Logarithmic Corrections to Scaling in the Two Dimensional $XY$--Model

TL;DR

This paper demonstrates that in the 2D XY model, standard Kosterlitz-Thouless scaling must be modified by logarithmic corrections for consistency, with analytical and numerical identification of these corrections for specific heat and susceptibility.

Contribution

It analytically and numerically identifies logarithmic corrections to scaling in the 2D XY model, challenging the conventional Kosterlitz-Thouless predictions.

Findings

Logarithmic corrections are necessary for consistent scaling.

Analytical derivation of corrections for specific heat.

Numerical confirmation of corrections for susceptibility.

Abstract

By expressing thermodynamic functions in terms of the edge and density of Lee--Yang zeroes, we relate the scaling behaviour of the specific heat to that of the zero field magnetic susceptibility in the thermodynamic limit of the $XY$--model in two dimensions. Assuming that finite--size scaling holds, we show that the conventional Kosterlitz--Thouless scaling predictions for these thermodynamic functions are not mutually compatable unless they are modified by multiplicative logarithmic corrections. We identify these logarithmic corrections analytically in the case of the specific heat and numerically in the case of the susceptibility. The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.