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Physical Review

Ising Model: Field-Theoretic and Functional-Integral Aspects

Eugene Helfand

Published Apr 10, 1969• 2 citations
Publisher

Abstract

A functional-integral formulation of the Ising model is used as a link between the usual approach in terms of summation over spin variables and field-theory-like formulas. The latter take the form of Feynman-diagram expansions, Dyson integral equations, or Schwinger functional-derivative field equations. Basic to the theory is a spinless-nucleon-like Green's function, related to the Ising spin-spin correlation function. By means of an infinite renormalization technique, a mesonlike propagator, related to local energy-energy correlations, is bootstrapped. Finally, a vertex function, associated with energy-spin-spin correlations, is introduced. To zeroth order this theory bears similarity to the spherical model, but vital differences are also noted. A brief discussion is presented of the relation of methods employed in field theory for the treatment of the infrared divergence, and approximations which might be of value for critical correlations.

DOI
10.1103/physrev.180.600

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A functional-integral formulation of the Ising model is used as a link between the usual approach in terms of summation over spin variables and field-theory-like formulas. The latter take the form of Feynman-diagram expansions, Dyson integral equations, or Schwinger functional-derivative field equations. Basic to the theory is a spinless-nucleon-like Green's function, related to the Ising spin-spin correlation function. By means of an infinite renormalization technique, a mesonlike propagator, related to local energy-energy correlations, is bootstrapped. Finally, a vertex function, associated with energy-spin-spin correlations, is introduced. To zeroth order this theory bears similarity to the spherical model, but vital differences are also noted. A brief discussion is presented of the relation of methods employed in field theory for the treatment of the infrared divergence, and approximations which might be of value for critical correlations.
We developed a systematic non-perturbative method base on Dyson-Schwinger theory and the derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at T c . The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry.
The Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi-invariants, which play the role of the one-particle density or the one-particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite-dimensional (infinite-range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi-invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
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.