Series of effective-field approximations and coherent anomaly in Kosterlitz-Thouless transitions

Abstract

A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi n(Kc(n)-K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y0)/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu 0, nu < nu 0=1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta c, of the spin-spin correlation function at the critical point is found to be eta c=1/(4+1/ pi 2).