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.

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