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Heveling R, Wang J, Steinigeweg R, Gemmer J. Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states. Phys Rev E 2022; 105:064112. [PMID: 35854572 DOI: 10.1103/physreve.105.064112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2021] [Accepted: 04/13/2022] [Indexed: 06/15/2023]
Abstract
Fluctuation theorems are cornerstones of modern statistical mechanics and their standard derivations routinely rely on the crucial assumption of a canonical equilibrium state. Yet rigorous derivations of certain fluctuation theorems for microcanonical states and pure energy eigenstates in isolated quantum systems are still lacking and constitute a major challenge to theory. In this work we tackle this challenge and present such a derivation of an integral fluctuation theorem (IFT) by invoking two central and physically natural conditions, i.e., the so-called "stiffness" and "smoothness" of transition probabilities. Our analytical arguments are additionally substantiated by numerical simulations for archetypal many-body quantum systems, including integrable as well as nonintegrable models of interacting spins and hard-core bosons on a lattice. These simulations strongly suggest that "stiffness" and "smoothness" are indeed of vital importance for the validity of the IFT for microcanonical and pure states. Our work contrasts with recent approaches to the IFT based on Lieb-Robinson speeds and the eigenstate thermalization hypothesis.
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Affiliation(s)
- Robin Heveling
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Jiaozi Wang
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Robin Steinigeweg
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
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2
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Wang J, Lamann MH, Richter J, Steinigeweg R, Dymarsky A, Gemmer J. Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time. PHYSICAL REVIEW LETTERS 2022; 128:180601. [PMID: 35594081 DOI: 10.1103/physrevlett.128.180601] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2021] [Accepted: 04/14/2022] [Indexed: 06/15/2023]
Abstract
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
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Affiliation(s)
- Jiaozi Wang
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Mats H Lamann
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Jonas Richter
- Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Robin Steinigeweg
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Anatoly Dymarsky
- Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny 141701, Russia
- Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 121205, Russia
- Department of Physics, University of Kentucky, Lexington, Kentucky 40506, USA
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
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3
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Fritzsch F, Prosen T. Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions. Phys Rev E 2021; 103:062133. [PMID: 34271691 DOI: 10.1103/physreve.103.062133] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2021] [Accepted: 05/24/2021] [Indexed: 11/07/2022]
Abstract
The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi)energy eigenbasis. Here we study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits in dependence of the frequency associated with the corresponding eigenstates. We provide an exact asymptotic expression for the spectral function, i.e., the second moment of this frequency resolved distribution. The latter is obtained from the decay of dynamical correlations between local operators which can be computed exactly from the elementary building blocks of the dual-unitary circuits. Comparing the asymptotic expression with results obtained by exact diagonalization we find excellent agreement. Small fluctuations at finite system size are explicitly related to dynamical correlations at intermediate times and the deviations from their asymptotical dynamics. Moreover, we confirm the expected Gaussian distribution of the matrix elements by computing higher moments numerically.
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Affiliation(s)
- Felix Fritzsch
- Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana SI-1000, Slovenia
| | - Tomaž Prosen
- Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana SI-1000, Slovenia
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4
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Knipschild L, Engel A, Gemmer J. Stiffness of probability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates. Phys Rev E 2021; 103:062139. [PMID: 34271673 DOI: 10.1103/physreve.103.062139] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2020] [Accepted: 04/01/2021] [Indexed: 11/07/2022]
Abstract
We consider closed quantum systems which are driven such that only negligible heating occurs. If driving only affects small parts of the system, it may nonetheless be strong. Our analysis aims at clarifying under which conditions the Jarzynski relation (JR) holds in such setups, if the initial states are microcanonical or even energy eigenstates. We find that the validity of the JR for the microcanonical initial state hinges on an exponential density of states and on stiffness. The latter indicates an independence of the probability density functions (PDFs) of work of the energy of the respective microcanonical initial state. The validity of the JR for initial energy eigenstates is found to additionally require smoothness. The latter indicates an independence of the work PDFs of the specific energy eigenstates within a microcanonical energy shell. As the validity of the JR for pure initial energy eigenstates has no analog in classical systems, we consider it a genuine quantum phenomenon.
