In this Thesis, we investigate problems of thermalization in closed (or isolated) and open quantum many-body systems, which are deeply tied to the foundations of quantum statistical mechanics.Such studies have attracted growing interest thanks to recent experimental realizations of controllable quantum many-body systems, e.g., cold atoms and ions.We firstly discuss thermalization in closed quantum many-body systems.The eigenstate thermalization hypothesis (ETH) has been proposed as a sufficient condition for a system to be described by the microcanonical ensemble after a long-time dynamics.The ETH is expected to hold true for generic systems except for some systems such as many-body localized (MBL) systems with strong disorder.On the other hand, the reason why the ETH generically holds truehas not yet been fully understood.We here rigorously show that the typicality argument, which has long been thought to justify the ETH, does not hold true for a realistic setup with few-body properties of observables and Hamiltonians.We secondly discuss thermalization in open quantum many-body systems.State-of-the-art experiments (e.g., experiments under continuous measurement) have now enabled us to control dissipation, which may enrich properties of closed systems.As one of the fundamental problems of thermalization in open quantum systems, we study the MBL in non-Hermitian systems, which are relevant for continuously measured systems.We show that the transition between delocalized (ETH) and MBL phases occurs even for non-Hermitian many-body Hamiltonians and affects spectral and dynamical properties.In particular, we find that a novel real-complex transition occurs upon MBL and affects the dynamical stability in non-Hermitian interacting systems with asymmetric hopping.Just as delocalized phases in closed systems are characterized by Hermitian random matrices, those in open quantum systems are characterized by non-Hermitian random matrices.On the other hand, in contrast with the Hermitian cases, the universality in non-Hermitian random matrices and its relation to symmetries are not well understood.In Hermitian random matrices, there are three universality classes of local spectral statistics dependent on time-reversal symmetry (TRS).However, TRS is known not to alter universality in non-Hermitian random matrices.We here show that three different universal level-spacing distributions appear even for non-Hermitian random matrices when we consider transposition symmetry instead of TRS.This result serves as a basis for characterizing nonintegrability and delocalization in open quantum systems with symmetry.asymmetric hopping . . . .
In this Thesis, we investigate problems of thermalization in closed (or isolated) and open quantum many-body systems, which are deeply tied to the foundations of quantum statistical mechanics.Such studies have attracted growing interest thanks to recent experimental realizations of controllable quantum many-body systems, e.g., cold atoms and ions.We firstly discuss thermalization in closed quantum many-body systems.The eigenstate thermalization hypothesis (ETH) has been proposed as a sufficient condition for a system to be described by the microcanonical ensemble after a long-time dynamics.The ETH is expected to hold true for generic systems except for some systems such as many-body localized (MBL) systems with strong disorder.On the other hand, the reason why the ETH generically holds truehas not yet been fully understood.We here rigorously show that the typicality argument, which has long been thought to justify the ETH, does not hold true for a realistic setup with few-body properties of observables and Hamiltonians.We secondly discuss thermalization in open quantum many-body systems.State-of-the-art experiments (e.g., experiments under continuous measurement) have now enabled us to control dissipation, which may enrich properties of closed systems.As one of the fundamental problems of thermalization in open quantum systems, we study the MBL in non-Hermitian systems, which are relevant for continuously measured systems.We show that the transition between delocalized (ETH) and MBL phases occurs even for non-Hermitian many-body Hamiltonians and affects spectral and dynamical properties.In particular, we find that a novel real-complex transition occurs upon MBL and affects the dynamical stability in non-Hermitian interacting systems with asymmetric hopping.Just as delocalized phases in closed systems are characterized by Hermitian random matrices, those in open quantum systems are characterized by non-Hermitian random matrices.On the other hand, in contrast with the Hermitian cases, the universality in non-Hermitian random matrices and its relation to symmetries are not well understood.In Hermitian random matrices, there are three universality classes of local spectral statistics dependent on time-reversal symmetry (TRS).However, TRS is known not to alter universality in non-Hermitian random matrices.We here show that three different universal level-spacing distributions appear even for non-Hermitian random matrices when we consider transposition symmetry instead of TRS.This result serves as a basis for characterizing nonintegrability and delocalization in open quantum systems with symmetry.asymmetric hopping . . . .
We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the eigenstate thermalization hypothesis (ETH) and the resulting single-eigenstate statistical mechanics. We then focus on a class of systems that fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson-localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can forever locally remember information about their local initial conditions and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL) and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveal dynamically stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and can occur at high energy and low spatial dimensionality, where equilibrium ordering is forbidden.
