Quantized vortices in arbitrary dimensions and the normal-to-superfluid phase transition

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 a superfluid 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 vortical superflow and Ampµere-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. Extending these results to systems containing multiple vortices is elementary due to the linearity of the theory. The energy for multiple vortices is thus a sum of self-energies and power-law interaction terms. The statistical mechanics of a system containing vortices is addressed via the grand canonical partition function. A renormalization-group analysis in which the low energy excitations are integrated approximately, is used to compute certain critical coefficients. The exponents obtained via this approximate procedure are compared with values obtained previously by other means. For dimensions higher than three the superfluid density is found to vanish as the critical temperature is approached from below.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2010

Author dc:creator

• Bora, Florin

Contributors dc:contributor

• Stack, John D.
• Mason, Nadya
• Goldbart, Paul M.
• Vishveshwara, Smitha

• superfluidity
• quantized vortices
• N-dimensional superfluidity
• vortex
• manifolds

Statement dc:rights

• Copyright 2010 Florin Bora

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/16096

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/16096