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arXiv:1606.08419 · Physical review. B./Physical review. B

Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition

Corentin Bertrand, Antonio M. Garcı́a-Garcı́a

Published Oct 4, 2016• 130 citations
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Abstract

We study a one-dimensional XXZ spin chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder-dependent exponent. This is consistent with the existence of an anomalous Thouless energy in the spectrum. In noninteracting disordered metals, this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case, it seems related to a more intricate anomalous diffusion process. This interpretation is not fully consistent with recent claims that for intermediate disorder, level statistics are described by a plasma model with power-law decaying interactions whose number variance grows slower than linear. As disorder is further increased, still on the metallic side, the Thouless energy is gradually washed out. In the range of sizes we can explore, level statistics are scale invariant and approach Poisson statistics at the many-body localization transition. Slightly below the many-body localization transition, spectral correlations, well described by critical statistics, are quantitatively similar to those of a high-dimensional, noninteracting, disordered conductor at the Anderson transition.

DOI
10.1103/physrevb.94.144201

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We study a one-dimensional XXZ spin chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder-dependent exponent. This is consistent with the existence of an anomalous Thouless energy in the spectrum. In noninteracting disordered metals, this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case, it seems related to a more intricate anomalous diffusion process. This interpretation is not fully consistent with recent claims that for intermediate disorder, level statistics are described by a plasma model with power-law decaying interactions whose number variance grows slower than linear. As disorder is further increased, still on the metallic side, the Thouless energy is gradually washed out. In the range of sizes we can explore, level statistics are scale invariant and approach Poisson statistics at the many-body localization transition. Slightly below the many-body localization transition, spectral correlations, well described by critical statistics, are quantitatively similar to those of a high-dimensional, noninteracting, disordered conductor at the Anderson transition.
Within the Thouless energy scale, many-body Hamiltonians behave like random matrices. We study the Thouless energy of the disordered Heisenberg quantum spin chain. This model has become the standard model for the study of many-body localization and the transition to an ergodic phase for weak disorder. We focus on the ergodic side of the Many-Body Localization transition. The Thouless energy extracted from long range spectral statistics and from the power spectrum of the fluctuations of the full momentum distribution coincide. This scale behaves non-monotonically and is peaked at an intermediate value of the disorder. An analysis of the scaling behavior of the Thouless energy and its comparison with the number of populated eigenstates after a quench suggests that this scale is not large enough to guarantee thermalization in the thermodynamic limit.
The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.
We study the ergodic side of the many-body localization transition in its standard model, the disordered Heisenberg quantum spin chain. We show that the Thouless energy, extracted from long-range spectral statistics and the power-spectrum of the full momentum distribution fluctuations, is not large enough to guarantee thermalization. We find that both estimates coincide and behave nonmonotonically, exhibiting a strong peak at an intermediate value of the disorder. Furthermore, we show that nonthermalizing initial conditions occur well within the ergodic phase with larger probability than expected. Finally, we propose a mechanism, driven by the Thouless energy and the presence of anomalous events, for the transition to the localized phase.
Disordered quantum systems feature an energy scale known as the Thouless energy. For energy ranges below this scale, the properties of the energy spectrum can be described by random matrix theory. Above this scale a different behavior sets in. For a metallic system it was shown long ago by Altshuler and Shklovskii [Sov. Phys. JETP 64, 127 (1986)] that the number variance should increase as a power law with power dependent on only the dimensionality of the system. Although tantalizing hints at this behavior were seen in previous numerical studies, it is quite difficult to verify this prediction using the standard local unfolding methods. Here we use a different unfolding method, i.e., singular value decomposition, and establish a connection between the power law behavior of the scree plot (the singular values ranked by their amplitude) and the power law behavior of the number variance. Thus, we are able to numerically verify Altshuler and Shklovskii's prediction for disordered three-, four-, and five-dimensional single-electron Anderson models on square lattices in the metallic regime. The same method could be applied to systems such as the Sachdev-Ye-Kitaev model and various interacting many-body models for which the many-body localization occurs. It was recently reported that such systems exhibit a Thouless energy, and analyzing the spectrum's behavior on larger scales is of much current interest.
Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered J_1 J 1 - J_2 J 2 model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.