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Journal of the Physical Society of Japan

Evidence of a Spin-Glass Phase Transition ofa Site-Random Ising Model in Two Dimensions

Takayuki Shirakura, Fumitaka Matsubara

Published Jul 15, 1995• 11 citations
Publisher

Abstract

We study a site-random Ising (SRI) model in two dimensions (2D) using a Monte Carlo method. It is found that the Binder parameter g L 's for different sizes of the lattice L come together at almost the same temperature as in the ± J model in 3D. Finite-size scaling analyses give results which are compatible with the assumption of a finite, nonzero transition temperature ( T c ≠0), although the scaling temperature range is not large. We suggest that in the SRI model a spin glass (SG) phase transition occurs at T c ≠0 even in 2D.

DOI
10.1143/jpsj.64.2338

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We study a site-random Ising (SRI) model in two dimensions (2D) using a Monte Carlo method. It is found that the Binder parameter g L 's for different sizes of the lattice L come together at almost the same temperature as in the ± J model in 3D. Finite-size scaling analyses give results which are compatible with the assumption of a finite, nonzero transition temperature ( T c ≠0), although the scaling temperature range is not large. We suggest that in the SRI model a spin glass (SG) phase transition occurs at T c ≠0 even in 2D.
We study a short-range site-random Ising model in three dimensions (3D) using a Monte Carlo (MC) method and find a phase transition to a spin glass (SG) phase at a very high temperature ( T c ≃2.83 J). Values of the critical exponents estimated from a finite-size scaling analysis are different from those in both the 3D∓J\mp J∓J Ising model and the 3D Ising ferromagnet. We suggest that the SG phase in this model is a new SG phase which is characterized by a nonzero SG order parameter q ≠0.
We investigate low temperature properties of a random Ising model with +J+J+J and $\ensuremath{-}aJ$ $(a\ensuremath{\ne}1)$ bonds in two dimensions using a cluster heat bath method. It is found that the Binder parameters gL{g}_{L}gL​ for different sizes of the lattice come together at almost the same temperature, implying the occurrence of the spin glass (SG) phase transition. From results of finite size scaling analyses, we suggest that the SG phase really occurs at low temperatures, which is characterized by a power law decay of spin correlations.
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .
We re-examine the spin-glass (SG) phase transition of the ± J Heisenberg models with and without random anisotropy D in three dimensions ( d = 3) using two complementary 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 θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T SG ∼ 0.19 J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ SG for D = 0 is obtained using almost the same transition temperature of T SG ∼ 0.18 J . Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.
We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.