Abstract
We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, and , of the spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.
Highlights
The study of spin glass models in an external field started more than 40 years ago [1], but has demonstrated to be an extremely challenging problem
According to the analytical results derived in the previous section, a spin glass model in a paramagnetic phase with a non-zero field, i.e. with h > 0 and Tc (h) < T < Tc (h = 0), is expected to undergo a phase transition to a spin glass phase as the overlap pd between two real replicas is decreased to a value p∗d
We present some results for ρ = 1.2 for comparison and the main results for ρ = 1.4, which are in the interesting non-mean field region and far from the lower critical dimension
Summary
The study of spin glass models in an external field started more than 40 years ago [1], but has demonstrated to be an extremely challenging problem. A similar problem in the analysis of Monte Carlo data measured from a spin glass model in a field was found in [16], where it was found that a standard finite size scaling analysis was unable to identify the correct critical point in the presence of a field, while the same analysis works perfectly in the absence of the external field. This conditioned analysis provides reliable estimates for the location of the critical dAT line Tc (h). In that work, the authors focused mainly on the Replica Symmetry Breaking (RSB) solutions, while we are mostly interested in identifying eventual phase transitions taking place in the paramagnetic phase when pd < qEA
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