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Physical Review Letters

Quantum dynamics and statistics of vortices in two-dimensional superfluids

F. D. M. Haldane, Yong-Shi Wu

Published Dec 23, 1985• 131 citations
Publisher

Abstract

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 $^{4}\mathrm{He}$ films.

DOI
10.1103/physrevlett.55.2887

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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.
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.
Abstract Within the past decade there has been a resurgence of interest in the dynamics of superfluid and superconducting vortices. One source of motivation for this topic may be attributed to the discovery of the quantum Hall effect (von Klitzing et al., 1980), which led to the suggestion that elementary excitations in 2D systems may obey nonstandard statistics (Wilczek, 1982; Halperin, 1984). Haldane and Wu (1985) pointed out a powerful analogy between the dynamics of a vortex in a 2D superfluid film and the electron motion in a cyclotron orbit. critical value.
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.
Abstract Vorticity in two-dimensional superfluids is subject to intense research efforts due to its role in quantum turbulence, dissipation and the BKT phase transition. Interaction of sound and vortices is of broad importance in Bose–Einstein condensates and superfluid helium. However, both the modelling of the vortex flow field and of its interaction with sound are complicated hydrodynamic problems, with analytic solutions only available in special cases. In this work, we develop methods to compute both the vortex and sound flow fields in an arbitrary two-dimensional domain. Further, we analyse the dispersive interaction of vortices with sound modes in a two-dimensional superfluid and develop a model that quantifies this interaction for any vortex distribution on any two-dimensional bounded domain, possibly non-simply connected, exploiting analogies with fluid dynamics of an ideal gas and electrostatics. As an example application we use this technique to propose an experiment that should be able to unambiguously detect single circulation quanta in a helium thin film.