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

Abstract

We study the anisotropic Heisenberg spin-glass model on 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 $\overline{J}$. The $(kT∕\overline{J})\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Delta}}_{0}$ phase diagram is obtained, where $T$ is the temperature, ${\ensuremath{\Delta}}_{0}$ is the first moment of the probability distribution for the uniaxial anisotropy, and $k$ is the Boltzmann constant. For the Ising model $({\ensuremath{\Delta}}_{0}=1)$, there is a spin-glass phase at low temperatures (high $\overline{J}$) and a paramagnetic phase at high temperatures (low $\overline{J}$). For the isotropic Heisenberg model $({\ensuremath{\Delta}}_{0}=0)$, our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with ${\ensuremath{\Delta}}_{0}$, as expected, but goes to zero at a finite value of the anisotropy parameter, namely ${\ensuremath{\Delta}}_{0}={\ensuremath{\Delta}}_{c}\ensuremath{\sim}0.59$. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for ${\ensuremath{\Delta}}_{c}<{\ensuremath{\Delta}}_{0}<1$, belongs to the same universality class as the transition for the Ising spin glass.