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

Basudha Roy; Adway Kumar Das; Anandamohan Ghosh; Ivan M. Khaymovich

doi:10.1103/s2d6-wkz9

paper

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

arXiv:2506.15283 · doi:10.1103/s2d6-wkz9

Abstract

The $β$ -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 $β$ -ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time, $t_{R} \equiv E_{c}^{- 1}$ , which is parametrically smaller than the Heisenberg time, $t_{H}$ , given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, $τ_{R} \equiv t_{R} / t_{H} ≪ 1$ is equal to the Dyson index, $β$ . We show that the energy correlations are absent within a temporal window $t_{R} < t < t_{H}$ , 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 $β$ -ensemble by a spatially local mapping to the Anderson model.

arXiv comment11 pages, 8 figures

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Anomalous energy correlations and spectral form factor in the nonergodic phase of the β-ensemble

Abstract

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Cited by in corpus (1)

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