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arXiv:2107.04353 · Physical review. B./Physical review. B

Probing the metallic energy spectrum beyond the Thouless energy scale using singular value decomposition

Richard Berkovits

Published Aug 23, 2021• 17 citations
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Abstract

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.

DOI
10.1103/physrevb.104.054207

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•Jan 12, 2020•1 citation
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.
In recent years, it became clear that the metallic regime of systems that exhibit a many-body localization (MBL) behavior shows properties that are quite different than the vanilla metallic region of the single-particle Anderson regime. Here we show that the large-scale energy spectrum of a canonical microscopical model featuring MBL displays a nonuniversal behavior at intermediate scales, which is distinct from the deviation from universality seen in the single-particle Anderson regime. The crucial step in revealing this behavior is a global unfolding of the spectrum performed using the singular value decomposition (SVD) which takes into account the sample to sample fluctuations of the spectra. The spectrum properties may be observed directly in the singular value amplitudes via the scree plot, or by using the SVD to unfold the spectra and then perform a number of states variance calculation. Both methods reveal an intermediate scale of energies that follow super-Poissonian statistics.
We employ singular value decomposition (SVD) to study the eigenvalue spectra of random spin systems. By SVD, eigenvalue spectrum is decomposed into orthonormal modes Wk{W}_{k}Wk​ with weight ${\ensuremath{\lambda}}_{k}$. We show that the scree plot (${\ensuremath{\lambda}}_{k}$ with respect to kkk) in the ergodic phase contains two branches that both follow power law ${\ensuremath{\lambda}}_{k}\ensuremath{\sim}{k}^{\ensuremath{-}\ensuremath{\alpha}}$ but with different exponents $\ensuremath{\alpha}$. By evaluating Wk{W}_{k}Wk​, we verified the part of ${\ensuremath{\lambda}}_{k}$ with k>kThk>{k}_{\text{Th}}k>kTh​ is universal that follows random matrix theory, where kTh{k}_{\text{Th}}kTh​ is related to the Thouless energy. We further demonstrate that $\ensuremath{\alpha}$ corresponds only to the exponential part of the level spacing distribution while being insensitive to the level repulsion, or equivalently the system's symmetry. Consequently, $\ensuremath{\alpha}$ gives an underestimation for the many-body localization transition point, which suggests a nonergodic behavior that may be attributed to the Griffiths regime.
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.
The short time (large energy) behavior of the Sachdev-Ye-Kitaev model (SYK) is one of the main reasons for the growing interest garnered by this model. True chaotic behavior sets in at the Thouless time, which can be extracted from the energy spectrum. In order to do so, it is necessary to unfold the spectrum, i.e., to filter out global tendencies. Using a simple ensemble average for unfolding results in a parametically low estimation of the Thouless energy. By examining the behavior of the spectrum as the distribution of the matrix elements is changed into a log-normal distribution, it is shown that the sample-to-sample level spacing variance determines this estimation of the Thouless energy. Using the singular value decomposition method, which filters out these global sample-to-sample fluctuations, the Thouless energy becomes parametrically much larger, essentially of the order of the band width. It is shown that the SYK model is non-self-averaging even in the thermodynamic limit which must be taken into account in considering its short time properties.
The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a nonergodic extended phase between the ergodic extended metallic phase and the localized phase. Such a phase is relevant to many physical models from the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity to the interacting many-body localization in condensed-matter physics and quantum computing. This phase is characterized by fractal behavior of the wave functions and a postulated correlated miniband structure of the energy spectrum. Here we will seek evidence for the latter in the spectrum. Since this behavior is expected on intermediate energy scales, spectral rigidity or number variance is a natural way to tease it out. Nevertheless, due to the Thouless energy and ambiguities in the unfolding procedure, the results are inconclusive. On the other hand, by using the singular-value decomposition method, clear evidence for a super-Poissonian behavior in this regime emerges, consistent with a picture of correlated minibands.
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.