Spherical 2+p spin-glass model: An analytically solvable model with a glass-to-glass transition

We present the detailed analysis of the spherical s+p spin-glass model with two competing interactions: among p spins and among s spins. The most interesting case is the 2+p model with p >= 4 for which a very rich phase diagram occurs, including, next to the paramagnetic and the glassy phase represented by the one step replica symmetry breaking ansatz typical of the spherical p-spin model, another two amorphous phases. Transitions between two contiguous phases can also be of a different kind. The model can thus serve as a mean-field representation of amorphous-amorphous transitions (or transitions between undercooled liquids of different structure). The model is analytically solvable everywhere in the phase space, even in the limit where the infinite replica symmetry breaking ansatz is required to yield a thermodynamically stable phase.