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.

Typos and presentation corrected, refernces added

References in corpus (6)

• Quantitative analyses of empirical fitness landscapes
• Many-body mobility edge in a mean-field quantum spin glass
• Thermodynamics and Universality for Mean Field Quantum Spin Glasses
• Existence of replica-symmetry breaking in quantum glasses
• Spectral Analysis of the Quantum Random Energy Model
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Cited by in corpus (3)

• Spectral Analysis of the Quantum Random Energy Model
• Scaling analysis and renormalization group on the mobility edge in the quantum random energy model
• The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems