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International Journal of Modern Physics C

3D ISING NONUNIVERSALITY: A MONTE CARLO STUDY

Melanie Schulte, Caroline Drope

Published Aug 1, 2005• 12 citations
Publisher

Abstract

We investigate as a member of the Ising universality class the Next-Nearest Neighbor Ising model without external field on a simple cubic lattice by using the Monte Carlo Metropolis Algorithm. The Binder cumulant and the susceptibility ratio, which should be universal quantities at the critical point, were shown to vary for small negative next-nearest neighbor interactions.

DOI
10.1142/s0129183105007844

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Henk W. J. Blöte, Erik Luijten, Jouke R. Heringa

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).

arXiv:cond-mat/9509016•Journal of Physics A Mathematical and General•Nov 21, 1995•308 citations
We investigate as a member of the Ising universality class the Next-Nearest Neighbor Ising model without external field on a simple cubic lattice by using the Monte Carlo Metropolis Algorithm. The Binder cumulant and the susceptibility ratio, which should be universal quantities at the critical point, were shown to vary for small negative next-nearest neighbor interactions.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin-1 / 2 models with nearest-neighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some "equivalent-neighbor" models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y(t)=1.5868(3), y(h)=2.4816(1), and y(i)=-0.821(5), respectively. The Binder ratio is estimated as Q=<m(2)>(2)/<m(4)>=0.62 341(4).
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin- 1 2 Ising model with nearest-neighbor interactions, a spin- 1 2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), yh =2.4815 (15) and y i = -0.82 (6). The universal ratio Q = #m 2 # 2 /#m 4...
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).