An electric-field representation of the harmonic XY model

TL;DR

This paper introduces a lattice electric-field representation of the 2D harmonic XY model, revealing how local interactions produce effective long-range correlations and elucidating the role of spin vortices and fluctuations in phase transitions.

Contribution

It formulates a novel electric-field framework for the HXY model, connecting spin vortices to Coulomb charges and explaining temperature-dependent behaviors.

Findings

HXY model acts as a lattice electric-field analogue of the Coulomb gas.

The model exhibits a temperature-dependent vacuum permittivity.

Global spin-twist excitations are ergodically excluded at low temperatures.

Abstract

The two-dimensional harmonic XY (HXY) model is a spin model in which the classical spins interact via a piecewise parabolic potential. We argue that the HXY model should be regarded as the canonical classical lattice spin model of phase fluctuations in two-dimensional condensates, as it is the simplest model that guarantees the modular symmetry of the experimental systems. Here we formulate a lattice electric-field representation of the HXY model and contrast this with an analogous representation of the Villain model and the two-dimensional Coulomb gas with a purely rotational auxiliary field. We find that the HXY model is a spin-model analogue of a lattice electric-field model of the Coulomb gas with an auxiliary field, but with a temperature-dependent vacuum (electric) permittivity that encodes the coupling of the spin vortices to their background spin-wave medium. The spin vortices map to the Coulomb charges, while the spin-wave fluctuations correspond to auxiliary-field fluctuations. The coupling explains the striking differences in the high-temperature asymptotes of the specific heats of the HXY model and the Coulomb gas with an auxiliary field. Our results elucidate the propagation of effective long-range interactions throughout the HXY model (whose interactions are purely local) by the lattice electric fields. They also imply that global spin-twist excitations (topological-sector fluctuations) generated by local spin dynamics are ergodically excluded in the low-temperature phase. We discuss the relevance of these results to condensate physics.