly journals Coexistence of Different Scaling Laws for the Entanglement Entropy in a Periodically Driven System

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 6
Author(s):  
Tony J. G. Apollaro ◽  
Salvatore Lorenzo
Keyword(s):  
Ground State ◽  
Scaling Laws ◽  
Entanglement Entropy ◽  
Finite Size ◽  
Simultaneous Occurrence ◽  
Many Body ◽  
Periodically Driven ◽  
Quantum Ising Model ◽  
Long Time ◽  
Many Body Systems

The out-of-equilibrium dynamics of many body systems has recently received a burst of interest, also due to experimental implementations. The dynamics of observables, such as magnetization and susceptibilities, and quantum information related quantities, such as concurrence and entanglement entropy, have been investigated under different protocols bringing the system out of equilibrium. In this paper we focus on the entanglement entropy dynamics under a sinusoidal drive of the tranverse magnetic field in the 1D quantum Ising model. We find that the area and the volume law of the entanglement entropy coexist under periodic drive for an initial non-critical ground state. Furthermore, starting from a critical ground state, the entanglement entropy exhibits finite size scaling even under such a periodic drive. This critical-like behaviour of the out-of-equilibrium driven state can persist for arbitrarily long time, provided that the entanglement entropy is evaluated on increasingly subsytem sizes, whereas for smaller sizes a volume law holds. Finally, we give an interpretation of the simultaneous occurrence of critical and non-critical behaviour in terms of the propagation of Floquet quasi-particles.

scholarly journals Quantum many-body attractors

2020 ◽  
Author(s):  
Berislav Buca ◽  
Archak Purkayastha ◽  
Giacomo Guarnieri ◽  
Mark Mitchison ◽  
Dieter Jaksch ◽  
...  
Keyword(s):  
Classical Limit ◽  
Initial Conditions ◽  
Quantum Level ◽  
Many Body ◽  
Dynamical Symmetries ◽  
Oscillatory Dynamics ◽  
Quantum Magnet ◽  
Interacting Systems ◽  
Symmetry Operators ◽  
Many Body Systems

Abstract Real-world complex systems often show robust, persistent oscillatory dynamics, e.g.~non-trivial attractors. On the quantum level this behaviour has only been found in semi-classical or weakly correlated systems under restrictive assumptions. However, strongly interacting systems without classical limits, e.g.~electrons on a lattice or spins, typically relax quickly to a stationary state (trivial attractors). This raises the puzzling question of how non-trivial attractors can arise from the quantum laws. Here, we introduce strictly local dynamical symmetries that lead to extremely robust and persistent oscillations in quantum many-body systems without a classical limit. Observables that do not have overlap with the symmetry operators can relax, losing memory of their initial conditions. The remaining observables enter complex dynamical cycles, signalling the emergence of a quantum many-body attractor. We provide a recipe for constructing Hamiltonians featuring local dynamical symmetries. As an example, we introduce the spin lace – a model of a quasi-1D quantum magnet.

scholarly journals Coexistence of Different Scaling Laws for the Entanglement Entropy in a Periodically Driven System

2018 ◽  
Author(s):  
Tony J. G. Apollaro ◽  
Salvatore Lorenzo
Keyword(s):  
Ground State ◽  
Scaling Laws ◽  
Entanglement Entropy ◽  
Finite Size ◽  
Simultaneous Occurrence ◽  
Many Body ◽  
Periodically Driven ◽  
Quantum Ising Model ◽  
Long Time ◽  
Many Body Systems

The out-of-equilibrium dynamics of many body systems has recently received a burst of interest, also due to experimental implementations. The dynamics of both observables, such as magnetization and susceptibilities, and quantum information related quantities, such as concurrence and entanglement entropy, have been investigated under different protocols bringing the system out of equilibrium. In this paper we focus on the entanglement entropy dynamics under a sinusoidal drive of the tranverse magnetic field in the 1D quantum Ising model. We find that the area and the volume law of the entanglement entropy coexist under periodic drive for an initial non-critical ground state. Furthermore, starting from a critical ground state, the entanglement entropy exhibits finite size scaling even under such a periodic drive. This critical-like behaviour of the out-of-equilibrium driven state can persist for arbitrarily long time, provided that the entanglement entropy is evaluated on increasingly subsytem sizes, whereas for smaller sizes a volume law holds. Finally, we give an interpretation of the simultaneous occurrence of critical and non-critical behaviour in terms of the propagation of Floquet quasi-particles.

