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

Abstract

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 ${\Delta }_c\le {\Delta }_0<1$, belongs to the same universality class as the transition for the Ising spin glass.

12 pages, 2 figures; larger version is provided, which will be published in Phs. Rev. E