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Canadian Journal of Physics

Ising model critical exponent η from a criticality equation

B. Frank, L. Macot, K.V. Bassias, Meir Danino

Published Oct 1, 1989• 2 citations
Publisher

Abstract

A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.

DOI
10.1139/p89-163

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A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.
Previous work on difference equations for the Ising model in d dimensions suggests that a useful ansatz for the k-space pair correlation function is G(k)∼1/[1−AJ(k)/J(0)]1−η/2, where A is the quantity that appears in the criticality equation A=1 at T=Tc, the transition temperature. This ansatz is in accord with the usual definition of η. The scaling relation γ=ν(2−η) follows automatically. Using the i−δ equations, the criticality equation and the U equations together with the above ansatz, the theory predicts, for the two-dimensional Ising model on the square lattice, η=0.2249, Tc/TMF=0.5854 and U2 (critical)=0.1731. The known values are, respectively, 0.2500, 0.5673 and 0.1768.
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.
The i–δ relations and the criticality condition, which are used in recent theoretical calculations of the Ising-model transition temperature and critical exponent η, are tested against exact calculations for the two-dimensional lattices: square, plane triangular, and honeycomb. They are found to be valid to within 2%. They are also tested against series results for the three-dimensional hydrogen peroxide lattice, with which they are shown to be consistent to within 3.3%. A new derivation is given of the i–δ relations.
The rational values beta = 5 / 16 , delta =5, alpha = 1 / 8 , nu = 5 / 8 , and eta =0 of the critical exponents for the three-dimensional spin 1 / 2 Ising ferromagnet are shown to be consistent with a recently-derived linearised equation for the Ising two-spin correlation function. This equation arises from an exact integral representation which previously, on application of a straightforward approximation for higher-order cumulant averages, was shown to yield transition temperatures T c to within 2% of their correct values for the cubic lattices. In the search for beta , the requirement is imposed that the correction terms for the magnetisation and for M 2 , a quantity closely related to the system internal energy, be analytic in some power of T c -T (which turns out to be 1 / 8 ) when the external magnetic field h is zero.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.