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

TL;DR

This paper derives the exact asymptotic behavior of the XY-model's spin correlation function at the Kosterlitz-Thouless transition, confirming the critical exponent $=1/4$ through sine-Gordon and Coulomb gas theories.

Contribution

It provides an exact derivation of the large-distance correlation function and critical exponent at the transition, using sine-Gordon and Coulomb gas formalisms, improving upon previous scaling arguments.

Findings

Exact asymptotic correlation function at transition

Critical exponent =1/4 confirmed

Method applicable to vortex correlations

Abstract

We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent $\eta=1/4$ at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which 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 description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at $\beta^2=8\pi$, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure.