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Physical review. B, Condensed matter

Perturbation calculation around the two-dimensional Ising model

Lee-Fen Ko

Published Jun 1, 1988• 1 citation
Publisher

Abstract

The dispersion expansions for the two-point electric correlation functions in the eight-vertex model are calculated to first order in the four-spin coupling in the scaling limit. This is a systematic study of perturbation around the Ising system in the massive regime; features having general relevance are emphasized.
DOI
10.1103/physrevb.37.9889

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The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, the critical indices obey $\frac{\ensuremath{\gamma}}{{\ensuremath{\gamma}}_{0}}=\frac{\ensuremath{\beta}}{{\ensuremath{\beta}}_{0}}=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}}=1\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}$, $\frac{\ensuremath{\delta}}{{\ensuremath{\delta}}_{0}}=\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}=1$, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model.
We calculate the two-particle contribution to the spectral weight function of the electric correlation functions of the eight-vertex model in the scaling limit to first order in the four-spin coupling. Both T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​ are considered. In both cases the result is remarkably simple. For T>TcT>{T}_{c}T>Tc​, the effect of the emergence of the bound state from the two-particle continuum is seen. The corresponding results for the Ashkin-Teller model and the XYZ\mathrm{XYZ}XYZ spin chain are also given.
The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.