On the 'naive' mean-field equations for spin glasses

Abstract

A model is introduced for which the 'naive' mean-field equations (NMFE) for an Ising system, m_i=tanh ( beta Sigma _jJ_ijm_j+ beta h_i), became exact in any dimension. The model is solved for the infinite-range Ising spin glass (i) starting from the Hamiltonian and using the replica method without replica symmetry breaking, and (ii) starting directly from the NMFE using the method of Sompolinsky (1983). The solution is qualitatively similar to that of the Sherrington-Kirkpatrick model (1975). The Glauber model (1963) for the dynamics of the system is also discussed; the spin autocorrelation function exhibits a t^-1/2 decay everywhere in the ordered phase.