Journal of the Physical Society of Japan

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

Takayuki Shirakura, Fumitaka Matsubara

Published Oct 15, 1996• 16 citations

Abstract

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.

DOI: 10.1143/jpsj.65.3138

Related papers

A new method to calculate the spin-glass order parameter of the two-dimensional ±JIsing model

Hidetsugu Kitatani, Akira Sinada

arXiv:cond-mat/9908370Journal of Physics A Mathematical and GeneralMay 5, 20009 citations

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

Takayuki Shirakura, Fumitaka Matsubara

Journal of the Physical Society of JapanJul 15, 199511 citations

Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Takayuki Shirakura, Fumitaka Matsubara

arXiv:cond-mat/9707059Physical Review LettersOct 13, 199723 citations

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

J. Houdayer, Alexander K. Hartmann

arXiv:cond-mat/0402036Physical Review BJul 1, 2004133 citations

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

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 .

Journal of the Physical Society of JapanSep 1, 200010 citations

A Peculiar Behavior of the Overlap Distribution in a Heisenberg Spin Glass Model in Three Dimensions

Takayuki Shirakura, Fumitaka Matsubara

Progress of Theoretical Physics SupplementJan 1, 20052 citations

Journal of the Physical Society of Japan

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

Takayuki Shirakura, Fumitaka Matsubara

Published Oct 15, 1996• 16 citations

Publisher

Abstract

DOI: 10.1143/jpsj.65.3138

Related papers

A new method to calculate the spin-glass order parameter of the two-dimensional ±JIsing model

Hidetsugu Kitatani, Akira Sinada

A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.

arXiv:cond-mat/9908370Journal of Physics A Mathematical and GeneralMay 5, 20009 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.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 JapanJul 15, 199511 citations

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.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/9707059Physical Review LettersOct 13, 199723 citations

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

J. Houdayer, Alexander K. Hartmann

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that ${T}_{c}=0$ . All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that ${T}_{c}=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

arXiv:cond-mat/0402036Physical Review BJul 1, 2004133 citations

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

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 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 .

Journal of the Physical Society of JapanSep 1, 200010 citations

A Peculiar Behavior of the Overlap Distribution in a Heisenberg Spin Glass Model in Three Dimensions

Takayuki Shirakura, Fumitaka Matsubara

We have studied the phase transition of the ±J Heisenberg model in three dimensions. Using the exchange Monte Calro method, we have investigated the spin-overlap distribution near the SG transition temperature. By selecting the order parameter as the maximum value of the spin overlap of the diagonal components under the uniform rotations, the calculated overlap distribution suggests the appearance of a peculiar type of replica-symmetry breaking which is similar to the one found in the chirality-overlap distribution by Hukushima and Kawamura. This result may explain absence of any crossing of the Binder parameter of the spin at a finite SG transition temperature.We have studied the phase transition of the ±J Heisenberg model in three dimensions. Using the exchange Monte Calro method, we have investigated the spin-overlap distribution near the SG transition temperature. By selecting the order parameter as the maximum value of the spin overlap of the diagonal components under the uniform rotations, the calculated overlap distribution suggests the appearance of a peculiar type of replica-symmetry breaking which is similar to the one found in the chirality-overlap distribution by Hukushima and Kawamura. This result may explain absence of any crossing of the Binder parameter of the spin at a finite SG transition temperature.

Progress of Theoretical Physics SupplementJan 1, 20052 citations

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.

{T}_{c}=0

. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

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 .