Quantized vortices and superflow in arbitrary dimensions: Structure, energetics and dynamics

TL;DR

This paper generalizes the structure, energetics, and dynamics of quantized vortices in superfluids to arbitrary dimensions, revealing how vortex geometry influences their behavior and energy in higher-dimensional spaces.

Contribution

It introduces a framework for analyzing vortices of various co-dimensions in any dimension, extending superfluid vortex theory beyond three dimensions.

Findings

Vortex superflow can be constructed using an analogy with magnetostatics.

Energy expressions simplify for large, smooth vortices.

Vortex motion depends on both extrinsic and intrinsic geometric properties.

Abstract

The structure and energetics of superflow around quantized vortices, and the motion inherited by these vortices from this superflow, are explored in the general setting of the superfluidity of helium-four in arbitrary dimensions. The vortices may be idealized as objects of co-dimension two, such as one-dimensional loops and two-dimensional closed surfaces, respectively, in the cases of three- and four-dimensional superfluidity. By using the analogy between vorticial superflow and Ampere-Maxwell magnetostatics, the equilibrium superflow containing any specified collection of vortices is constructed. The energy of the superflow is found to take on a simple form for vortices that are smooth and asymptotically large, compared with the vortex core size. The motion of vortices is analyzed in general, as well as for the special cases of hyper-spherical and weakly distorted hyper-planar vortices. In all dimensions, vortex motion reflects vortex geometry. In dimension four and higher, this includes not only extrinsic but also intrinsic aspects of the vortex shape, which enter via the first and second fundamental forms of classical geometry. For hyper-spherical vortices, which generalize the vortex rings of three dimensional superfluidity, the energy-momentum relation is determined. Simple scaling arguments recover the essential features of these results, up to numerical and logarithmic factors.