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

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 $\mathrm{\Delta }E\sim O(J)$ for $L\to \mathrm{\infty }$, and energy barriers $\mathrm{\Delta }W$ between the ground state and those metastable states are $\mathrm{\Delta }W\sim O(J{L}^{\theta })$ 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.