Exact Calculation of the Vortex-Antivortex Interaction Energy in the Anisotropic 3D XY-model

TL;DR

This paper presents an exact method to compute vortex-antivortex interaction energy in the anisotropic 3D XY-model, extending dual transformation techniques from 2D, revealing a transition from logarithmic to linear interaction dependence on separation.

Contribution

The authors extend the dual transformation method to the 3D XY-model for the first time, providing an explicit formula for vortex-antivortex interaction energy considering anisotropy.

Findings

Interaction energy scales as ln r at small separations.

Interaction energy becomes proportional to r at large separations.

Results align with previous variational upper bound calculations.

Abstract

We have developed an exact method to calculate the vortex-antivortex interaction energy in the anisotropic 3D-XY model. For this calculation, dual transformation which is already known for the 2D XY-model was extended. We found an explicit form of this interaction energy as a function of the anisotropic ratio and the separation $r$ between the vortex and antivortex located on the same layer. The form of interaction energy is $\ln r$ at the small $r$ limi t but is proportional to $r$ at the opposite limit. This form of interaction energ y is consistent with the upper bound calculation using the variational method by Cataudella and Minnhagen.