Vortex knots in tangled quantum eigenfunctions

TL;DR

This paper demonstrates that vortex lines in quantum wavefunctions can form complex knots, with the likelihood increasing with vortex length, revealing a topological similarity across wave chaos, polymers, and condensates.

Contribution

It shows that knotted vortex structures are common in quantum eigenfunctions, linking topology in quantum systems to other complex 3D wave phenomena.

Findings

Knotting probability increases with vortex length.

A variety of knots occur in quantum vortex filaments.

Knotted vortices are frequent even at low energies.

Abstract

Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA. Here we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic 3-dimensional harmonic oscillator and the 3-sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex 3-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose-Einstein condensates.