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arXiv:1610.00811 · Annalen der Physik

Localization and chaos in a quantum spin glass model in random longitudinal fields: Mapping to the localization problem in a Bethe lattice with a correlated disorder

Alexander L. Burin

Published Mar 21, 2017• 28 citations
arXivHTMLPDF

Abstract

The analytical solution of a many‐body localization problem in a quantum Sherrington‐Kirkpatrick spin glass model in a random longitudinal field is proposed matching the problem with a model of Anderson localization in a Bethe lattice. The localization transition is dramatically sensitive to the relationship between interspin interaction and random field revealing different regimes in which the interaction can either suppress or enhance the delocalization. The localization is enhanced by decreasing the temperature and the localization transition shows a remarkable universality in a spin glass phase. The observed trends should be qualitatively relevant for other systems showing many‐body localization. image

DOI
10.1002/andp.201600292

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The analytical solution of a many‐body localization problem in a quantum Sherrington‐Kirkpatrick spin glass model in a random longitudinal field is proposed matching the problem with a model of Anderson localization in a Bethe lattice. The localization transition is dramatically sensitive to the relationship between interspin interaction and random field revealing different regimes in which the interaction can either suppress or enhance the delocalization. The localization is enhanced by decreasing the temperature and the localization transition shows a remarkable universality in a spin glass phase. The observed trends should be qualitatively relevant for other systems showing many‐body localization. image
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.
We examine the stability of marginally Anderson localized phase transitions between localized phases to the addition of many-body interactions, focusing in particular on the spin-glass to paramagnet transition in a disordered transverse field Ising model in one dimension. We find evidence for a perturbative instability of localization at finite energy densities once interactions are added, i.e. evidence for the relevance of interactions - in a renormalization group sense - to the non-interacting critical point governed by infinite randomness scaling. We introduce a novel diagnostic, the "susceptibility of entanglement", which allows us to perturbatively probe the effect of adding interactions on the entanglement properties of eigenstates, and helps us elucidate the resonant processes that can cause thermalization. The susceptibility serves as a much more sensitive probe, and its divergence can detect the perturbative beginnings of an incipient instability even in regimes and system sizes for which conventional diagnostics point towards localization. We expect this new measure to be of independent interest for analyzing the stability of localization in a variety of different settings.
We present a thorough numerical study of the Richardson model with quenched disorder (a fully connected XX model with longitudinal random fields). We find that for any value of the interaction the eigenstates occupy an exponential number of sites on the unperturbed Fock space but that single-spin observables do not thermalize, as tested by a microcanonical version of the Edwards-Anderson order parameter. We therefore do not observe many-body localization in this model. We find a relation between the inverse participation ratio and the average Hamming distance between spin configurations covered by a typical eigenstate for which we hypothesize a remarkably simple form for the thermodynamic limit. We also studied the random process defined by the spread of a typical eigenstate on configuration space, highlighting several similarities with hopping on percolated hypercube, a process used to mimic the slow relaxation of spin glasses. A nearby nonintegrable model is also considered where delocalization is instead observed, although the presence of a phase transition at infinite temperature is questionable.
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.
Anderson localization - the suppression of wave propagation by interference in a sufficiently disordered medium - is one of the most remarkable phenomena in condensed matter physics. These lecture notes provide a pedagogical introduction to Anderson localization on the Bethe lattice, an infinite tree with fixed connectivity and no loops, and one of the few settings in which the transition can be studied exactly. The problem is important for two main reasons. First, the Anderson model on the Bethe lattice provides a mean-field, infinite-dimensional limit in which the transition point and critical behavior can be determined exactly. Second, it provides a simplified description of the high-dimensional configuration spaces of interacting quantum systems, and thus a useful toy model for many-body localization. We focus on the Green's function formalism and the cavity method. After recalling standard diagnostics of localization based on transport, spectral statistics, and eigenfunction moments, we show that they are encoded in the statistics of the Green's functions, and that the full distribution of the local density of states is the natural order parameter of the transition. We derive the exact self-consistent cavity equations on the Bethe lattice, use them to characterize the two phases and critical behavior - including an analytic solution at large connectivity - and describe population-dynamics algorithms that solve them with high precision. We explain how an exponentially large correlation volume and the localized nature of the critical point account for the strong finite-size effects on random regular graphs, which led to a long debate over a possible intermediate non-ergodic phase. Finally, we discuss the mapping onto directed polymers in random media, interpreting localization as a freezing transition dominated by a few rare resonant paths.