Exactly Solvable Model of a Quantum Spin Glass

A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low $T$. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.