scholarly journals Page curve from non-Markovianity

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Kaixiang Su ◽  
Pengfei Zhang ◽  
Hui Zhai
Keyword(s):  
Many Body ◽  
Thermal Entropy ◽  
Initial Stage ◽  
Environment Temperature ◽  
Exactly Solvable ◽  
Long Time ◽  
Kitaev Model ◽  
Many Body Systems ◽  
Near Future ◽  
System Entropy

Abstract In this paper, we use the exactly solvable Sachdev-Ye-Kitaev model to address the issue of entropy dynamics when an interacting quantum system is coupled to a non-Markovian environment. We find that at the initial stage, the entropy always increases linearly matching the Markovian result. When the system thermalizes with the environment at a sufficiently long time, if the environment temperature is low and the coupling between system and environment is weak, then the total thermal entropy is low and the entanglement between system and environment is also weak, which yields a small system entropy in the long-time steady state. This manifestation of non-Markovian effects of the environment forces the entropy to decrease in the later stage, which yields the Page curve for the entropy dynamics. We argue that this physical scenario revealed by the exact solution of the Sachdev-Ye-Kitaev model is universally applicable for general chaotic quantum many-body systems and can be verified experimentally in near future.

scholarly journals On intermediate statistics across many-body localization transition

2021 ◽  
Author(s):  
Bitan De ◽  
Piotr Sierant ◽  
Jakub Zakrzewski
Keyword(s):  
Joint Probability ◽  
Single Parameter ◽  
Joint Probability Distribution ◽  
Many Body ◽  
Localization Transition ◽  
Physical Systems ◽  
Level Statistics ◽  
Interacting Systems ◽  
Many Body Systems ◽  
Two Parameter

Abstract The level statistics in the transition between delocalized and localized {phases of} many body interacting systems is {considered}. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically {via Monte Carlo method}. The resulting higher order spacing ratios are compared with data coming from different {quantum many body systems}. It is found that this Pechukas-Yukawa distribution compares favorably with {$\beta$--Gaussian ensemble -- a single parameter model of level statistics proposed recently in the context of disordered many-body systems.} {Moreover, the Pechukas-Yukawa distribution is also} only slightly inferior to the two-parameter $\beta$-h ansatz shown {earlier} to reproduce {level statistics of} physical systems remarkably well.

Brownian Motion

2021 ◽  
pp. 66-84
Author(s):  
Robert W. Batterman
Keyword(s):  
Brownian Motion ◽  
Many Body ◽  
Hydrodynamic Description ◽  
Representative Volume ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem ◽  
Representative Volume Elements ◽  
Many Body Systems

This chapter discusses the phenomenon of Brownian motion and Einstein’s pioneering arguments that explained various aspects of it. It shows how Einstein presented two arguments that relate directly to the themes of this book. The first is the upscaling or homogenization to effective continuum parameters from correlational structures in representative volume elements at mesoscales. Einstein’s argument is shown to answer the question about autonomy raised in Chapter 2. The second relates to the Fluctuation-Dissipation theorem. This theorem justifies the mesoscale hydrodynamic description of many-body systems and Einstein provided the first statement of the theorem.

Mathematical analysis of time flow

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Rudolf Hilfer
Keyword(s):  
Mathematical Analysis ◽  
Stationary Solutions ◽  
Bounded Mean Oscillation ◽  
Many Body ◽  
C Algebras ◽  
Mean Oscillation ◽  
Time Flow ◽  
Long Time ◽  
Many Body Systems ◽  
Definition Of

AbstractThe mathematical analysis of time flow in physical many-body systems leads to the study of long-time limits. This article discusses the interdisciplinary problem of local stationarity, how stationary solutions can remain slowly time dependent after a long-time limit. A mathematical definition of almost invariant and nearly indistinguishable states on C*-algebras is introduced using functions of bounded mean oscillation. Rescaling of time yields generalized time flows of almost invariant and macroscopically indistinguishable states, that are mathematically related to stable convolution semigroups and fractional calculus. The infinitesimal generator is a fractional derivative of order less than or equal to unity. Applications of the analysis are given to irreversibility and to a physical experiment.

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