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Science Technology and Engineering

GPU Accelerated Monte Carlo Algorithm of Ising Model on Triangular Lattice

Wei Zhang

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

In the statistical model,the efficiency of most Monte Carlo algorithm reduces quickly near the critical point.In the analysis of traditional local algorithms,a GPU-based parallel simulation algorithm on the triangular lattice Ising model,which greatly improves the efficiency of the Monte Carlo simulation,is raised.For the model with the size of 1 024×1 024,a speedup of 69 is achieved.Besides,the critical behavior is analyzed,a high-precision critical point(βc=0.274 66(1)) and critical exponents(yt= 1.01(2),yh= 1.875 6(3)) of triangular lattice Ising model are obtained,which implies the effectiveness of the GPU algorithm.

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We test the performance of the Monte Carlo renormalization method in the context of the Ising model on a triangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalization imply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of the optimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor critical Hamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are more accurate than those obtained with the majority rule.

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MONTE CARLO SIMULATIONS OF THE ISING MODEL ON THREE-DIMENSIONAL RANDOM LATTICE USING THE CLUSTER ALGORITHM

JI DA-REN, J. B. Zhang, (1)浙江大学,杭州310027; (2)浙江近代物理中心,杭州310027

In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained is βc=0. 075±0. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.

Acta Physica Sinica•Jan 1, 1993•0 citations

Phase transitions and critical properties in the antiferromagnetic Ising model on a layered triangular lattice with allowance for intralayer next-nearest-neighbor interactions

М. К. Бадиев, Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович

The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.

Journal of Experimental and Theoretical Physics•Oct 1, 2016•9 citations
In the statistical model,the efficiency of most Monte Carlo algorithm reduces quickly near the critical point.In the analysis of traditional local algorithms,a GPU-based parallel simulation algorithm on the triangular lattice Ising model,which greatly improves the efficiency of the Monte Carlo simulation,is raised.For the model with the size of 1 024×1 024,a speedup of 69 is achieved.Besides,the critical behavior is analyzed,a high-precision critical point(βc=0.274 66(1)) and critical exponents(yt= 1.01(2),yh= 1.875 6(3)) of triangular lattice Ising model are obtained,which implies the effectiveness of the GPU algorithm.
We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L = 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T c = 3.6403(2). A convincing finite-size scaling analysis of the model yields ν = 0.9995(21), β/ν = 0.12400(17), γ/ν = 1.75223(22), γ'/ν = 1.7555(22), α/ν = 0.00077(420) (scaling) and α/ν = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
We test the performance of the Monte Carlo renormalization method in the context of the Ising model on a triangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalization imply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of the optimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor critical Hamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are more accurate than those obtained with the majority rule.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained is βc=0. 075±0. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.