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Canadian Journal of Physics

Green's function approach to the Ising model

K. C. Lee, Robert Barrie

Published Apr 1, 1969• 9 citations
Publisher

Abstract

It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.

DOI
10.1139/p69-097

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It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.
We exactly solve the ferromagnetic spin- 1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number z. Nonlocal correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.
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.
The rigorous evolution equations for the one-body Green's functions (g>,g<{g}^{>},{g}^{<}g>,g<) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, NNN (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (g>,g<{g}^{>},{g}^{<}g>,g<).
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
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.