A Green's Function Approach to the Heisenberg Model of One-, Two- and Three-Dimensional Systems

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Abstract
The second order Green’s function is applied to discussion of the paramagnetic phase of the Heisenberg ferromagnet in the simple cubic, square and linear lattices. The decoupling parameter β is introduced to ensure the self-consistency requirement which is represented in the form ≪S-0S+0 >= ½ in the case of spin one-half. The high-temperature expansions yielded the results that the decoupling parameter is given by β= 1 - 1/6zτ+ (6z + 5)/62z2τ2 + O(τ-3), the nearest neighbour correlation function by C1 = 1/4zτ(1 - 1/2zτ+ (z-5)/12z2τ2 + O(τ-3)) and the inverse of the susceptibility by x-1 = 2τ- 1 + 1/zτ+ O(τ-2). From the numerical solutions for β, C1, x-1, in the simple cubic lattice the infinite susceptibility turns up at the finite temperature τc = 0.308, but the specific heat does not have a peak near τc. In the square and linear lattice, the susceptibility does not involve an anomaly and the specific heat has a rather broad maximum at a finite temperature. Therefore we conclude that the finite τc does not appear.

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