Many-body localization-delocalization transition in the quantum Sherrington-Kirkpatrick model

Abstract

We analyze the many-body localization- (MBL) to-delocalization transition in the Sherrington-Kirkpatrick (SK) model of Ising spin glass in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$. Based on energy-resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating the MBL phase, which is nonergodic and nonthermal, from the delocalized phase, which is ergodic and thermal. The range of the delocalized regime increases with an increase in the strength of $\mathrm{\ensuremath{\Gamma}}$, and eventually for $\mathrm{\ensuremath{\Gamma}}$ larger than ${\mathrm{\ensuremath{\Gamma}}}_{\text{CP}}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase while the delocalized phase shows extensive Renyi entropy. We further obtain the spin-glass transition curve in the energy density $\ensuremath{\epsilon}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ plane from the collapse of the eigenstate spin susceptibility. We demonstrate that in most of the parameter regime, the spin-glass transition occurs close to the MBL transition, indicating that the spin-glass phase is nonergodic and nonthermal while the paramagnetic phase is delocalized and thermal.