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# Self-averaging behavior at the metal-insulator transition of many-body quantum systems out of equilibrium .

TL;DR: In this article, the authors consider two local and two non-local quantities in real space and show that their self-averaging behavior is highly dependent on the observable itself and on the time scale, but the picture simplifies substantially as they approach localization.

Abstract: An observable of a disordered system is self-averaging when its properties do not depend on the specific realization considered. Lack of self-averaging, on the other hand, implies that sample to sample fluctuations persist no matter how large the system is. The latter scenario is often found in the vicinity of critical points, such as at the metal-insulator transition of interacting many-body quantum systems. Much attention has been devoted to these systems at equilibrium, but little is known about their self-averaging behavior out of equilibrium, which is the subject of this work. We consider two local and two non-local quantities in real space that are of great experimental and theoretical interest. In the metallic phase, we show that their self-averaging behavior is highly dependent on the observable itself and on the time scale, but the picture simplifies substantially as we approach localization. In this phase, the local quantities are self-averaging at any time, while the non-local ones are non-self-averaging at all time scales.

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TL;DR: Dynamical signatures of quantum chaos are studied in one of the most relevant models in many-body quantum mechanics, the Bose-Hubbard model, whose high degree of symmetries yields a large number of invariant subspaces and degenerate energy levels.

Abstract: We study dynamical signatures of quantum chaos in one of the most relevant models in many-body quantum mechanics, the Bose-Hubbard model, whose high degree of symmetries yields a large number of invariant subspaces and degenerate energy levels. The standard procedure to reveal signatures of quantum chaos requires classifying the energy levels according to their symmetries, which may be experimentally and theoretically challenging. We show that this classification is not necessary to observe manifestations of spectral correlations in the temporal evolution of the survival probability, which makes this quantity a powerful tool in the identification of chaotic many-body quantum systems.

29 citations

### Cites background from "Self-averaging behavior at the meta..."

...The temporal evolution of the survival probability is naturally fuzzy due to quantum fluctuations, whose relative size compared with their average value do not diminish with the dimension of the system, they are not self-averaging at any scale [24, 27]....

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TL;DR: In this paper, the authors show that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices, and they also analyze the participation ratio, Renyi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiment with ion traps.

Abstract: Despite its importance to experiments, numerical simulations, and the development of theoretical models, self-averaging in many-body quantum systems out of equilibrium remains underinvestigated. Usually, in the chaotic regime, self-averaging is taken for granted. The numerical and analytical results presented here force us to rethink these expectations. They demonstrate that self-averaging properties depend on the quantity and also on the time scale considered. We show analytically that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices. We also analyze the participation ratio, Renyi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiments with ion traps. We find that self-averaging holds at short times for the quantities that are local in space, while at long times, self-averaging applies for quantities that are local in time. Various behaviors are revealed at intermediate time scales.

29 citations

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TL;DR: In this article, the authors consider the specific effects of a bias on anomalous diffusion, and discuss the generalizations of Einstein's relation in the presence of disorder, and illustrate the theoretical models by describing many physical situations where anomalous (non-Brownian) diffusion laws have been observed or could be observed.

3,383 citations

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TL;DR: In this paper, the authors provide a brief introduction to quantum thermalization, paying particular attention to the eigenstate thermalization hypothesis (ETH) and the resulting single-eigenstate statistical mechanics.

Abstract: 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...

1,945 citations

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TL;DR: This experiment experimentally observed this nonergodic evolution for interacting fermions in a one-dimensional quasirandom optical lattice and identified the MBL transition through the relaxation dynamics of an initially prepared charge density wave.

Abstract: Many-body localization (MBL), the disorder-induced localization of interacting particles, signals a breakdown of conventional thermodynamics because MBL systems do not thermalize and show nonergodic time evolution. We experimentally observed this nonergodic evolution for interacting fermions in a one-dimensional quasirandom optical lattice and identified the MBL transition through the relaxation dynamics of an initially prepared charge density wave. For sufficiently weak disorder, the time evolution appears ergodic and thermalizing, erasing all initial ordering, whereas above a critical disorder strength, a substantial portion of the initial ordering persists. The critical disorder value shows a distinctive dependence on the interaction strength, which is in agreement with numerical simulations. Our experiment paves the way to further detailed studies of MBL, such as in noncorrelated disorder or higher dimensions.

1,454 citations

### "Self-averaging behavior at the meta..." refers methods or result in this paper

...The spin autocorrelation function is equivalent to the density imbalance used in experiments with cold atoms [30] and the connected spin-spin correlation function is measured in experiments with ion traps [31]....

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...This quantity is similar to the density imbalance between even and odd sites measured in experiments with cold atoms [30]....

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TL;DR: In this paper, exact diagonalization is used to explore the many-body localization transition in a random-field spin-1/2 chain, showing that this quantum phase transition at nonzero temperature might be showing infinite-randomness scaling with a dynamic critical exponent.

Abstract: We use exact diagonalization to explore the many-body localization transition in a random-field spin-1/2 chain. We examine the correlations within each many-body eigenstate, looking at all high-energy states and thus effectively working at infinite temperature. For weak random field the eigenstates are thermal, as expected in this nonlocalized, ``ergodic'' phase. For strong random field the eigenstates are localized with only short-range entanglement. We roughly locate the localization transition and examine some of its finite-size scaling, finding that this quantum phase transition at nonzero temperature might be showing infinite-randomness scaling with a dynamic critical exponent $z\ensuremath{\rightarrow}\ensuremath{\infty}$.

1,270 citations

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TL;DR: This work applies a variable-range Ising spin chain Hamiltonian and aVariable-range XY spin chainHamiltonian to a far-from-equilibrium quantum many-body system and observes its time evolution, determining the spatial and time-dependent correlations, extracting the shape of the light cone and measuring the velocity with which correlations propagate through the system.

Abstract: The maximum speed with which information can propagate in a quantum many-body system directly affects how quickly disparate parts of the system can become correlated and how difficult the system will be to describe numerically. For systems with only short-range interactions, Lieb and Robinson derived a constant-velocity bound that limits correlations to within a linear effective ‘light cone’. However, little is known about the propagation speed in systems with long-range interactions, because analytic solutions rarely exist and because the best long-range bound is too loose to accurately describe the relevant dynamical timescales for any known spin model. Here we apply a variable-range Ising spin chain Hamiltonian and a variable-range XY spin chain Hamiltonian to a far-from-equilibrium quantum many-body system and observe its time evolution. For several different interaction ranges, we determine the spatial and time-dependent correlations, extract the shape of the light cone and measure the velocity with which correlations propagate through the system. This work opens the possibility for studying a wide range of many-body dynamics in quantum systems that are otherwise intractable.

669 citations