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 $ε_\mathrm{MBL}$ dual to $T_\mathrm{MBL}$. We show that the mobility edge (ME), numerically estimated from the spread of local perturbations and the optical conductivity, indeed coincides with $ε_\mathrm{MBL}$.