Thermalization in open systems coupled to macroscopic environments is usually analyzed from the perspective of relaxation of the reduced state of the system to the equilibrium state. Less emphasis is given to the change of the state of the bath. However, as previously shown for some specific models, during the thermalization the environment may undergo a nontrivial dynamics, indicated by the the change of its von Neumann entropy, at time scales even longer than the relaxation time of the system; here such a behavior is nicknamed as post-thermalization. We provide a more detailed analysis of this phenomenon by simulating the full dynamics of a variety of systems together with their environment. In particular, the post-thermalization is qualitatively explained as a result of reconversion of the initially built up correlation between the system and the bath into the correlation between the degrees of freedom in the environment. We also present exemplary systems in which such a reconversion is suppressed due to non-Markovian dynamics or the presence of interactions.
The observable long-time behavior of an isolated many-body system after a quantum quench is considered, i.e., an eigenstate (or an equilibrium ensemble) of some pre-quench Hamiltonian $H$ serves as initial condition which then evolves in time according to some post-quench Hamiltonian $H_p$. Absence of thermalization is analytically demonstrated for a large class of quite common pre- and post-quench spin Hamiltonians. The main requirement is that the pre-quench Hamiltonian must exhibit a $Z_2$ (spin-flip) symmetry, which would be spontaneously broken in the thermodynamic limit, though we actual…
Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent high-temperature re…
The interplay of fluctuations, ergodicity, and disorder in many-body interacting systems has been striking attention for half a century, pivoted on two celebrated phenomena: Anderson localization predicted in disordered media, and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence observed in a nonlinear system. The destruction of Anderson localization by nonlinearity and the recovery of ergodicity after long enough computational times lead to more questions. This thesis is devoted to contributing to the insight of the nonlinear system dynamics in and out of equilibrium. Focusing mainly on the GP latt…
We employ equation of motion techniques to study the non-equilibrium dynamics in a lattice model of weakly interacting spinless fermions. Our model provides a simple setting for analyzing the effects of weak integrability breaking perturbations on the time evolution after a quantum quench. We establish the accuracy of the method by comparing results at short and intermediate times to time-dependent density matrix renormalization group computations. For sufficiently weak integrability-breaking interactions we always observe prethermalization plateaux, where local observables relax to non-therma…
We study dynamics of entanglement in the thermalization process of an isolated quantum many-body system. We propose a simple setup for measuring the propagation speed of entanglement entropy (EE) in numerical simulations and apply it to the integrable/non-integrable spin models in 1D - the transverse Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising (ECI) model. We find that two distinct time-scales $t^\ast$ and $t_{\rm {diff}}$ arise in the dynamics of EE in the thermalization process: the former represents the time-scale for the saturation of EE and the latter ch…
We numerically investigate the relaxation dynamics in an isolated quantum system of interacting bosons trapped in a double-well potential after an integrability breaking quench. Using the statistics of the spectrum, we identify the postquench Hamiltonian as nonchaotic and close to integrability over a wide range of interaction parameters. We demonstrate that the system exhibits thermalization in the context of the eigenstate thermalization hypothesis (ETH). We also explore the possibility of an initial state to delocalize with respect to the eigenstates of the postquench Hamiltonian even for e…
Novel cluster spin model with interactions and disorder is introduced and studied. In specific type of interactions, we find an extensive number of local integrals of motion (LIOMs), which are a modified version of the stabilizers in quantum information, i.e., mutually commuting operators specifying all quantum states in the system. These LIOMs can be defined for any strength of the interactions and disorder, and are of compact-support instead of exponentially-decaying tail. Hence, even under the presence of interactions, integrability is held, and all energy eigenstates are labeled by these L…
The light cone spreading of correlations following a quantum quench is obtained from first principles. Fully taking into account quantum and interaction effects, the derivation shows how light cone dynamics does not require peculiar properties of the post-quench state.
Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption, and compute the time dependence of main chaos indicators - Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermaliz…
Tripartite mutual information (TMI) is an efficient observable to quantify the ability of scrambler for unitary time-evolution operator with quenched many-body Hamiltonian. In this paper, we give numerical demonstrations of the TMI in disorder-free (translational invariant) spin models with 3-body and 4-body multiple-spin interactions. The dynamical behavior of the TMI of these models does {\it not} exhibit linear light-cone for sufficiently strong interactions. In early-time evolution, the TMI displays distinct negative increase behavior fitted by a logarithmic-like function. This is in contr…
Coherence-induced deep thermalization transition in random permutation quantum dynamics
National University of Singapore（新加坡国立大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
We examine the time evolution of the entanglement spectrum of a small subsystem of a non-integrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the "entanglement energy", and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on time…
The Eigenstate Thermalization Hypothesis (ETH) explains emergence of the thermodynamic equilibrium by assuming a particular structure of 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 physical operators can be described by RMT, more precisely at which energy scale strict RMT description applies, is however not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matri…
Thermalizing quantum systems are conventionally described by statistical mechanics at equilibrium. However, not all systems fall into this category, with many body localization providing a generic mechanism for thermalization to fail in strongly disordered systems. Many-body localized (MBL) systems remain perfect insulators at non-zero temperature, which do not thermalize and therefore cannot be described using statistical mechanics. In this Colloquium we review recent theoretical and experimental advances in studies of MBL systems, focusing on the new perspective provided by entanglement and …
It was recently observed that in certain thermalizing many-body systems, measuring the complement of a subsystem that thermalized to infinite temperature in a suitable orthonormal basis gives rise to approximate quantum state k-designs as post-measurement states on the subsystem. We prove that this emergence of approximate k-designs holds true in every large system where some small subsystem is close to being maximally mixed. On a technical level we show that any high-dimensional purification of an approximately maximally mixed state induces an approximate quantum state k-design when measured …
If and how an isolated quantum system thermalizes despite its unitary time evolution is a long-standing, open problem of many-body physics. The eigenstate thermalization hypothesis (ETH) postulates that thermalization happens at the level of individual eigenstates of a system's Hamiltonian. However, the ETH requires stringent conditions to be validated, and it does not address how the thermal state is reached dynamically from an inital non-equilibrium state. We consider a Bose-Einstein condensate (BEC) trapped in a double-well potential with an initial population imbalance. We find that the sy…
Universality, Robustness, and Limits of the Eigenstate Thermalization Hypothesis in Open Quantum Systems
Universidade de Lisboa（里斯本大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Technische Universität Dresden（德累斯顿工业大学）；University of Cambridge（剑桥大学）
The fully frustrated ladder - a quasi-1D geometrically frustrated spin one half Heisenberg model - is non-integrable with local conserved quantities on rungs of the ladder, inducing the fragmentation of the Hilbert space into sectors composed of singlets and triplets on rungs. We explore the far-from-equilibrium dynamics of this model through the entanglement entropy and out-of-time-ordered correlators (OTOC). The post-quench dynamics of the entanglement entropy is highly anomalous as it shows clear non-damped revivals that emerge from short connected chunks of triplets and whose persistence i…
We consider the time evolution of entanglement entropy after a global quench in a strongly coupled holographic system, whose subsequent equilibration is described in the gravity dual by the gravitational collapse of a thin shell of matter resulting in a black hole. In the limit of large regions of entanglement, the evolution of entanglement entropy is controlled by the geometry around and inside the event horizon of the black hole, resulting in regimes of pre-local- equilibration quadratic growth (in time), post-local-equilibration linear growth, a late-time regime in which the evolution does …
We derive in detail several universal features in the time evolution of entanglement entropy and other nonlocal observables in quenched holographic systems. The quenches are such that a spatially uniform density of energy is injected at an instant in time, exciting a strongly coupled CFT which eventually equilibrates. Such quench processes are described on the gravity side by the gravitational collapse of a thin shell that results in a black hole. Various nonlocal observables have a unified description in terms of the area of extremal surfaces of different dimensions. In the large distance lim…
We consider an open system near a quantum critical point that is suddenly moved towards the critical point. The bath-dominated diffusive non-equilibrium dynamics after the quench is shown to follow scaling behavior, governed by a critical exponent that emerges in addition to the known equilibrium critical exponents. We determine this exponent and show that it describes universal prethermalized coarsening dynamics of the order parameter in an intermediate time regime. Implications of this quantum critical prethermalization are a powerlaw rise of order and correlations after an initial collapse …
Probing many-body localization crossover in quasiperiodic Floquet circuits on a quantum processor
RIKEN Center for Computational Science（理化学研究所计算科学中心）；RIKEN Center for Quantum Computing（理化学研究所量子计算中心）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences（理化学研究所交叉理论与数学科学中心）；RIKEN Pioneering Research Institute（理化学研究所先驱研究所）；Jožef Stefan Institute（约热·斯特凡研究所）；University of Ljubljana（卢布尔雅那大学）
How quantum information is scrambled in the global degrees of freedom of non-equilibrium many-body systems is a key question to understand local thermalization. Here we propose that the scaling of the mutual information between two intervals of fixed length as a function of their distance is a diagnostic tool for scrambling after a quantum quench. We consider both integrable and non-integrable one dimensional systems. In integrable systems, the mutual information exhibits an algebraic decay with the distance between the intervals, signalling weak scrambling. This behavior may be qualitatively …
The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has lately received considerable attention. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits, aspects of which are conjectured to be universal to ergodic many-body systems, even without randomness. Here we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U$(1)$ charge and argue, with numerical and analytical e…
The central philosophy of statistical mechanics (stat-mech) and random-matrix theory of complex systems is that while individual instances are essentially intractable to simulate, the statistical properties of random ensembles obey simple universal "laws". This same philosophy promises powerful methods for studying the dynamics of quantum information in ideal and noisy quantum circuits -- for which classical description of individual circuits is expected to be generically intractable. Here, we review recent progress in understanding the dynamics of quantum information in ensembles of random qu…
