The Binder Cumulant at the Kosterlitz-Thouless Transition

TL;DR

This paper investigates the Binder cumulant at the Kosterlitz-Thouless transition using Monte Carlo simulations, determining fixed point values, corrections, and a new method for accurately locating the transition temperature.

Contribution

It introduces a combined analysis method of the Binder cumulant and correlation length to precisely determine the transition temperature on small lattices.

Findings

Fixed point value of the Binder cumulant determined

Leading logarithmic correction coefficient calculated

New method accurately finds transition temperature on small lattices

Abstract

We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L=4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component phi^4 model on the lattice.