Solution of the random field $XY$ magnet on a fully connected graph

Abstract

We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude $h$. The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the disorder-averaged ordered state reflects the symmetry of the underlying distribution. The phase boundary has a multicritical point which separates a locus of continuous transitions (for small values of $h$) from a locus of first order transitions (for large $h$). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the multicritical points in the two cases. We find that the locus of continuous transitions is given by the same equation for a family of quadriperiodic distributions, which includes the distributions (a) and (b). However, the location of the multicritical point and the nature of ordered state depend on the form of the distribution. The disorder-averaged ground state energy is found exactly, and the specific heat is shown to approach a constant as temperature approaches zero.

replaced with the version accepted for publication in J. Phys. A : Math. Theor

References in corpus (7)

• Critical phenomena in complex networks
• The large deviation approach to statistical mechanics
• Critical behavior of colloid-polymer mixtures in random porous media
• First-order transition features of the 3D bimodal random-field Ising model
• Phase Diagram of the 3D Bimodal Random-Field Ising Model
• Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions
• XY Neural Networks

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• Phase transitions in the Blume-Capel model with trimodal and Gaussian random fields