Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

Published Sep 1, 2000• 10 citations
Publisher

Abstract

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy \(\overline{\Delta F(T)}\) 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, \(\overline{\Delta F(T)}\) 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, \(\overline{\Delta F(T)}\) 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 .

DOI
10.1143/jpsj.69.2798

Related papers

Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Takayuki Shirakura, Fumitaka Matsubara

We investigate low temperature properties of a random Ising model with $+J$ and $\ensuremath{-}aJ$ $(a\ensuremath{\ne}1)$ bonds in two dimensions using a cluster heat bath method. It is found that the Binder parameters ${g}_{L}$ 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.

arXiv:cond-mat/9707059•Physical Review Letters•Oct 13, 1997•23 citations

Numerical studies of Ising spin glasses in two, three, and four dimensions

R. N. Bhatt, A. P. Young

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

Physical review. B, Condensed matter•Apr 1, 1988•301 citations

Finite-size scaling analysis of exact ground states for ± J spin glass models in two dimensions

Naoki Kawashima, Heiko Rieger

With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.

arXiv:cond-mat/9612116•Europhysics Letters (EPL)•Jul 1, 1997•81 citations

Finite-size scaling in two-dimensional Ising spin-glass models

Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

We study the finite-size behavior of two-dimensional spin-glass models. We consider the $\ifmmode\pm\else\textpm\fi{}J$ model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of $T{L}^{1/\ensuremath{\nu}}$ in the $\ifmmode\pm\else\textpm\fi{}J$ model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as ${T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}{\ensuremath{\theta}}_{S}}$ with ${\ensuremath{\theta}}_{S}\ensuremath{\approx}0.5$.

arXiv:1106.5720•Physical Review E•Nov 16, 2011•21 citations

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

Takayuki Shirakura, Fumitaka Matsubara

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.

Journal of the Physical Society of Japan•Jul 15, 1995•11 citations

Reexamination of the SG Transition in the Two-Dimensional ±JIsing Model

Takayuki Shirakura, Fumitaka Matsubara

Using the exchange Monte Carlo method, we reexamine the ± J Ising spin glass (SG) transition in two dimensions. We calculate the SG susceptibility, the Binder cumulant and the distribution function of spin overlap at low temperatures. In contrast to our expectations, better finite-size scaling plots are obtained for a finite, nonzero transition temperature than for a zero transition temperature.

Journal of the Physical Society of Japan•Oct 15, 1996•16 citations
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy (\overline{\Delta F(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, (\overline{\Delta F(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, (\overline{\Delta F(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 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 present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.
We study the finite-size behavior of two-dimensional spin-glass models. We consider the $\ifmmode\pm\else\textpm\fi{}J$ model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of $T{L}^{1/\ensuremath{\nu}}$ in the $\ifmmode\pm\else\textpm\fi{}J$ model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as ${T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}{\ensuremath{\theta}}_{S}}$ with ${\ensuremath{\theta}}_{S}\ensuremath{\approx}0.5$.
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.