Identifying mobility edge from finite temperature spectral form factor

Abstract

The spectral form factor (SFF) is a measure of energy correlations and has been widely used to identify the transition from the ergodic to the localized phase in interacting many-body quantum systems. In this work, we show that in a disordered Heisenberg spin-$\frac{1}{2}$ model, the finite temperature SFF can be used to generate a canonical phase diagram exhibiting a critical temperature $T_{\mathrm{MBL}}$. Using simple ideas of statistical mechanics, we obtain the critical energy density $\hat{I}{\mu }_{\mathrm{MBL}}$ dual to $T_{\mathrm{MBL}}$. We show that the mobility edge, numerically estimated from the spread of local perturbations and the optical conductivity, indeed coincides with $\hat{I}{\mu }_{\mathrm{MBL}}$.

12 pages, 7 figures