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.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.

arXiv:cond-mat/0404251Apr 10, 20040 citations

Absence of Spin Glass Ordering in Some Random Spin Systems

Yukiyasu Ozeki, Hidetoshi Nishimori

We discuss the improvement of the imperfect argument of Schuster for the absence of spin glass ordering at and below four dimensions in short-range continuous spin systems, and propose a condition for his method to be justified. Assuming satisfaction of the above condition, we apply the method to long-range interactions to reproduce the phase diagram drawn by a renormalization group approach. In particular the three-dimensional RKKY system is found to be at the lower critical dimension. Effects of exchange and uniaxial anisotropies are discussed. The latter anisotropy is shown not to induce transverse spin glass ordering if the system is not ordered in the , x y -plane by the exchange interaction.We discuss the improvement of the imperfect argument of Schuster for the absence of spin glass ordering at and below four dimensions in short-range continuous spin systems, and propose a condition for his method to be justified. Assuming satisfaction of the above condition, we apply the method to long-range interactions to reproduce the phase diagram drawn by a renormalization group approach. In particular the three-dimensional RKKY system is found to be at the lower critical dimension. Effects of exchange and uniaxial anisotropies are discussed. The latter anisotropy is shown not to induce transverse spin glass ordering if the system is not ordered in the , x y -plane by the exchange interaction.

Journal of the Physical Society of JapanDec 15, 198810 citations

Critical aspects of three-dimensional anisotropic spin-glass models

Theodoros Papakonstantinou, Nikolaos G. Fytas, Anastasios Malakis +1 more

arXiv:1410.8397The European Physical Journal BApr 1, 20153 citations

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.