Sine-Gordon/Coulomb Gas Soliton Correlation Functions and an Exact Evaluation of the Kosterlitz-Thouless Critical Exponent

摘要

We present an exact derivation for the asymptotic large distance behavior ofthe spin two-point correlation function in the XY-model. This allows for theexact obtainment of the critical exponent $\eta=1/4$ at the Kosterlitz-Thoulesstransition that occurs in this model and in the 2D neutral Coulomb gas andwhich has been previously obtained by scaling arguments. In order to do that,we use the language of sine-Gordon theory to obtain a Coulomb Gas descriptionof the XY-model spin correlation function, which becomes identified with thesoliton correlator of that theory. Using a representation in terms of bipolarcoordinates we obtain an exact expression for the asymptotic large distancebehavior of the relevant correlator at $\beta^2=8\pi$, which corresponds to theKosterlitz-Thouless transition. The result is obtained by approaching thispoint from the plasma (high-temperature) phase of the gas. The vortexcorrelator of the XY-model is also obtained using the same procedure.