Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they become Poissonian.
Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they become Poissonian.
Since the seminal work of Anderson, localization has been recognized as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea has acquired even more prominence in the last decade as it has been argued that localization---dubbed many-body localization (MBL) in this context---can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localization is in the statistical properties of the spectrum---spectral statistics---which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localization, their derivation from first principles in the presence of nontrivial interactions has been lacking. Here we provide the example of a simple class of quantum many-body systems---which we dub strongly localized quantum circuits---that are interacting and localized and where the spectral statistics can be characterized exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale do they become Poissonian.
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.
Recently it has been suggested that many-body localization (MBL) can occur in translationally-invariant systems, and candidate 1D models have been proposed. We find that such models typically exhibit strong finite-size effects due to the presence of two or more vastly different energy scales, including an artificial separation of density of states into bands separated by large energy gaps. By carefully choosing the parameters of the models such that finite-size effects are minimized, we perform extensive numerical studies of small systems using exact diagonalization. Based on the variety of diagnostics, including entanglement properties and the behavior of local observables, we find the systems exhibit thermal (ergodic), rather than MBL-like behavior. Our results suggest that MBL in translationally-invariant systems with two very different energy scales is less robust than perturbative arguments suggest, possibly pointing to the importance of non-perturbative effects which induce delocalization in the thermodynamic limit.
This article reviews recent progress in understanding the physics of many-body localisation (MBL) in disordered and interacting quantum many-body systems, from the perspective of ergodicity breaking on the associated Fock space. This approach to MBL is underpinned by mapping the dynamics of the many-body system onto that of a fictitious single particle on the high-dimensional, correlated and disordered Fock-space graph; yet, as we elaborate, the problem is fundamentally different from that of conventional Anderson localisation on high-dimensional or hierarchical graphs. We discuss in detail the nature of eigenstate correlations on the Fock space, both static and dynamic, and in the ergodic and many-body localised phases as well as in the vicinity of the MBL transition. The latter in turn sheds light on the nature of the transition, and motivates a scaling theory for it in terms of Fock-space based quantities. We also illustrate how these quantities can be concretely connected to real-space observables. An overview is given of several analytical and numerical techniques which have proven important in developing a comprehensive picture. Finally, we comment on some open questions in the field of MBL where the Fock-space approach is likely to prove insightful.
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
We use numerical exact diagonalization to analyze which aspects of the many-body localization phenomenon survive in an imperfectly isolated setting, when the system of interest is weakly coupled to a thermalizing environment. We show that widely used diagnostics (such as many-body level statistics and expectation values in exact eigenstates) cease to show signatures of many-body localization above a critical coupling that is exponentially small in the size of the environment. However, we also identify alternative diagnostics for many-body localization, in the spectral functions of local operators. Diagnostics include a discrete spectrum and a hierarchy of energy gaps, including a universal gap at zero frequency. These alternative diagnostics are shown to be robust, and continue to show signatures of many-body localization as long as the coupling to the bath is weaker than the characteristic energy scales in the system. We also examine how these signatures disappear when the coupling to the environment becomes larger than the characteristic energy scales of the system.