Three-dimensional Heisenberg spin-glass models with and without random anisotropy

We re-examine the spin-glass (SG) phase transition of the ±J Heisenberg models with and without random anisotropy D in three dimensions (d = 3) using two complementary 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 θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T_SG ∼ 0.19J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ_SG for D = 0 is obtained using almost the same transition temperature of T_SG ∼ 0.18J. Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.