Numerical Studies of the Two Dimensional XY Model with Symmetry Breaking Fields

TL;DR

This paper numerically investigates the phase transitions in the two-dimensional XY model with symmetry-breaking fields, revealing how anisotropy influences transition types and multicritical points, relevant for surface reconstruction phenomena.

Contribution

It provides a detailed numerical analysis of phase transitions and multicritical points in the XY model with four and eight fold symmetry-breaking fields, confirming theoretical predictions.

Findings

Identification of continuous and discontinuous phase transitions.

Location of the multicritical point separating different regimes.

Verification of theoretical predictions through numerical methods.

Abstract

We present results of numerical studies of the two dimensional XY model with four and eight fold symmetry breaking fields. This model has recently been shown to describe hydrogen induced reconstruction on the W(100) surface. Based on mean-field and renormalization group arguments,we first show how the interplay between the anisotropy fields can give rise to different phase transitions in the model. When the fields are compatible with each other there is a continuous phase transition when the fourth order field is varied from negative to positive values. This transition becomes discontinuous at low temperatures. These two regimes are separated by a multicritical point. In the case of competing four and eight fold fields, the first order transition at low temperatures opens up into two Ising transitions. We then use numerical methods to accurately locate the position of the multicritical point, and to verify the nature of the transitions. The different techniques used include Monte Carlo histogram methods combined with finite size scaling analysis, the real space Monte Carlo Renormalization Group method, and the Monte Carlo Transfer Matrix method. Our numerical results are in good agreement with the theoretical arguments.