Spectral Analysis of the Quantum Random Energy Model

Abstract

The quantum random energy model (QREM) is a random matrix of Anderson-type which describes effects of a transversal magnetic field on Derrida’s spin glass. The model exhibits a glass phase as well as a classical and a quantum paramagnetic phase. We analyze in detail the low-energy spectrum and establish a localization-delocalization transition for the corresponding eigenvectors of the QREM. Based on a combination of random matrix and operator techniques as well as insights in the random geometry, we derive next-to-leading order asymptotics for the ground-state energy and eigenvectors in all regimes of the parameter space. Based on this, we also deduce the next-to-leading order of the free energy, which turns out to be deterministic and on order one in the system size in all phases of the QREM. As a result, we determine the nature of the fluctuations of the free energy in the spin glass regime.

The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems

“…Let us conclude the main part of the paper with a general outlook. Having derived precise low-energy asymptotics of eigenvalues in terms of quadratic approximations and using techniques as in [35,Sec. 5], our techniques should extend to derive the fluctuations of the free energy not only at 𝛽 = ∞ but also for finite temperature.…”

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

“…Based on this, replica symmetry breaking approximations to the free energy have been found early on [19,50,52] (see also [9,45] and references therein). However, aside from the case of hierarchical transversal models of generalised random energy type [22,29,30,31,32], a closed-form variational formula for the free energy has been absent. The purpose of this paper is to present a first version of such a variational formula of Parisi type for a large class of transversal field models, which includes the quantum Sherrington-Kirkpatrick (QSK) model.…”