Absence of Replica Symmetry Breaking in a Region of the Phase Diagram of the Ising Spin Glass

Abstract

We prove that the distribution functions of magnetization and spin glass order parameter coincide on the Nishimori line in the phase diagram of the $\pm J$ Ising model in any dimension. This implies absence of replica symmetry breaking because the distribution function of magnetization consists only of two delta functions, suggesting the same simple structure for the distribution of spin glass order parameter. It then follows that the mixed (glassy) phase, where the ferromagnetic order coexists with complex phase space, should lie, if any, below the Nishimori line. We also argue that the AT line to mark the onset of RSB with a continous distribution of the spin glass order parameter, if any again, would start with an infinite slope from the multicritical point where paramagnetic, ferromagnetic and spin glass phases merge.