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Physical review. B, Condensed matter

Intermediate statistics for vortices in superfluid films

Jon Magne Leinaas, Jan Myrheim

Published Jun 1, 1988• 95 citations
Publisher

Abstract

A quantization of the classical theory of a two-vortex system in a superfluid film is carried out. This is done by finding the representations of an algebra of observables which respect the fact that the vortices are identical objects. The representations depend on an undetermined parameter \ensuremath{\theta}, which may be interpreted as defining intermediate types of statistics for the vortices. We relate our approach to a previous discussion, where only two values for \ensuremath{\theta} were claimed to be possible.

DOI
10.1103/physrevb.37.9286

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A quantization of the classical theory of a two-vortex system in a superfluid film is carried out. This is done by finding the representations of an algebra of observables which respect the fact that the vortices are identical objects. The representations depend on an undetermined parameter \ensuremath{\theta}, which may be interpreted as defining intermediate types of statistics for the vortices. We relate our approach to a previous discussion, where only two values for \ensuremath{\theta} were claimed to be possible.
The phase change of the wave function of a superfluid film as a vortex moves around a closed path counts the number of superfluid particles enclosed by that path. This result is used to investigate whether such vortices obey ``fractional statistics.'' We conclude that this is not the case in compressible superfluids such as 4He^{4}\mathrm{He}4He films.
We give a general review of recent developments in the theory of vortices in superfluids and superconductors, discussing why the dynamics of vortices is important, and why some key results are still controversial. We discuss work that we have done on the dynamics of quantized vortices in a superfluid. Despite the fact that this problem has been recognized as important for forty years, there is still a lot of controversy about the forces on and masses of quantized vortices. We think that one can get unambiguous answers by considering a broken symmetry state that consists of one vortex in an infinite ideal system. We argue for a Magnus force that is proportional to the superfluid density, and we find that the effective mass density of a vortex in a neutral superfluid is divergent at low frequencies. We have generalized some of the results for a neutral superfluid to a charged system.
The coupling of superfluid film to a moving vortex is a gauge coupling entirely dictated by topology. From the definition of a linking number, one can define a gauge field ${\mathrm{A}}^{\mathrm{\ensuremath{\mu}}}$ , whose (2+1)-dimensional curl is the vortex three-current ${\mathrm{J}}^{\mathrm{\ensuremath{\mu}}}$ , and to which the superfluid is minimally coupled. We compute the superfluid density and current response to a moving vortex. Exploiting the analogy to (2+1)-dimensional electrodynamics, we compute the effective vortex mass M(\ensuremath{\omega}) and find that it is logarithmically divergent in the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, with a constant imaginary part, yielding a super-Ohmic dissipation in the presence of an oscillating superflow. Numerical integration of the nonlinear Schr\"odinger equation supports these conclusions. The interaction of vortices with impurities coupling to the density also is discussed.