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Communications in Theoretical Physics

Topological Vortices in Superfluid Films

Junping Wang, Yi-Shi Duan

Published Jul 1, 2005• 2 citations
Publisher

Abstract

We study the topological structure of the vortex system in a superfluid film. Explicit expressions for the vortex density and velocity field as functions of the superfluid order parameter are derived. The evolution of vortices is also studied from the topological properties of the superfluid order parameter field.
DOI
10.1088/6102/44/1/160

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The superfluid flow velocity is proportional to the gradient of the phase of the superfluid order parameter, leading to the quantization of circulation around a vortex core. In this work, we study the dynamics of a superfluid film on the surface of a torus. Such a compact surface allows only configurations of vortices with zero net vorticity. We derive analytic expressions for the flow field, the total energy, and the time-dependent dynamics of the vortex cores. The local curvature of the torus and the presence of noncontractable loops on this multiply connected surface alter both the superfluid flow and the vortex dynamics. Finally, we consider more general surfaces of revolution, called toroids.
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.
We present a general method for calculating vortex creation and annihilation rates at arbitrary boundaries in two-dimensional superfluids and give explicit results for the cases of parallel film edges and cylinders. We explain why cylindrical boundaries may provide a reasonable description of vortex shedding and pinning in regions of a superfluid film where the superfluid density is reduced locally to zero, for example, by substrate imperfections. The relation of our work to other recent contributions is discussed, and the current status of the comparison between theoretical predictions and experimental results for the decay of persistent currents in superfluid films is discussed and clarified.
Using the phi-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the delta function indicates that the vortices can only be generated from the zero points of Phi and are quantized in terms of the Hopf indices and Brouwer degrees. The phi-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.
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.