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Journal of Physics A Mathematical and General

The (2+1)D Ising model on a triangular lattice

Chris J. Hamer

Published Feb 21, 1986• 18 citations
Publisher

Abstract

The (2+1)D Ising model on a triangular lattice is explored for lattices of up to 5*5 sites. Finite-size scaling techniques are used to estimate the critical parameters x c =0.2096(2), nu =0.627(4), beta =0.322(6), gamma =1.236(8). These agree with previous estimates, within errors, and with the hypotheses of universality and hyperscaling. The behaviour at low temperatures is also studied, and theoretical predictions of finite-size scaling at a first-order transition are confirmed.

DOI
10.1088/0305-4470/19/3/023

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Physical review. B, Condensed matter•Mar 1, 1993•8 citations
The (2+1)D Ising model on a triangular lattice is explored for lattices of up to 5*5 sites. Finite-size scaling techniques are used to estimate the critical parameters x c =0.2096(2), nu =0.627(4), beta =0.322(6), gamma =1.236(8). These agree with previous estimates, within errors, and with the hypotheses of universality and hyperscaling. The behaviour at low temperatures is also studied, and theoretical predictions of finite-size scaling at a first-order transition are confirmed.
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.
The method of 'symmetrized' stochastic truncation is applied to the (2+1)-dimensional Ising model, to calculate the ground-state energy, mass gap, magnetization and susceptibility on lattices of sizes up to 6*6 sites. The method is shown to be a powerful and accurate Monte Carlo technique, provided the lattice size is not too large, and the model is suitably 'convergent'. Finite-size scaling estimates of the critical indices are nu =0.627(2), alpha =0.12(2), beta =0.324(3), and delta =1.23(1). An estimate of the combination gamma / nu +2 beta / nu =3.000(1) is obtained, in outstanding agreement with the expected hyperscaling relation.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.