Many-Body Mobility Edge in a Mean-Field Quantum Spin Glass

The quantum random energy model provides a mean-field description of the equilibrium spin glass transition. We show that it further exhibits a many-body localization–delocalization (MBLD) transition when viewed as a closed quantum system. The mean-field structure of the model allows an analytically tractable description of the MBLD transition using the forward-scattering approximation and replica techniques. The predictions are in good agreement with the numerics. The MBLD transition lies at energy density significantly above the equilibrium spin glass transition, indicating that the closed system dynamics freezes well outside of the traditional glass phase. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

Figure 1

(a) The canonical phase diagram of the QREM in $\mathrm{\Gamma }$-$T$ plane. Dashed lines correspond to first order thermodynamic transitions due to the crossing of free energies found in the replica treatment. Solid line corresponds to the MBLD transition at $T_{\mathrm{MBL}}=1/2\mathrm{\Gamma }$. Red (blue) shaded region is localized (ergodic). (b) The microcanonical phase diagram in the $\mathrm{\Gamma }$-$\epsilon$ plane. Shaded regions correspond to support of many-body spectrum. Blue dots are an estimate of the transition from finite-size crossing points of $[r]$ at fixed $\mathrm{\Gamma }$.

Figure 2

(a) Critical value of level spacing ratio $r_c$ as a function of ${\epsilon }_c$ parametrizing the phase boundary. $r_c$ is estimated from the $N$-independent crossing point of $[r]$ as shown in the inset. Inset: finite-size crossovers of mean level gap ratio $[r]$ as a function of energy density $\epsilon$ at fixed transverse field $\mathrm{\Gamma }=0.25$ at sizes $N=8,10,12,14$. (b) Finite-size crossovers of mean level gap ratio $[r]$ as a function of rescaled transverse field $\mathrm{\Gamma }\sqrt{N}\log N$ at zero energy. The solid vertical line gives the finite-size estimate of ${\mathrm{\Gamma }}_c$ from Eq. (6). The horizontal dashed line at $[r]=0.38$ (0.53) indicates the expected value for Poisson (GOE) level statistics.

Figure 3

(a) Density plot of $P(\delta M_n)$ as a function of energy density $\epsilon$ at $\mathrm{\Gamma }\approx 0.28$. Vertical line cuts are histograms across disorder and within narrow energy density windows. (b) Finite-size crossover of variance of $\delta M_n$ as a function of energy density.