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Affiliation(s)
- Lars Knipschild
- Department of Physics, University of Osnabrück, 49069 Osnabrück, Germany
| | - Andreas Engel
- Institute of Physics, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, 49069 Osnabrück, Germany
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5
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Richter J, Dymarsky A, Steinigeweg R, Gemmer J. Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies. Phys Rev E 2020; 102:042127. [PMID: 33212586 DOI: 10.1103/physreve.102.042127] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Accepted: 09/29/2020] [Indexed: 11/07/2022]
Abstract
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
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Affiliation(s)
- Jonas Richter
- Department of Physics, University of Osnabrück, Osnabrück, Germany
| | - Anatoly Dymarsky
- Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny, Russia.,Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow, Russia.,Department of Physics, University of Kentucky, Lexington, Kentucky, USA
| | | | - Jochen Gemmer
- Department of Physics, University of Osnabrück, Osnabrück, Germany
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Nation C, Porras D. Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories. Phys Rev E 2020; 102:042115. [PMID: 33212596 DOI: 10.1103/physreve.102.042115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2020] [Accepted: 09/12/2020] [Indexed: 06/11/2023]
Abstract
We investigate theoretically the emergence of classical statistical physics in a finite quantum system that is either totally isolated or otherwise subjected to a quantum measurement process. We show via a random matrix theory approach to nonintegrable quantum systems that the set of outcomes of the measurement of a macroscopic observable evolve in time like stochastic variables, whose variance satisfies the celebrated Einstein relation for Brownian diffusion. Our results show how to extend the framework of eigenstate thermalization to the prediction of properties of quantum measurements on an otherwise closed quantum system. We show numerically the validity of the random matrix approach in quantum chain models.
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Affiliation(s)
- Charlie Nation
- Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
- Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom
| | - Diego Porras
- Institute of Fundamental Physics, CSIC, Calle Serrano 113b, 28006 Madrid, Spain
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Knipschild L, Gemmer J. Modern concepts of quantum equilibration do not rule out strange relaxation dynamics. Phys Rev E 2020; 101:062205. [PMID: 32688607 DOI: 10.1103/physreve.101.062205] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2019] [Accepted: 03/10/2020] [Indexed: 11/07/2022]
Abstract
Numerous pivotal concepts have been introduced to clarify the puzzle of relaxation and/or equilibration in closed quantum systems. All of these concepts rely in some way on specific conditions on Hamiltonians H, observables A, and initial states ρ or combinations thereof. We numerically demonstrate and analytically argue that there is a multitude of pairs H,A that meet said conditions for equilibration and generate some typical expectation-value dynamics, which means 〈A(t)〉∝f(t) approximately holds for the vast majority of all initial states. Remarkably we find that, while restrictions on the f(t) exist, they do not at all exclude f(t) that are rather adverse or strange regarding thermal relaxation.
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Affiliation(s)
- Lars Knipschild
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
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8
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Richter J, Jin F, Knipschild L, De Raedt H, Michielsen K, Gemmer J, Steinigeweg R. Exponential damping induced by random and realistic perturbations. Phys Rev E 2020; 101:062133. [PMID: 32688487 DOI: 10.1103/physreve.101.062133] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Accepted: 06/03/2020] [Indexed: 06/11/2023]
Abstract
Given a quantum many-body system and the expectation-value dynamics of some operator, we study how this reference dynamics is altered due to a perturbation of the system's Hamiltonian. Based on projection operator techniques, we unveil that if the perturbation exhibits a random-matrix structure in the eigenbasis of the unperturbed Hamiltonian, then this perturbation effectively leads to an exponential damping of the original dynamics. Employing a combination of dynamical quantum typicality and numerical linked cluster expansions, we demonstrate that our theoretical findings for random matrices can, in some cases, be relevant for the dynamics of realistic quantum many-body models as well. Specifically, we study the decay of current autocorrelation functions in spin-1/2 ladder systems, where the rungs of the ladder are treated as a perturbation to the otherwise uncoupled legs. We find a convincing agreement between the exact dynamics and the lowest-order prediction over a wide range of interchain couplings.
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Affiliation(s)
- Jonas Richter
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Fengping Jin
- Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Lars Knipschild
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Hans De Raedt
- Zernike Institute for Advanced Materials, University of Groningen, NL-9747AG Groningen, The Netherlands
| | - Kristel Michielsen
- Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany
- RWTH Aachen University, D-52056 Aachen, Germany
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Robin Steinigeweg
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
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9
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Heitmann T, Richter J, Schubert D, Steinigeweg R. Selected applications of typicality to real-time dynamics of quantum many-body systems. ACTA ACUST UNITED AC 2020. [DOI: 10.1515/zna-2020-0010] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
Loosely speaking, the concept of quantum typicality refers to the fact that a single pure state can imitate the full statistical ensemble. This fact has given rise to a rather simple but remarkably useful numerical approach to simulate the dynamics of quantum many-body systems, called dynamical quantum typicality (DQT). In this paper, we give a brief overview of selected applications of DQT, where particular emphasis is given to questions on transport and thermalization in low-dimensional lattice systems like chains or ladders of interacting spins or fermions. For these systems, we discuss that DQT provides an efficient means to obtain time-dependent equilibrium correlation functions for comparatively large Hilbert-space dimensions and long time scales, allowing the quantitative extraction of transport coefficients within the framework of, e. g., linear response theory (LRT). Furthermore, it is discussed that DQT can also be used to study the far-from-equilibrium dynamics resulting from sudden quench scenarios, where the initial state is a thermal Gibbs state of the pre-quench Hamiltonian. Eventually, we summarize a few combinations of DQT with other approaches such as numerical linked cluster expansions or projection operator techniques. In this way, we demonstrate the versatility of DQT.
