Mean-field Monte Carlo approach to the Sherrington-Kirkpatrick model with asymmetric couplings

An alternative method is applied to the nonequilibrium zero temperature dynamics of a Sherrington-Kirkpatrick model with nonsymmetric random exchange couplings ${\mathit{J}}_{\mathit{i}\mathit{j}}$≠${\mathit{J}}_{\mathit{j}\mathit{i}}$. Based on exact stochastic single spin equations for the infinite system, this mean-field Monte Carlo approach avoids the strong finite size effects of the usual simulations for this problem. Varying the average symmetry η=[${\mathit{J}}_{\mathit{i}\mathit{j}}$ ${\mathit{J}}_{\mathit{j}\mathit{i}}$]/[${\mathit{J}}_{\mathit{i}\mathit{j}}^2$], we find a clear transition from ergodic dynamics at η<${\mathrm{\eta }}_{\mathit{c}}$=0.825 to a phase (η>${\mathrm{\eta }}_{\mathit{c}}$) where a finite fraction of the spins freezes. Only in the fully symmetric case η=1 is the entire system frozen.