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arXiv:2302.00703 · SciPost Physics

Spectral statistics of a minimal quantum glass model

Richard Barney, Michael Winer, Christopher L. Baldwin, Brian G. Swingle, Victor M. Galitski

Published Sep 7, 2023• 13 citations
arXivHTMLPDF

Abstract

Glasses have the interesting feature of being neither integrable nor fully chaotic. They thermalize quickly within a subspace but thermalize much more slowly across the full space due to high free energy barriers which partition the configuration space into sectors. Past works have examined the Rosenzweig-Porter (RP) model as a minimal quantum model which transitions from localized to chaotic behavior. In this work we generalize the RP model in such a way that it becomes a minimal model which transitions from glassy to chaotic behavior, which we term the “Block Rosenzweig-Porter” (BRP) model. We calculate the spectral form factors of both models at all timescales larger than the inverse spectral width. Whereas the RP model exhibits a crossover from localized to ergodic behavior at the Thouless timescale, the new BRP model instead crosses over from glassy to fully chaotic behavior, as seen by a change in the steepness of the ramp of the spectral form factor.

DOI
10.21468/scipostphys.15.3.084

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Glasses have the interesting feature of being neither integrable nor fully chaotic. They thermalize quickly within a subspace but thermalize much more slowly across the full space due to high free energy barriers which partition the configuration space into sectors. Past works have examined the Rosenzweig-Porter (RP) model as a minimal quantum model which transitions from localized to chaotic behavior. In this work we generalize the RP model in such a way that it becomes a minimal model which transitions from glassy to chaotic behavior, which we term the “Block Rosenzweig-Porter” (BRP) model. We calculate the spectral form factors of both models at all timescales larger than the inverse spectral width. Whereas the RP model exhibits a crossover from localized to ergodic behavior at the Thouless timescale, the new BRP model instead crosses over from glassy to fully chaotic behavior, as seen by a change in the steepness of the ramp of the spectral form factor.
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy ETE_TET​, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around ETE_TET​, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10−4010^{-40}10−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
The Rosenzweig-Porter model is a single-parameter random matrix ensemble that supports an ergodic, fractal, and localized phase. The names of these phases refer to the properties of the (midspectrum) eigenstates. This work focuses on the long-range spectral statistics of the recently introduced unitary equivalent of this model. By numerically studying the Thouless time obtained from the spectral form factor, it is argued that long-range spectral statistics can be used to probe the transition between the ergodic and the fractal phases. The scaling of the Thouless time as a function of the model parameters is found to be similar to the scaling of the spreading width of the eigenstates. Provided that the transition between the fractal and the localized phases can be probed through short-range level statistics, such as the average ratio of consecutive level spacings, this work establishes that spectral statistics are sufficient to probe both transitions present in the phase diagram. Published by the American Physical Society 2024
Abstract It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations — the exponential of the spin glass “complexity” as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.
Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the $\ensuremath{\beta}$ ensemble, known for its structural simplicity. However, eigenvector properties of the $\ensuremath{\beta}$ ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the $\ensuremath{\beta}$ ensemble and find that the Anderson transition occurs at $\ensuremath{\gamma}=1$ and ergodicity breaks down at $\ensuremath{\gamma}=0$ if we express the repulsion parameter as $\ensuremath{\beta}={N}^{\ensuremath{-}\ensuremath{\gamma}}$. Thus other than the Rosenzweig-Porter ensemble (RPE), the $\ensuremath{\beta}$ ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values $(0<\ensuremath{\gamma}<1)$. We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the $\ensuremath{\beta}$ ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the $\ensuremath{\beta}$ ensemble behave differently than the latter models.
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.
One of the central problems in many-body physics, both classical and quantum, is the relations between different notions of chaos. Ergodicity, mixing, operator growth, the eigenstate thermalization hypothesis, and spectral chaos are defined in terms of completely different objects in different contexts, don't necessarily co-occur, but still seem to be manifestations of closely related phenomena. In this dissertation, we study the relation between two notions of chaos: thermalization and spectral chaos. We define a quantity called the Total Return Probability (TRP) which measures how a system forgets its initial state after time TTT, and show that it is closely connected to the Spectral Form Factor (SFF), a measure of chaos deriving from the energy level spectrum of a quantum system. The main thrust of this work concerns hydrodynamic systems -- systems where locality prevents charge or energy from spreading quickly, this putting a throttle on thermalization. We show that the detailed spacings of energy levels closely capture the dynamics of these locally conserved charges. We also study spin glasses, a phase of matter where the obstacle to thermalization comes not from locality but from the presence of too many neighbors. Changing one region requires changing nearby regions which requires changing nearby-to-nearby regions, until only catastrophic realignments of the whole system can fully explore phase space. In spin glasses we find our clearest analytic link between thermalization and spectral statistics. We analytically calculate the spectral form factor in the limit of large system size and show it is equal to the TRP. Finally, in the conclusion, we discuss some ideas for the future of both the SFF and the TRP.