Cyclotron motion of a quantized vortex in a superfluid

TL;DR

This paper develops a microscopic theory describing the cyclotron motion of quantized vortices in superfluids, revealing their energy spectrum, degeneracy, and effective mass, and drawing analogies with electron Landau levels.

Contribution

It introduces a variational approach to model vortex dynamics, deriving an integral equation for superposition weights and analyzing vortex energy levels and effective mass.

Findings

Vortex energy levels are highly degenerate, similar to Landau levels.

The energy gap depends on quantized circulation and vortex size.

The vortex effective mass diverges logarithmically with vortex size.

Abstract

In two dimensions a microscopic theory providing a basis for the naive analogy between a quantized vortex in a superfluid and an electron in a uniform magnetic field is presented. Following the variational approach developed by Peierls, Yoccoz, and Thouless, the cyclotron motion of a vortex is described by the many-body wave function, which is a linear combination of Feynman wave functions centered at different positions. An integral equation for the weighting functions of the superposition is derived by minimizing the energy functional. The matrix elements of the kernel are the overlaps between any two displaced Feynman wave functions. A numerical study is conducted for a bosonic superfluid based on a Hartree ground state. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with energy, and scales with the size of the vortex. The fluid density is finite at the vortex axis and the vorticity is distributed in the core region. The effective mass of a quantized vortex defined by the inverse of the energy-level spacing is shown to be logarithmically divergent with the size of the vortex.