TY - JOUR

T1 - Spectral Analysis of the Quantum Random Energy Model

AU - Manai, Chokri

AU - Warzel, Simone

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85163138606&partnerID=8YFLogxK

U2 - 10.1007/s00220-023-04743-4

DO - 10.1007/s00220-023-04743-4

M3 - Article

AN - SCOPUS:85163138606

SN - 0010-3616

VL - 402

SP - 1259

EP - 1306

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -