Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble

Abstract

The $\beta$-ensemble is a prototypical model of a single particle system on a one-dimensional disordered lattice with inhomogeneous nearest neighbor hopping. Corresponding nonergodic phase has an anomalous critical energy scale, $E_c$: correlations are present above and absent below $E_c$ as reflected in the number variance. We study the dynamical properties of the $\beta$-ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time, $t_{\mathrm{R}}\equiv E_c^{-1}$, which is parametrically smaller than the Heisenberg time, $t_{\mathrm{H}}$, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, ${\tau }_{\mathrm{R}}\equiv t_{\mathrm{R}}/t_{\mathrm{H}}\ll 1$ is equal to the Dyson index, $\beta$. We show that the energy correlations are absent within a temporal window $t_{\mathrm{R}}<t<t_{\mathrm{H}}$, which we term as the correlation void. This is in contrast to the mechanism of equilibration in a typical many-body system. We analytically explain the qualitative behavior of the number variance and the spectral form factor of the $\beta$-ensemble by a spatially local mapping to the Anderson model.

11 pages, 8 figures

References in corpus (42)

• Thermalization and its mechanism for generic isolated quantum systems
• Anderson Transitions
• Many-body localization edge in the random-field Heisenberg chain
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice
• Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
• Chaos, Complexity, and Random Matrices
• Anomalous thermalization in ergodic systems
• Introduction to Random Matrices - Theory and Practice
• From Random Matrices to Stochastic Operators
• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• Dynamics of magnetization at infinite temperature in a Heisenberg spin chain
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• The Lévy-Rosenzweig-Porter random matrix ensemble
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• Nonexponential fidelity decay in isolated interacting quantum systems
• Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase
• Many-body quantum chaos in stroboscopically-driven cold atoms
• Measuring Spectral Form Factor in Many-Body Chaotic and Localized Phases of Quantum Processors
• Dynamical quantum ergodicity from energy level statistics
• Scale-Invariant Survival Probability at Eigenstate Transitions
• Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States
• Scale-invariant critical dynamics at eigenstate transitions
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Tuning the phase diagram of a Rosenzweig-Porter model with fractal disorder
• Probing multi-mobility edges in quasiperiodic mosaic lattices
• Random Cantor sets and mini-bands in local spectrum of quantum systems
• Uncovering Local Integrability in Quantum Many-Body Dynamics
• Disorder-tunable entanglement at infinite temperature
• Proposal for many-body quantum chaos detection
• Coexistence of extended and localized states in finite-sized mosaic Wannier-Stark lattices
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Chaos due to symmetry-breaking in deformed Poisson ensemble
• Electronic states of a disordered 2D quasiperiodic tiling: from critical states to Anderson localization
• Robust non-ergodicity of ground state in the $\beta$ ensemble
• Reducing dynamical fluctuations and enforcing self-averaging by opening many-body quantum systems
• Finite-Range Coulomb Gas Models of Banded Random Matrices and Quantum Kicked Rotors
• Transport in deformed centrosymmetric networks
• Eigenstate Correlations in Dual-Unitary Quantum Circuits: Partial Spectral Form Factor
• Quantum multifractality as a probe of phase space in the Dicke model
• Emergent multifractality in power-law decaying eigenstates
• Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos

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