Topological Landau-Ginzburg theory for vortices in superfluid 4He.

We propose a new Landau-Ginzburg theory for arbitrarily shaped vortex strings in superfluid He. The theory contains a topological term and directly describes vortex dynamics. We introduce gauge fields in order to remove singularities from the Landau-Ginzburg order parameter of the superfluid, so that two kinds of gauge symmetries appear, making the continuity equation and conservation of the total vorticity manifest. The topological term gives rise to the Berry phase term in the vortex mechanical actions. Since the existence of quantized vortices was predicted by Onsager and Feynman, vortices have been observed in superfluid He and He, and in superconductor systems. At low temperature in superfluid helium the quantized vortex obeys the classical hydrodynamical law that the vortex moves with the local velocity of the fluid, while the vorticity quantization comes from the fact that the superfluid is a quantum state described by a wavefunction. The vortex dynamics is governed by classical hydrodynamics and the quantum aspects of the system is governed by a non-linear Schrödinger equation which is equivalent to the Landau-Ginzburg theory for superfluid developed by Ginzburg, Pitaevskii and Gross (GPG) [1]. By inserting a suitable form for the phase of the field by hand on a case-by-case basis it has been shown that a Landau-Ginzburg theory produces vortex dynamics [2, 3]. However there is no satisfactory theory which describes both vortex dynamics and quantum properties of the vortex. In this paper we propose a topological Landau-Ginzburg theory for vortices in superfluid He. The characteristic features of our formalism are as follows. (i) We introduce a gauge field, Aμ, in the GPG theory in such a way that Aμ carries the singularities in the phase of the Landau-Ginzburg order parameter: the phase therefore becomes single valued. In order not to change physical observables we introduce Aμ gauge covariantly, and we choose the condition that the dual field strength of Aμ coincides with the vorticity tensor. This condition is imposed by using a rank two antisymmetric tensor Lagrange multiplier, Bμν . There are two kinds of gauge symmetries, which lead to the continuity equation and to the conservation of the total vorticity. (ii) A topological term, “BF term” (εBμνFρλ), where Fμν is the field strength, Fμν = ∂μAν − ∂νAμ, and a coupling term, BμνJ , to the vorticity tensor J are required to reproduce the Berry phase term. Because the BF term does not couple to the 3+1 dimensional metric, it is a so-called topological term. In general the BF term is used in evaluating linking numbers which are topological numbers counting how many times a string and a membrane are entangled in 3+1 dimensions [4], while the Berry phase term in the vortex mechanical action is similar to the Hopf term which counts the instanton number in the O(3) nonlinear sigma model [5]. The topological BF term is a generalization of the Chern-Simons term which plays an important role in the study of the fractional quantized Hall effect and anyon systems in 2+1 dimensions [5]. It is desirable to include such a topological term since a vortex is a topological excitation in the sense that it is not obtained by a continuous deformation from the ground state. (iii) The vorticity tensor, whose time components, J, correspond to a vorticity vector, has general form so that it can describe arbitrarily shaped vortex strings or rings. Regularization, if needed, involves regularizing only this tensor, so that the density ρ and the velocity, v, never become singular. (iv) Since the vorticity tensor contains vortex coordinates explicitly, this action directly leads to the equation of motion of vortices as well as the field equations for the order parameter. This action also reproduces the correct vortex mechanical action which contains the Berry phase term.