Three Dimensional Heisenberg Spin Glass Models with and without Random Anisotropy

TL;DR

This study reexamines the spin glass phase transition in three-dimensional Heisenberg models with and without random anisotropy, using new and existing methods to clarify the existence and temperature of the transition.

Contribution

It introduces a new lattice stiffness-based method and provides evidence supporting the occurrence of a spin glass transition in the 3D Heisenberg model without anisotropy.

Findings

Positive stiffness exponent indicating transition presence

Transition temperature estimated around 0.18-0.19J

Supports the existence of a spin glass phase transition in 3D Heisenberg models

Abstract

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 $d = 3$.