Spherical analogues of some non-Ising spin glass models: Exact solutions

Abstract

We consider several models of non-Ising spin glass for which we can naturally formulate an analogue of spherical conditions well known for the Ising model. We calculate the partition functions of the obtained models exactly. The random matrix method and the replica method yield identical results.Here, we consider continuous analogues of some known discrete non-Ising models of spin glass, namely, models of the spin-1 spin glass, quadruple glass, and the Potts spin glass with an arbitrary number of states. In these models, we can naturally introduce a condition analogous to the well-known spherical condition for the continuous analogue of the Ising model [1], which consequently allows solving these new "spherical" models exactly. The essential result is that exact results for these models obtained with the random matrix theory coincide with the results obtained using the replica method here.Although it is well known that spherical models are quite nonrealistic from the physical standpoint, they are rather important because they are rare examples of multiparticle systems for which partition functions can be calculated explicitly.The spherical approximation was first formulated by Berlin and Kac in [1], where discrete Ising spins with S 2 i = 1 were replaced with continuous spins satisfying the global condition i S 2 i = N , where the summation ranges all N sites of a lattice. The obtained problem was solved explicitly for the nearestneighbor interaction on various three-dimensional lattices.Soon after the appearance of the first theoretical papers on spin glasses, a spherical variant of the Sherrington-Kirkpatrick model [2], i.e., a model with an infinite radius of random interactions, was proposed, and its exact solution was found [3]. Models of spin glass with soft spins are in general quite popular in the kinetic approach to spin glasses because the presence of one-site terms allows describing the dynamical behavior of the glass. The spherical model of p-spin glass of quasi-Ising spins (see, e.g., [4], [5]), whose behavior provides a scenario of the liquid-glass transition, is the best-known model.In [6], we proposed an exactly solvable Potts spin glass model with three states and with additional conditions generalizing the spherical conditions for Ising spins. We there obtained an exact solution using random matrix theory. We also showed that the replica method gives the same solution. The obtained solution is replica-symmetric and stable, and we therefore do not need to break the replica symmetry. This distinguishes the spherical variant from just the Potts model with three states in which the replica-symmetric solution is stable only in a finite temperature range near the phase transition to the glass state [7], [8].This paper is a straightforward continuation of [6] in two directions. First, we consider spherical models of a Potts spin glass with an arbitrary number of states. Second, we break the Potts symmetry of the model in [6], which provides the possibility to study spherical analogues of the axial quad...