Identifying mobility edge from finite temperature spectral form factor

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 $T_{\mathrm{M}\mathrm{B}\mathrm{L}}$. Using simple ideas of statistical mechanics, we obtain the critical energy density ${\epsilon }_{\mathrm{M}\mathrm{B}\mathrm{L}}$ dual to $T_{\mathrm{M}\mathrm{B}\mathrm{L}}$. We show that the mobility edge, numerically estimated from the spread of local perturbations and the optical conductivity, indeed coincides with ${\epsilon }_{\mathrm{M}\mathrm{B}\mathrm{L}}$.