The eigenstate thermalization hypothesis (ETH) has been highly influential in explaining thermodynamic behavior of closed quantum systems. As of yet, it is unclear whether and how the ETH applies to non-Hermitian systems. Here, we introduce a framework that extends the ETH to non-Hermitian systems, within which expectation values of local operators reproduce statistical and scaling predictions known from Hermitian ETH. We illustrate the validity of the framework on non-Hermitian random-matrix and Sachdev-Ye-Kitaev models. Further, we show numerically how the static ETH predictions become imprinted onto the dynamics of local observables. Finally, we present a prescription for observing both ETH-obeying and ETH-violating regimes in an optical-lattice experiment that implements a disordered interacting Hatano-Nelson model. Our results generalize the celebrated ETH to the non-Hermitian setting, and they show how it affects the system dynamics, and how the salient signatures can be observed in present-day cold-atom experiments.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
The emergence of statistical mechanics from quantum dynamics is conventionally understood through quantum chaos, the Eigenstate Thermalization Hypothesis (ETH), and canonical typicality, which together explain the thermal behavior of local observables in quantum many-body systems. We provide a distinct mechanism in which, for a quantum many-body system under strong random perturbation, the chaotic behavior of the system's eigenvectors and a constraint on the total energy directly generate a canonical ensemble for the entire system. The canonical ensemble occurs as the first moment of the Porter-Thomas distribution of the system's chaotic eigenstates, and the constraint on the total energy sets the temperature in the ensemble. This yields canonically distributed expectation values without invoking ETH, a microcanonical ensemble, or a subsystem-bath partition for local and global observables. We numerically demonstrate the mechanism for the 1D transverse-field Ising model by showing canonically distributed eigenstate projectors and spin-spin correlators with distinctive temperature dependence of the correlation length. An implementation of this thermalization approach on a quantum computer can be utilized for thermal state preparation.
"Under certain conditions, an interacting system can defy the concept of thermalization, a keystone in our understanding of physical processes. Many-body localization (MBL) is a phase of matter in which thermalization does not apply and ergodicity is broken. This striking behavior, which can appears on closed, interacting, quantum systems subject to strong disorder, has been the focus of a large body of theoretical, numerical, and experimental work in recent years. In this thesis we numerically study several aspects of MBL and the ergodic-MBL transition. Chapter 1 introduces the concept of thermalization in quantum systems, which relies on the idea of the eigenstate thermalization hypothesis (ETH). I then present localization as a phenomenon that breaks ETH, both in its single-particle as in its many-body versions. I discuss some of the main aspects that are known about MBL, as well as some of its open questions, some of which we tackle in later chapters. Chapter 2 discusses the main numerical methods used in this thesis for the study of MBL. I provide both theoretical background as well as discuss some more practical matters, such as their advantages and disadvantages, or details on the software developed/used in my studies. In Chapter 3 I present our work on MBL from the point of view of single-particle orbitals. In this work we access a complete set of approximate integrals of motion of a one-dimensional system by computing the one-particle orbitals (OPO) of highly-excited MBL energy eigenstates, which are obtained through the shift-and-invert matrix product state (SIMPS) algorithm. We then study the properties of the OPOs over large systems, up to L = 64. We find that the OPOs drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the OPOs decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find that the probability distribution of the strength of the large-range coupling constants of the number operators generated by the OPOs approach a log-uniform distribution at strong disorder. In Chapter 4 I present our work on the hybridization of eigenstates in either the MBL or the ergodic phase, as well as at the transition. We do so by adiabatially evolving highly excited eigenstates of the Hamiltonian and measuring their their hybridization process with other eigenstates of the system. This hybridization, which dresses the eigenstate and has the potential of bringing it out of the MBL phase through the transition, is a consequence of the ""collisions"" of eigenstates in energy, which avoid level crossings every time their energy gap is small. The hybridization of eigenstates with each other involves only local regions of the system in the MBL phase, ignores locality in the ergodic phase, and is range-independent at the transition. This range independence suggests the proliferation of long-range resonances at the transition, as well as the divergence of a localization length. In Chapter 5 I present our studies on the typical and extreme (atypically strong) correlations across a one-dimensional system in the ergodic-MBL phase diagram. While typical correlations decay exponentially with range in the MBL phase, in the ergodic phase they are constant and independent of the range. Surprisingly, we identify a moderate region of the phase in which typical correlations decay as a stretched exponential with range r, and in particular as exp[-r^(1/2)] at the transition, a decay that is reminiscent of the random singlet phase. Moreover, at the transition the distribution of the logarithm of the correlations show vanishing even excess moments and non-zero range-invariant odd excess moments. This distinct behavior at the transition is in contrast with ergodic and MBL phenomenologies. In addition, we study the extreme correlations in the system. Our results suggest that strong long-range correlations proliferate at the transition, in contrast with a decay with range in the MBL phase and the lack of strong long-range correlations in the ergodic phase. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, which has been proposed as a robust order parameter in the ergodic-MBL phase diagram. We find that this probability is non-zero deep in the MBL phase, but vanishes at moderate disorder, well above the transition, thus not providing a proper order parameter. Chapter 6 I summarize the work presented with an emphasis on context and perspective."