A conjectured scenario for order-parameter fluctuations in spin glasses

We study order-parameter fluctuations (OPF) in disordered systems by considering the behaviour of some recently introduced parameters G,Gc which have proven very useful in locating phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and Gc (for connected disorder averages) take the respective universal values 3 and 13 31 in the T → 0 limit for any finite volume provided the ground state is unique and there is no gap in the ground-state local-field distributions, conditions which are met in generic spin-glass models with continuous couplings and no gap at zero coupling. This makes G,Gc ideal parameters to locate phase transitions in disordered systems much like the Binder cumulant for ordered systems. We check our results by exactly computing OPF in a simple example of uncoupled spins in the presence of random fields and the one-dimensional Ising spin glass. At finite temperatures, we discuss under which conditions the value 3 for G may be recovered by conjecturing different scenarios depending on whether OPF are finite or vanish in the infinite-volume limit. In particular, we discuss replica equivalence and its natural consequence limV→∞ G(V, T ) = 3 when OPF are finite. As an example of a model where OPF vanish and replica equivalence does not give information aboutGwe study the Sherrington–Kirkpatrick spherical spin-glass model by performing numerical simulations for small sizes. Again we find results compatible with G = 3 in the spin-glass phase.