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Affiliation(s)
- Tjark Heitmann
- Department of Physics , University of Osnabrück , Osnabrück , D-49069 , Germany
| | - Jonas Richter
- Department of Physics , University of Osnabrück , Osnabrück , D-49069 , Germany
| | - Dennis Schubert
- Department of Physics , University of Osnabrück , Osnabrück , D-49069 , Germany
| | - Robin Steinigeweg
- Department of Physics , University of Osnabrück , Osnabrück , D-49069 , Germany
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10
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Alhambra ÁM, Riddell J, García-Pintos LP. Time Evolution of Correlation Functions in Quantum Many-Body Systems. PHYSICAL REVIEW LETTERS 2020; 124:110605. [PMID: 32242711 DOI: 10.1103/physrevlett.124.110605] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2019] [Revised: 11/24/2019] [Accepted: 02/27/2020] [Indexed: 06/11/2023]
Abstract
We give rigorous analytical results on the temporal behavior of two-point correlation functions-also known as dynamical response functions or Green's functions-in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times ⟨A(t)B⟩_{β}→⟨A⟩_{β}⟨B⟩_{β}, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For autocorrelation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values and does not increase with system size. We give numerical examples that show that this bound is a good estimate in nonintegrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.
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Affiliation(s)
- Álvaro M Alhambra
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Jonathon Riddell
- Department of Physics & Astronomy, McMaster University 1280 Main St. W., Hamilton Ontario L8S 4M1, Canada
| | - Luis Pedro García-Pintos
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
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11
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Mierzejewski M, Vidmar L. Quantitative Impact of Integrals of Motion on the Eigenstate Thermalization Hypothesis. PHYSICAL REVIEW LETTERS 2020; 124:040603. [PMID: 32058771 DOI: 10.1103/physrevlett.124.040603] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2019] [Revised: 11/26/2019] [Indexed: 06/10/2023]
Abstract
Even though the eigenstate thermalization hypothesis (ETH) may be introduced as an extension of the random matrix theory, physical Hamiltonians and observables differ from random operators. One of the challenges is to embed local integrals of motion (LIOMs) within the ETH. Here we make steps towards a unified treatment of the ETH in integrable and nonintegrable models with translational invariance. Specifically, we focus on the impact of LIOMs on the fluctuations and structure of the diagonal matrix elements of local observables. We first show that nonvanishing fluctuations entail the presence of LIOMs. Then we introduce a generic protocol to construct observables, subtracted by their projections on LIOMs as well as products of LIOMs. The protocol systematically reduces fluctuations and/or the structure of the diagonal matrix elements. We verify our arguments by numerical results for integrable and nonintegrable models.
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Affiliation(s)
- Marcin Mierzejewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
| | - Lev Vidmar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
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12
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Richter J, Lamann MH, Bartsch C, Steinigeweg R, Gemmer J. Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory. Phys Rev E 2019; 100:032124. [PMID: 31639999 DOI: 10.1103/physreve.100.032124] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2019] [Indexed: 06/10/2023]
Abstract
We consider a realistic nonequilibrium protocol, where a quantum system in thermal equilibrium is suddenly subjected to an external force. Due to this force, the system is driven out of equilibrium and the expectation values of certain observables acquire a dependence on time. Eventually, upon switching off the external force, the system unitarily evolves under its own Hamiltonian and, as a consequence, the expectation values of observables equilibrate towards specific constant long-time values. Summarizing our main results, we show that, in systems which violate the eigenstate thermalization hypothesis (ETH), this long-time value exhibits an intriguing dependence on the strength of the external force. Specifically, for weak external forces, i.e., within the linear response regime, we show that expectation values thermalize to their original equilibrium values, despite the ETH being violated. In contrast, for stronger perturbations beyond linear response, the quantum system relaxes to some nonthermal value which depends on the previous nonequilibrium protocol. While we present theoretical arguments which underpin these results, we also numerically demonstrate our findings by studying the real-time dynamics of two low-dimensional quantum spin models.
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Affiliation(s)
- Jonas Richter
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Mats H Lamann
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Christian Bartsch
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
- Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany
| | - Robin Steinigeweg
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49069 Osnabrück, Germany
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