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 Diagram of the ± J Ising Model in Two Dimensions

Yukiyasu Ozeki, Hidetoshi Nishimori

Published Sep 1, 1987• 68 citations
Publisher

Abstract

The ± J Ising model with asymmetric probability weight of ± bonds in two dimensions is studied in the p - T plane by the numerical transfer-matrix method. We calculate finite-width susceptibilities and finite-width correlation lengths of long strips of various sizes up to 14×10 5 . Finite-size scaling analysis suggests that the random antiphase state exists adjacent to the ferromagnetic phase. The boundary separating these phases is vertical (parallel to the temperature axis) in the phase diagram. The weak universality seems to hold along the ferromagnetic critical line.

DOI
10.1143/jpsj.56.3265

Related papers

The Critical Exponentνalong the Ferromagnetic-Nonferromagnetic Phase Boundary in the Two-Dimensional ±JIsing Model

Hidetsugu Kitatani, Takehiko Oguchi

The value of the critical exponent ν of the two-dimensional ± J Ising model along the ferromagnetic-nonferromagnetic phase boundary in the p - T plane ( p is the concentration of + J bonds; T is the temperature) has been calculated by the use of the transfer matrix method and the Roomany-Wyld approximation for long strips with finite width ( L ×10 5 : L ≤18), and has been found to take almost constant value, 1.0, at temperature above the Nishimori line. By the same method, a lower bound of the critical concentration between paramagnetic phase and the random antiphase state (RAS phase) has been calculated at some temperature below the Nishimori line in the p - T plane.

Journal of the Physical Society of Japan•May 15, 1992•30 citations

Ground State Properties of the ±JIsing Model in Two Dimensions

Yukiyasu Ozeki

We investigate the ground state properties of the ± J Ising model with asymmetric probability weight in two dimensions. We propose a numerical transfer matrix method which enables us to calculate directly the ground-state properties such as correlation functions, defect energy and entropy at T =0. We find that the spin glass correlation function decays in a power law like \(\tilde{g}(r)\sim{r}^{-\tilde{\eta}}\) with \(\tilde{\eta}\simeq0.21\) in the range \(0.5\leq{p}\lesssim0.85\), suggesting the weak universality of spin glass-the concentration of + J bond is denoted by p . At the ferromagnetic phase boundary locating around p =0.89, the ferromagnetic correlation function decays as \(g(r)\sim{r}^{-\tilde{\eta}}\) with η≃0.240±0.016, which suggests the weak universality of ferromagnetic critical exponent from the non-random critical point ( p =1, T =2.269···) to the ground sate counterpart. The behavior of correlation functions, and also that of defect energies indicate the existence of an extra-phase, called the random antiphase state, in \(0.85\lesssim{p}\lesssim0.89\).

Journal of the Physical Society of Japan•Oct 15, 1990•61 citations

Phase Diagram and Critical Exponents of the ±J Ising Model in Finite Dimensions by Monte Carlo Renormalization Group

Yukiyasu Ozeki, Hidetoshi Nishimori

We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.

Journal of the Physical Society of Japan•Apr 1, 1987•85 citations

Equilibrium properties of a two-dimensional random Ising model with a continuous distribution of interactions

H -F Cheung, W. L. McMillan

The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.

Journal of Physics C Solid State Physics•Dec 30, 1983•22 citations

Weak versus strong universality in the two-dimensional random-bond Ising ferromagnet

Fabio D. A. Aarao Reis, S. L. A. de Queiroz, Raimundo R. dos Santos

We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.

arXiv:cond-mat/9608083•Physical review. B, Condensed matter•Oct 1, 1996•35 citations

Ferromagnetic-Nonferromagnetic Phase Boundary on the Two-Dimensional ±JIsing Model

Hidetsugu Kitatani, Takehiko Oguchi

The ferromagnetic-nonferromagnetic phase boundary on the two-dimensional ± J Ising model in the p - T plane is determined by the use of the coherent-anomaly method (CAM) proposed by Suzuki, where p is the concentration of the + J bond, and T the temperature. The phase boundary below the Nishimori line has been found to be vertical at p =0.884±0.001 to the p -axis, which is consistent with the previous results obtained under the assumption of Suzuki's weak universality. The critical exponent γ p ( T ) has also been determined along the phase boundary, which shows a crossover tendency.

Journal of the Physical Society of Japan•Nov 15, 1990•41 citations
The ± J Ising model with asymmetric probability weight of ± bonds in two dimensions is studied in the p - T plane by the numerical transfer-matrix method. We calculate finite-width susceptibilities and finite-width correlation lengths of long strips of various sizes up to 14×10 5 . Finite-size scaling analysis suggests that the random antiphase state exists adjacent to the ferromagnetic phase. The boundary separating these phases is vertical (parallel to the temperature axis) in the phase diagram. The weak universality seems to hold along the ferromagnetic critical line.
The value of the critical exponent ν of the two-dimensional ± J Ising model along the ferromagnetic-nonferromagnetic phase boundary in the p - T plane ( p is the concentration of + J bonds; T is the temperature) has been calculated by the use of the transfer matrix method and the Roomany-Wyld approximation for long strips with finite width ( L ×10 5 : L ≤18), and has been found to take almost constant value, 1.0, at temperature above the Nishimori line. By the same method, a lower bound of the critical concentration between paramagnetic phase and the random antiphase state (RAS phase) has been calculated at some temperature below the Nishimori line in the p - T plane.
We investigate the ground state properties of the ± J Ising model with asymmetric probability weight in two dimensions. We propose a numerical transfer matrix method which enables us to calculate directly the ground-state properties such as correlation functions, defect energy and entropy at T =0. We find that the spin glass correlation function decays in a power law like g~(r)∼r−η~\tilde{g}(r)\sim{r}^{-\tilde{\eta}}g~​(r)∼r−η~​ with η~≃0.21\tilde{\eta}\simeq0.21η~​≃0.21 in the range 0.5≤p≲0.850.5\leq{p}\lesssim0.850.5≤p≲0.85, suggesting the weak universality of spin glass-the concentration of + J bond is denoted by p . At the ferromagnetic phase boundary locating around p =0.89, the ferromagnetic correlation function decays as g(r)∼r−η~g(r)\sim{r}^{-\tilde{\eta}}g(r)∼r−η~​ with η≃0.240±0.016, which suggests the weak universality of ferromagnetic critical exponent from the non-random critical point ( p =1, T =2.269···) to the ground sate counterpart. The behavior of correlation functions, and also that of defect energies indicate the existence of an extra-phase, called the random antiphase state, in 0.85≲p≲0.890.85\lesssim{p}\lesssim0.890.85≲p≲0.89.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.
We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.
The ferromagnetic-nonferromagnetic phase boundary on the two-dimensional ± J Ising model in the p - T plane is determined by the use of the coherent-anomaly method (CAM) proposed by Suzuki, where p is the concentration of the + J bond, and T the temperature. The phase boundary below the Nishimori line has been found to be vertical at p =0.884±0.001 to the p -axis, which is consistent with the previous results obtained under the assumption of Suzuki's weak universality. The critical exponent γ p ( T ) has also been determined along the phase boundary, which shows a crossover tendency.