Exact Solution of the Spherical Spin Glass Model Beyond Mean Field Theory

We present an exact solution of a Gaussian spin-glass model with infinite ranged interactions and a global spherical constraint at zero magnetic field. The replicated spin-glass Hamiltonian is mapped onto a Coulomb gas of logarithmically interacting particles confined by a peculiar single particle potential. The precise free energy is obtained by analyzing the Painlev\'e $\tau^{IV} [n]$ function in the $n\to 0$ limit, accounting for neglected fluctuations beyond the semi-circle density utilized in the large $N$ analysis of Kosterlitz, Thouless and Jones\cite{ktjonesPRL}. This is the first known exact solution of a spin-glass model beyond mean-field theory.