Effects of anisotropy on a quantum Heisenberg spin glass in a three-dimensional hierarchical lattice

We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction ($J_{ij}$), Gaussian and uniform, are used, with zero mean and width $\bar{J}$. The $(kT/\bar{J}) \times \Delta_0$ phase diagram is obtained, where $T$ is the temperature, $\Delta_0$ is the first moment of the probability distribution for the uniaxial anisotropy, and $k$ is the Boltzmann constant. For the Ising model ($\Delta_0=1$), there is a spin-glass phase at low temperatures (high $\bar{J}$) and a paramagnetic phase at high temperatures (low $\bar{J}$). For the isotropic Heisenberg model ($\Delta_0=0$), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with $\Delta_0$, as expected, but goes to zero at a finite value of the anisotropy parameter, namely $\Delta_0 = \Delta_c \sim 0.59$. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for $\Delta_c \leq \Delta_0 < 1$, belongs to the same universality class as the transition for the Ising spin glass.