Anti-Holomorphic Modes in Vortex Lattices

TL;DR

This paper develops a continuum theory for vortex lattice dynamics revealing anti-holomorphic edge modes, validated by numerical results, advancing understanding of vortex lattice excitations.

Contribution

It introduces a novel continuum framework for vortex lattice dynamics based on anti-holomorphic functions, connecting microscopic vortex behavior to macroscopic edge modes.

Findings

Discovery of power-law localized edge modes

Excellent agreement between numerical and theoretical results

Introduction of anti-holomorphic functions in vortex dynamics

Abstract

A continuum theory of linearized Helmholtz-Kirchoff point vortex dynamics about a steadily rotating lattice state is developed by two separate methods: firstly by a direct procedure, secondly by taking the long-wavelength limit of Tkachenko's exact solution for a triangular vortex lattice. Solutions to the continuum theory are found, described by arbitrary anti-holomorphic functions, and give power-law localized edge modes. Numerical results for finite lattices show excellent agreement to the theory.