A Migdal-Kadanoff renormalization-group approach for the XY spin glass

Abstract

The XY spin glass is studied in terms of the discrete p-state clock spin glass model, in the limit p → ∞, through the Migdal-Kadanoff renormalization-group approach. The interactions obey a Gaussian probability distribution with non-zero mean, acting on nearest-neighbour pairs of spins, defined on the sites of diamond hierarchical lattices. By numerically following the bond probability distribution, the phase diagrams of hierarchical lattices corresponding to approximations of d-dimensional Bravais lattices (d = 2, 3, 4, 5 and 6) are investigated. Within the scheme proposed here, the XY limit is attained at relatively small values of p, such that p = 20 represents a good approximation of the continuous limit. The lower critical dimension of the XY spin glass is estimated to lie in between 2 and 3. Some analogies with mean-field results are discussed, for increasing values of d.