Correlation functions for the $XY$ model in a Magnetic Field

TL;DR

This study analyzes spin-spin correlation functions in the 2D XY model under a magnetic field, identifying distinct vortex phases and their signatures through analytical and numerical methods, revealing long-range order and anisotropic behaviors.

Contribution

It provides a detailed analysis of correlation functions in different vortex phases of the XY model with a magnetic field, highlighting anisotropic effects and long-range order signatures.

Findings

Long-range order persists in all phases.

Correlation functions decay exponentially for transverse fluctuations.

Anisotropy in correlation length and prefactors is observed, especially under magnetic field.

Abstract

Recent studies of the two-dimensional, classical $XY$ magnet in a magnetic field suggest that it has three distinct vortex phases: a linearly confined phase, a logarithmically confined phase, and a free vortex phase. In this work we study spin-spin correlation functions in this model by analytical analysis and numerical simulations to search for signatures of the various phases. In all three phases, the order parameter is nonzero and $<\cos(\th({\bf r}_1))\cos(\th({\bf r}_2))>$ remains nonzero for $r \equiv |{\bf r}_1-{\bf r}_2|\rightarrow \infty$, indicating the expected long range order. The correlation function for transverse fluctuations of the spins, $C(r)=<\sin(\th({\bf r}_1))\sin(\th({\bf r}_2))>$, falls exponentially in all three phases. A renormalization group analysis suggests that the logarithmically confined phase should have a spatially anisotropic correlation length. In addition, there is a generic anisotropy in the prefactor which is always present. We find that this prefactor anisotropy becomes rather strong in the presence of a magnetic field, masking the effects of any anisotropy in the correlation length in the simulations.