Hidden Mattis Order and Dynamic Correlations in the Annealed Ising Spin Glass

The hidden Mattis phase of annealed spin-glass models was suggested by Kasai and Okiji 40 years ago but received relatively little attention until very recently. In terms of equilibrium configurations, the phase is characterized by a spin-glass type order but no thermodynamic signature across the transition. To better understand the static and dynamic correlations implied by the hidden order, we performed detailed numerical simulations of the Sherrington-Kirkpatrick (SK) Ising spin glass model with annealed coupling constants. Relaxation from the initial spin glass state towards the stationary state is studied when the interactions are allowed to evolve with time under the usual Metropolis updating scheme. The spectrum of the instantaneous interaction matrix is monitored during the simulation and our results generally agree with the analytical predictions . Above the transition, the spectrum remains essentially the same as that of the initial random matrix, but a gap appears in the low temperature Mattis phase that separates the largest eigenvalue from the rest of the spectrum. The autocorrelation of spins and of the eigenvector with the largest eigenvalue in the stationary state are studied. Our data show that the finite-size scaling exponent of the autocorrelation functions changes as temperature changes. At low temperatures the spin configuration and the eigenvector are closely correlated, and this correlation disappears above the transition. Hence the hidden Mattis order can be associated with the formation of a system-wide condensate of spins that drifts in configuration space. More complex dynamical behavior in the neighborhood of the transition will also be reported.