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Physical review. B, Condensed matter

Cluster Monte Carlo dynamics for the antiferromagnetic Ising model on a triangular lattice

G. M. Zhang, C. Z. Yang

Published Nov 1, 1994• 16 citations
Publisher

Abstract

Within the general cluster framework of Kandel, Ben-Av, and Domany, we develop a cluster algorithm for Monte Carlo simulations of the antiferromagnetic Ising model on a triangular lattice. The algorithm does not suffer from problems of metastability and is extremely efficient even at T=0, which allows us to extract the static exponent \ensuremath{\eta}=0.5 as well as the effective dynamical critical exponent of the algorithm z=0.64\ifmmode\pm\else\textpm\fi{}0.02.

DOI
10.1103/physrevb.50.12546

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Within the general cluster framework of Kandel, Ben-Av, and Domany, we develop a cluster algorithm for Monte Carlo simulations of the antiferromagnetic Ising model on a triangular lattice. The algorithm does not suffer from problems of metastability and is extremely efficient even at T=0, which allows us to extract the static exponent \ensuremath{\eta}=0.5 as well as the effective dynamical critical exponent of the algorithm z=0.64\ifmmode\pm\else\textpm\fi{}0.02.
We have developed a cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The method does not suffer from problems of metastability, and is extremely efficient even at T=0, as we demonstrated previously. Here we describe results of more extensive simulations of the model at T=0 which allow us to extract the effective dynamical critical exponent of the algorithm, z\ensuremath{\approxeq}0.55, as well as the static exponent \ensuremath{\eta}=0.5. We also provide an argument that explains why our method works.
We review the development of Monte Carlo algorithms for the simulation of the Ising model with local dynamics. The multi lattice coding algorithm for the pure and for the site-disordered Ising model is described in detail. Application of these algorithms on fast vector computers have led to new progress concerning the critical dynamics of the pure Ising model leading to the critical exponent z = 2:05. The importance of disorder for the critical dynamics of the Ising model has been investigated recently. We review the numerical results and compare with theoretical predictions. A crossover scenario is found as for the static behavior with an effective dynamical exponent z = 1:75 fl. 1 Introduction The Ising model is a standard model in statistical physics for the description of a ferromagnet in an extremely simplified manner. It is assumed that quantum spin operators may be replaced by classical spins S i = \\Sigma 1 which are oriented parallel or antiparallel to a magnetic filed H: H...
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.
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.
Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.