# The Kosterlitz-Thouless Universality Class

## TL;DR

This paper investigates the Kosterlitz-Thouless universality class, highlighting the necessity of logarithmic corrections for consistency, and introduces new numerical methods to analyze phase transitions in related models.

## Contribution

It introduces a novel numerical approach involving Lee-Yang zeroes and index scaling to study phase transitions and logarithmic corrections in the Kosterlitz-Thouless class.

## Key findings

- Logarithmic corrections are essential for consistent essential scaling.
- The XY-model and step model share the same universality class.
- Numerical corrections differ from renormalization group predictions.

## Abstract

We examine the Kosterlitz-Thouless universality class and show that essential scaling at this type of phase transition is not self-consistent unless multiplicative logarithmic corrections are included. In the case of specific heat these logarithmic corrections are identified analytically. To identify those corresponding to the susceptibility we set up a numerical method involving the finite-size scaling of Lee-Yang zeroes. We also study the density of zeroes and introduce a new concept called index scaling. We apply the method to the XY-model and the closely related step model in two dimensions. The critical parameters (including logarithmic corrections) of the step model are compatable with those of the XY-model indicating that both models belong to the same universality class. This result then raises questions over how a vortex binding scenario can be the driving mechanism for the phase transition. Furthermore, the logarithmic corrections identified numerically by our methods of fitting are not in agreement with the renormalization group predictions of Kosterlitz and Thouless.