Three Dimensional Heisenberg Spin Glass Models with and without Random Anisotropy

We reexamine the spin glass (SG) phase transition of the $\pm J$ Heisenberg models with and without the random anisotropy $D$ in three dimensions ($d = 3$) using complementary two methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent $\theta$ has a positive value ($\theta > 0$) even when $D = 0$. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of $T_{\rm SG} \sim 0.19J$ for $D = 0$. On the other hand, a large scale MC simulation shows that, in contrary to the previous results, a scaling plot of the SG susceptibility $\chi_{\rm SG}$ for $D = 0$ is obtained using almost the same transiton temperature of $T_{\rm SG} \sim 0.18J$. Hence we believe that the SG phase transition occurs in the Heisenberg SG model in $d = 3$.Comment: 15 pages, 9 figures, to be published in J. Phys.

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 $\Delta E \sim O(J)$ 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.Comment: 4 pages, 6 figure

Binder Parameter of a Heisenberg Spin-Glass Model in Four Dimensions

We studied the phase transition of the $\pm J$ Heisenberg model with and without a random anisotropy on four dimensional lattice $L\times L\times L\times (L+1)$ $(L\leq 9)$. We showed that the Binder parameters $g(L,T)$'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists, $g(L,T)$ exhibits a steep negative dip near the spin-glass phase transition temperature $T_{\rm SG}$ similarly to the $p-$state infinite-range Potts glass model with $p \geq 3$, in which the one-step replica-symmetry-breaking (RSB) occurs. We speculated that a one-step RSB-like state occurs below $T_{\rm SG}$, which breaks the usual crossing behavior of $g(L,T)$.Comment: 4 pages including 4 figures, submitted to Phys. Rev.