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Journal of Physics C Solid State Physics

Two-point correlations for Ising models

Y Achiam, J. Michael Kosterlitz

Published Nov 28, 1977• 11 citations
Publisher

Abstract

Renormalisation group methods are used to calculate to first order in epsilon =4-d the momentum dependent correlation function of an Ising-like model in the critical region for arbitrary momentum, temperature and external field. The results give an interpolation between various asymptotic limits.
DOI
10.1088/0022-3719/10/22/024

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A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below Tc{T}_{c}Tc​ than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.
With the aid of Nelson's method we derive expressions for the frequency-, momentum-, and temperature-dependent response function below a critical point in the Ising-type system. The scaling function is given within the renormalization-group formalism at one-loop order for zero external field. The comparison with the corresponding expressions above Tc{T}_{c}Tc​ and with the mean-field approximation is made. We also discuss the dynamic correlation function, focusing on the deviations from the Gaussian expression. The comparison with field-theoretical calculations is also given.