Logarithmic Corrections to Scaling in the $XY_2$--Model

TL;DR

This paper investigates the distribution of partition function zeroes in the 2D XY model, revealing logarithmic corrections to scaling behavior through numerical analysis, which can be applied to other models.

Contribution

It introduces a method to identify logarithmic corrections to scaling in the XY model and demonstrates their presence through numerical analysis.

Findings

Logarithmic corrections to scaling are necessary in the XY model.

Finite-size scaling formulas include these logarithmic corrections.

The method can be applied to other models to verify scaling behavior.

Abstract

We study the distribution of partition function zeroes for the $XY$--model in two dimensions. In particular we find the scaling behaviour of the end of the distribution of zeroes in the complex external magnetic field plane in the thermodynamic limit (the Yang--Lee edge) and the form for the density of these zeroes. Assuming that finite--size scaling holds, we show that there have to exist logarithmic corrections to the leading scaling behaviour of thermodynamic quantities in this model. These logarithmic corrections are also manifest in the finite--size scaling formulae and we identify them numerically. The method presented here can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.