On the nature of the ordering in Ising spin glasses

Abstract

Monte Carlo studies of the Ising square lattice and simple cubic lattice with a random (symmetric Gaussian) nearest-neighbor exchange are extended, with emphasis on the behavior at zero temperature and at very long “observation” times. Characterizing a ground-state spin configuration by a vector we found that projections of two ground states on each other are typically of order zero. We observe that the order parameterΨ decreases under the action of a homogeneous magnetic field and vanishes at a critical field. The zero-field susceptibility at zero temperature is found to be finite for both two and three dimensions. The anomalous slow relaxation observed in simulations of spin glasses is traced back to the high ground-state degeneracy. Two sources of anomalous relaxation are identified: (i) disappearance of large domains with (on the average) wrong orientation of the order parameter; and (ii) diffusion of order parameter orientation in a finite system with continuous symmetry of the order parameter. Case (i) is exemplified by computations on a two-dimensional Mattis spin glass model. We find that the observations of Bray and More cannot be maintained asfirm evidence against a phase transition, althoughfirm evidence in favor of a transition is also lacking. With the hypothesis that a transition occurs, a cluster description is used to derive some relations characterizing its singularities. Our Monte Carlo simulations give the field-dependence of the Edwards-Anderson order parameterq atT _f and give Chalupa's exponentδ _q as about 5 in two dimensions. Our scaling theory shows that a spin-glass transition may occur with finite susceptibilityχ _q , which offers a possible interpretation of the series-expansion results of Fisch and Harris.