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Journal of Physics A Mathematical and General

On the nature of the critical point in the three-spin triangular Ising model

Michael N. Barber

Published Nov 1, 1976• 14 citations
Publisher

Abstract

The nature of the critical point in the triangular Ising model with pure triplet interactions is discussed. In particular, it is shown that the correction terms to the asymptotic power law behaviour of thermodynamic quantities require the existence of a third relevant scaling field in accord with recent renormalization group calculations. The exponent of this field is found to be y 3 =7/8.

DOI
10.1088/0305-4470/9/11/003

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The nature of the critical point in the triangular Ising model with pure triplet interactions is discussed. In particular, it is shown that the correction terms to the asymptotic power law behaviour of thermodynamic quantities require the existence of a third relevant scaling field in accord with recent renormalization group calculations. The exponent of this field is found to be y 3 =7/8.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
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).
The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.
Below its critical temperature, the triangular Ising model with pure three-spin interactions has a spontaneous magnetization M. The first 13 terms of a series expansion for M are known. Here these are re-arranged into a form which can be extrapolated to all terms in a natural way. Thus a conjecture is obtained for M. It fits the scaling prediction beta =1/12. Another order parameter is similarly discussed, together with the difficulties encountered in applying the technique to the two-spin Ising model susceptibility.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).