Parisi States in a Heisenberg Spin-Glass Model in Three Dimensions

Abstract

We have studied low-lying metastable states of the $\pm J$ Heisenberg model in two ($d=2$) and three ($d=3$) dimensions having developed a hybrid genetic algorithm. We have found a strong evidence of the occurrence of the Parisi states in $d=3$ but not in $d=2$. That is, in $L^d$ lattices, there exist metastable states with a finite excitation energy of $ΔE \sim O(J)$ for $L \to \infty$, and energy barriers $ΔW$ between the ground state and those metastable states are $ΔW \sim O(JL^θ)$ with $θ> 0$ in $d=3$ but with $θ< 0$ in $d=2$. We have also found droplet-like excitations, suggesting a mixed scenario of the replica-symmetry-breaking picture and the droplet picture recently speculated in the Ising SG model.