Absence of Replica Symmetry Breaking in a Region of the Phase Diagram of the Ising Spin Gla.pdf

a r X i v : c o n d - m a t / 0 0 0 8 1 3 9 v 1 [ c o n d - m a t .d i s - n n ] 9 A u g 2 0 0 0 Absence of Replica Symmetry Breaking in a Region of the Phase Diagram of the Ising Spin Glass Hidetoshi Nishimori? and David Sherrington? ?Department of Physics, Tokyo Institute of Technology Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan ?Department of Physics, University of Oxford Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK 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 ±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. INTRODUCTION Existence and characteristics of the spin glass phase are actively investigated for finite-dimensional random spin systems. Closely related is the problem of the mixed (glassy) ferromagnetic phase. If the mean-field picture applies to finite-dimensional systems, the ferromagnetic phase would split up into two regions, one with a simple structure (the replica-symmetric (RS) phase in the mean-field framework) and the other with a complex phase space (replica-symmetry broken (RSB) state). The boundary between these two phases would be the finite-dimensional counterpart of the AT (de Almeida-Thouless) line found for the infinite-range SK (Sherrington- Kirkpatrick) model. Investigations of these and rel