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Physical Review

Green's Functions and the Ising Model

Ronald L. Bell

Published Mar 4, 1966• 6 citations
Publisher

Abstract

The results of Tahir-Kheli, Doman, and ter Haar for the correlation functions of the Ising ferromagnet are extended, and the nature of the possible approximate solutions for the magnetization, critical temperature, and short-range order are indicated for the spin-\textonehalf{} case.
DOI
10.1103/physrev.143.215

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"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."

•Jan 1, 1967•0 citations
The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and zzz nearest neighbors.
In this paper we present Green function technique applied to calculations of spin-spin correlations in systems governed by Ising Hamiltonian. This offers approximate yet reasonably accurate analytical results, as an alternative approach to direct computer simulation. Local spin operators are represented in terms of particle operators for fermions. Chain of Green functions for local magnetization on given site involves nearest neighbors and it is decoupled at the next-nearest-neighbors level. Therefore, still accounts at the lowest level for magnetic correlations between nearest neighbors. This technique was applied to binary alloy A x B 1-x with given atomic short range order. Calculations for stoichiometric x=1/2 composition show how temperature dependence of magnetization and spin-spin correlations are modified when atomic order parameter changes. We scanned this parameter from the case of ordered binary alloy, via random alloy, to clusterlike structure. In model calculations we (a) confirm characteristic λ -like shape of the magnetic specific heat near Curie temperature and (b) recover essential results of exact, two-dimensional Ising model. This supports our claim of usefulness of such analytical approach, which is easy and flexible to describe different geometries (clusters, thin films) and conditions imposed on spins located on specific atoms (spins partly pinned on the surface atoms).
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."