Undecidability in quantum thermalization

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

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Undecidability in quantum thermalization

10.21203/rs.3.rs-208752/v1 ◽

2021 ◽

Author(s):

Naoto Shiraishi ◽

Keiji Matsumoto

Keyword(s):

Turing Machine ◽

Thermal Equilibrium ◽

General Theorem ◽

Nearest Neighbour ◽

Many Body ◽

Initial State ◽

One Dimensional ◽

Universal Turing Machine ◽

Many Body Systems ◽

Neighbour Interaction

Abstract The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

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Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains

Nature Communications ◽

10.1038/s41467-021-24726-0 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Sebastian Scherg ◽

Thomas Kohlert ◽

Pablo Sala ◽

Frank Pollmann ◽

Bharath Hebbe Madhusudhana ◽

...

Keyword(s):

Density Wave ◽

Many Body ◽

Initial State ◽

Free System ◽

One Dimensional ◽

Ergodic Behavior ◽

Wide Range ◽

Clean System ◽

Initial Charge ◽

Many Body Systems

AbstractThe thermalization of isolated quantum many-body systems is deeply related to fundamental questions of quantum information theory. While integrable or many-body localized systems display non-ergodic behavior due to extensively many conserved quantities, recent theoretical studies have identified a rich variety of more exotic phenomena in between these two extreme limits. The tilted one-dimensional Fermi-Hubbard model, which is readily accessible in experiments with ultracold atoms, emerged as an intriguing playground to study non-ergodic behavior in a clean disorder-free system. While non-ergodic behavior was established theoretically in certain limiting cases, there is no complete understanding of the complex thermalization properties of this model. In this work, we experimentally study the relaxation of an initial charge-density wave and find a remarkably long-lived initial-state memory over a wide range of parameters. Our observations are well reproduced by numerical simulations of a clean system. Using analytical calculations we further provide a detailed microscopic understanding of this behavior, which can be attributed to emergent kinetic constraints.

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Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

Entropy ◽

10.3390/e22090984 ◽

2020 ◽

Vol 22 (9) ◽

pp. 984

Author(s):

Regina Finsterhölzl ◽

Manuel Katzer ◽

Andreas Knorr ◽

Alexander Carmele

Keyword(s):

Time Evolution ◽

Nearest Neighbor ◽

Matrix Product ◽

Body System ◽

Many Body ◽

Initial State ◽

Matrix Product States ◽

Open Quantum ◽

Many Body Systems ◽

The Many

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.

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Quantum echo dynamics in the Sherrington-Kirkpatrick model

SciPost Physics ◽

10.21468/scipostphys.9.2.021 ◽

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.

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Probing non-thermal density fluctuations in the one-dimensional Bose gas

SciPost Physics ◽

10.21468/scipostphys.3.3.023 ◽

2017 ◽

Vol 3 (3) ◽

Cited By ~ 12

Author(s):

Jacopo De Nardis ◽

Milosz Panfil ◽

Andrea Gambassi ◽

Leticia Cugliandolo ◽

Robert Konik ◽

...

Keyword(s):

Spatial Dimension ◽

Bose Gas ◽

Density Fluctuations ◽

Dynamical Structure ◽

Many Body ◽

Initial State ◽

One Dimensional ◽

Fluctuation Dissipation ◽

Quantum Integrable Models ◽

The One

Quantum integrable models display a rich variety of non-thermal excited states with unusual properties. The most common way to probe them is by performing a quantum quench, i.e., by letting a many-body initial state unitarily evolve with an integrable Hamiltonian. At late times these systems are locally described by a generalized Gibbs ensemble with as many effective temperatures as their local conserved quantities. The experimental measurement of this macroscopic number of temperatures remains elusive. Here we show that they can be obtained for the Bose gas in one spatial dimension by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuation-dissipation theorem that we provide. Our procedure allows us to completely reconstruct the stationary state of a quantum integrable system from state-of-the-art experimental observations.

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