Conditions for the Existence and Stability of the Continuous Attractor in the Classical XY Model with an Associative-Memory-Type Interaction

Abstract

We analyze the structure of attractors in the classical XY model with an associative-memory-type interaction by the statistical mechanical method. Previously, it was found that when patterns are uncorrelated, points on a path connecting two memory patterns in the space of the order parameters are solutions of the saddle point equations (SPEs) in the case that p is \(\mathcal{O}(1)\) irrespective of N and N ≫ 1, where p and N are the numbers of patterns and spins, respectively. This state is called the continuous attractor (CA). In this paper, we clarify the conditions for the existence and stability of the CA with and without the correlation a (0 ≤ a < 1) between any two patterns in the case that N ≫ 1 and the self-averaging property holds. We find that the CA exists for any p ≥ 2 when a = 0, but it exists only for p = 2 when 0 < a < 1 and for p = 3 when a < 1/3. For p = 2 and 3, and for a < 1, we analyze the SPEs and find all solutions and study their stabilities. We perform Markov chain Monte Carlo simu...