Short range intervortex forces

TL;DR

This paper derives an explicit formula for the short-range interaction energy of multiple vortices in the abelian Higgs model, revealing a $d^4$ dependence for vortex pairs and validating it with numerical and simulation results.

Contribution

It provides the first explicit formula for multi-vortex interaction energy in the near-coalescence regime, including spectral data and numerical coefficients.

Findings

Interaction energy varies as $d^4$ for vortex pairs.

Numerical coefficients computed for $n=2,3$ and various couplings.

Excellent agreement with full field theory simulations at small to moderate separations.

Abstract

An explicit formula for the interaction energy of $n$ vortices in the abelian Higgs (or Ginzburg-Landau) model is derived, valid in the regime where all vortices are close to one another. An immediate consequence of this formula is that the interaction energy of a vortex pair with separation $d$ varies as $d^4$, not $d^2$. The formula contains $n-1$ real coefficients which are fixed by certain spectral data of the Jacobi operator of the cocentred $n$-vortex. The coefficients are computed numerically for $n=2$ and $n=3$ for couplings $0.1\leq \lambda\leq 2.5$. The resulting short range interaction potentials are compared with the results of full field theory simulations for $\lambda=0.5$ and $\lambda=2$, with excellent agreement at small to moderate vortex separation.