Ground State Properties of the ±<i>J</i>Ising Model in Two Dimensions

Ozeki, Yukiyasu

doi:10.1143/jpsj.59.3531

Cited by 60 publications

(57 citation statements)

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“…We think, however, that these are not necessarily incompatible, because the previous authors have concluded only that their data are not incompatible with the prediction that T c 0, and have not ruled out the possibility of such low T c as estimated here. The thing that was certainly suggested by the previous studies is that, at T 0, no long-range order exists and the spin correlation decays according to the power law [5,6,8]. This is compatible with our result with T c fi 0.…”

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confidence: 93%

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Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Shirakura

Matsubara

1997

Phys. Rev. Lett.

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We investigate low temperature properties of a random Ising model with 1J and 2aJ ͑a fi 1͒ 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. [S0031-9007(97) Spin glasses have attracted great challenge for computational physics in these two decades. It is widely believed now in the bond-random Ising model that spin glass (SG) transitions occur at a finite, nonzero temperature T c fi 0 in three dimensions (3D) [1][2][3][4] and at zero temperature T c 0 in two dimensions (2D) [4][5][6][7][8]. Recently, the present authors [9] reexamined the SG phase transition of the 6J Ising model on a square lattice of L 3 L by means of an exchange Monte Carlo method [10] and found that the Binder parameters g L for L # 16 intersect at T fi 0. They also found that better finite-size scaling (FSS) fits of the spin glass susceptibility x SG are obtained when T c fi 0. These results imply the occurrence of the SG phase transition at T c fi 0. If so, it is quite interesting, because it disproves the belief of T c 0. However, there remain two problems which should be considered to suggest T c fi 0 in 2D. One is that g L for a smaller lattice almost saturates below a rather high temperature [7] and its saturation value slightly increases with L [9]. Therefore, it is difficult to see whether the intersection of g L for smaller L suggests the presence of the SG phase at T c fi 0 or is merely due to a finite size effect. The other is that it is still open whether or not the model really exhibits the nature of the SG phase at T , T c , because the estimated transition temperature T c is slightly lower than the lowest temperature which is reached in the simulation. The problems would be solved if we study the same model on bigger lattices at lower temperatures. The saturation of g L at rather high temperatures, however, may be removed if we treat an asymmetric random Ising model with 1J and 2aJ ͑a fi 1͒ bonds, because the energy gap of 2j1 2 ajJ in that model between the ground state and the lowest excitation state is much smaller than that of 4J in the 6J model [11], and, if the lattice is rather small, we may study equilibrium properties at any temperature using a cluster heat bath (CHB) method [12,13].In this Letter, we investigate low temperature properties of the asymmetric random Ising model on the square lattice of L 3 L ͑L # 18͒ using the CHB method. In fact, g L does not saturate down to a very low temperature. We find that as the temperature decreases, g L 's for different L meet at almost the same temperature and then increase together. This property rather resembles that of the 6J model in 3D in which the SG phase transition occurs at T c fi 0. We make the FSS an...

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confidence: 93%

“…Once the set of these 0031-9007͞97͞79(15)͞2887(4)$10.00functions ͕F l ͕͑s ͑l͒ i ͖͖͒ is obtained, the spin configurations of individual layers can be determined successively from the layer ͑L͒ to the layer ͑1͒ by using Eqs. (8) or (11) in Ref. [13] and random numbers.…”

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confidence: 99%

Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Shirakura

Matsubara

1997

Phys. Rev. Lett.

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“…This is in fair agreement with a number of previous estimates. 7,9,[14][15][16] Nevertheless, the important issue here is universality; that is whether or not is positive.…”

Section: Introductionmentioning

confidence: 99%

Domain wall entropy of the bimodal two-dimensional Ising spin glass

Aromsawa

Poulter

2007

Phys. Rev. B

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We report calculations of the domain wall entropy for the bimodal two-dimensional Ising spin glass in the critical ground state. The L * L system sizes are large with L up to 256. We find that it is possible to fit the variance of the domain wall entropy to a power function of L. However, the quality of the data distributions are unsatisfactory with large L > 96. Consequently, it is not possible to reliably determine the fractal dimension of the domain walls.Comment: 4 pages, 2 figures, submitted to PR

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“…There are two limiting regimes, with a size dependent crossover temperature T * (L) [10], a T < T * (L) ground state plus gap dominated regime and an effectively continuous energy level regime T > T * (L). There have been consistent estimates over decades from correlation function measurements [11,12], Monte Carlo renormalization-group measurements [13], transfer matrix calculations [14], numerical simulations [1,[15][16][17], and ground state measurements [18,19] showing that the anomalous dimension critical exponent η ≈ 0.20 in both regimes, indicating that the bimodal model is not in the same universality class as the continuous distribution models. However, it has also been claimed that the bimodal model in the T > T * (L) regime is in the same universality class as the Gaussian model, because for the bimodal model : "fits... lead to values of η that are very small, between 0 and 0.1, strongly suggestive of η = 0" [10], and "the data are not sufficiently precise to provide a precise determination of η, being consistent with a small value η ≤ 0.2, including η = 0" [20,21].…”

Section: Introductionmentioning

confidence: 95%

Ising spin glasses in two dimensions: Universality and nonuniversality

Lundow

Campbell

2017

Phys. Rev. E

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Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising Spin Glass (ISG) models in dimension two (P.H. Lundow and I.A. Campbell, Phys. Rev. E 93, 022119 (2016)). Here we further analyse the bimodal and Gaussian data together with data on two other continuous interaction distribution 2D ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "anti-diluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent η ≡ 0 but that to within the present numerical precision they share the same critical exponent ν also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and differ from one discrete distribution model to another. Discrete distribution ISG models in dimension two have non-zero values of the critical exponent η; they do not lie in a single universality class.

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Ground State Properties of the ±JIsing Model in Two Dimensions

Cited by 60 publications

References 19 publications

Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Low Temperature Phase of Asymmetric Spin Glass Model in Two Dimensions

Domain wall entropy of the bimodal two-dimensional Ising spin glass

Ising spin glasses in two dimensions: Universality and nonuniversality

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