Gauge Theory of the Ising Spin Glass

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 (SherringtonKirkpatrick) model. Investigations of these and related problems are almost exclusively carried out currently by numerical methods because of lack of reliable analytical methods. We show in the present contribution that simple symmetry arguments lead to a strong constraint on the possible location and shape of the AT line in the phase diagram of the Ising spin glass on an arbitrary lattice with arbitrary range of interactions. More precisely, it is possible to prove that there is nothing like RSB on the Nishimori line in the phase diagram and that the AT line, if it ever exists, should start as a vertical line from the multicritical point where paramagnetic, ferromagnetic and spin glass phases merge. Our method is an application of the gauge theory which has been used to derive exact energy and many other exact/rigorous results. GAUGE THEORY OF THE ISING SPIN GLASS Let us consider the ±J Ising model (Si = ±1) with the Hamiltonian H = − ∑