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Journal of Applied Physics

Green function ansatz for the Ising model

B. Frank

Published Mar 15, 1984• 4 citations
Publisher

Abstract

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.

DOI
10.1063/1.333700

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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.
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.
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.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
The modified definition of the correlation length for the four dimensional Ising model \xi \propto \epsilon^{-\nu} log^{1/6}\epsilon [1] predicts that the relaxation time for a quantity near the critical temperature T_c should diverge as \tau \propto \epsilon^{-\Delta} log^{\Delta/3}\epsilon where \epsilon, \nu, and \Delta are, respectively, the reduced temperature ( \epsilon=( T_c-T )/T_c, T_c = 6.68 ), the correlation length critical exponent, and a dynamical critical exponent. The \tau-data for the nonlinear relaxation of the order parameter computed on the Creutz cellular automaton [2,3], by using the finite-size lattices of linear dimension L = 12 and 14 as approximations to the infinite lattice for T.