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

The β ensemble is a prototypical model of a single-particle system on a one-dimensional disordered lattice with inhomogeneous nearest-neighbor hopping. The corresponding nonergodic phase has an anomalous critical energy scale, Ec: Correlations are present above and absent below Ec, 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 tR≡Ec−1, which is parametrically smaller than the Heisenberg time, tH, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, τR≡tR/tH≪1 is equal to the Dyson index, β. We show that the energy correlations are absent within a temporal window tRttH, which we term as the . 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.