Vortex Lattice Theory: A Particle Interaction Perspective

Abstract

Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete symmetries and can form in a random environment. We give an overview of an $N$-particle based Hamiltonian theory which, if formulated in terms of the $O(N^2)$ interparticle distances, leads to the analysis of a nonnormal “configuration” matrix whose nullspace structure determines the existence or nonexistence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths, in principle, can be classified and calculated by a random-walk scheme which systematically uses the $m$ smallest singular values as a ratchet mechanism to home in on lattices with $m$-dimensional nullspaces, where $0 < m \le N$. The resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice, including its size (as measured by the Frobenius or 2-norms), distance between the lattices (hence lattice density), robustness, and Shannon entropy as a quantitative measure of its level of disorder. This article gives an overview of vortex lattice theory from 1957 to the present, highlighting recent experiments in Bose-Einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix. We discuss some of the computational challenges associated with producing large $N$ lattices, the subtleties associated with understanding and exploiting complicated Hamiltonian energy surfaces in high dimensions, and we highlight ten important directions for future research in this area.