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.
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.
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.
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 with weight ${\ensuremath{\lambda}}_{k}$. We show that the scree plot (${\ensuremath{\lambda}}_{k}$ with respect to k) 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, we verified the part of ${\ensuremath{\lambda}}_{k}$ with k>kTh is universal that follows random matrix theory, where 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.
Motivated by recent debates around the many-body localization (MBL) problem, and in particular its stability against systemwide resonances, we investigate long-distance spin-spin correlations across the phase diagram of the random-field XXZ model, with a particular focus on the strong disorder regime. Building on state-of-the-art shift-invert diagonalization techniques, we study the high-energy behavior of transverse and longitudinal correlation functions, computed at the largest possible distance, for a broad range of disorder and interaction strengths. Our results show that while transverse correlations display a fairly stable exponential decay over the entire XXZ phase diagram, longitudinal correlations exhibit markedly different behavior, revealing distinct physical regimes. More precisely, we identify an intermediate disorder region where standard observables show well-converged MBL behavior [Colbois et al., Phys. Rev. Lett. 133, 116502 (2024)] while the distributions of longitudinal correlations reveal unexpected fat tails toward large values. These rare events strongly influence the average decay of longitudinal correlations, which we find to be algebraic in a broad region inside the supposed MBL phase, whereas the typical decay remains mostly exponential. At stronger disorder and weaker interactions, this intermediate regime is replaced by a more conventional exponential decay with short correlation lengths for both typical and average correlators, as expected for standard localization. Our findings shed light on the systemwide instabilities and raise important questions about the impact of such rare but large long-range correlations on the stability of the MBL phase. Finally, we discuss the possible fate of the intermediate region in the context of recent perspectives in the field.
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.
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.