It is proved that Girvin's method (see ibid., vol.11, p.L427 (1978)) for the calculation of the two-spin correlation function of the three-dimensional Ising model is based on the same approximation made by Frank and Mitran (1978).
It is noted that expressions for the T=Tcn-spin correlation functions of a two-dimensional Ising system obtained by continuum methods do agree with exact results for restricted geometries. This leads to the conjecture that the continuum result is true for all n and all geometries. Speculation regarding the behavior of these functions away from T=Tc are made.
Reply to Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J Ising model'
An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.
We exhibit a new method for calculating high-and low-temperature series expansions for diagonal spin-spin correlation functions in the d=2 Ising model. The method makes use of a differential equation obeyed by these functions.