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

TL;DR

This study investigates the existence of Parisi states in a 3D Heisenberg spin-glass model, revealing their presence in three dimensions but not in two, and discusses implications for spin-glass theories.

Contribution

The paper provides evidence for Parisi states in 3D Heisenberg spin glasses using a hybrid genetic algorithm, contrasting with their absence in 2D.

Findings

Parisi states occur in 3D but not in 2D Heisenberg spin glasses.

Metastable states have finite excitation energy in 3D.

Energy barriers scale with system size in 3D, but not in 2D.

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 $\Delta E \sim O(J)$ for $L \to \infty$, and energy barriers $\Delta W$ between the ground state and those metastable states are $\Delta W \sim O(JL^{\theta})$ with $\theta > 0$ in $d=3$ but with $\theta < 0$ 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.