Vortices in superconducting films are anyons

We point out that nontrivial quantumstatistics of vortices in planar superfluid systems has its origin in the Schrodinger equation rather then the Chern-Simons term. Vortices in superfluid helium films are not anyons because helium films are compressible. Quantum Hall effect layers admit vortices with nontrivial statistical interactions thanks to their incompressibility. We consider type II planar superconductors described by time-dependent nonlinear Schrodinger equation. Inside of a vortex core superconductor is in the normal state. Because of that vortices in superconducting films are anyons. We consider also vortex loops in 3 dimensions but a phase factor due to linking of vortex loops’ worldsheets is not a topological invariant. TPJU 9/95 cond-mat/9504??? Interactions of vortices in superfluids described by the nonlinear Schrodinger equation are by now a well studied problem [1, 2, 3, 4, 5]. What has not been studied in detail is the quantum theory of vortices. In [4] there has been proposed a topological Landau-Ginzburg theory for vortices in superfluid helium. The theory is equivalent to the standard nonlinear Schrodinger equation provided the topology of vortex lines is determined. The authors of [4] have derived, in 3-dimensional case, Berry phase picked up by the microscopic wave-function due to vortex-loops’ motion. In this paper we consider mainly planar systems which may be relevant to description of thin superfluid films. It is shown that there is a Berry phase proportial to the area enclosed by the vortex trajectory. Mutual interactions are considered and it appears that vortices might be anyons if a superfluid helium film were not compressible. Thus we try to find a model with an incompressible condensate. The phenomenological models for quantum Hall effect appear to be a well known example. However we argue that Chern-Simons term is not essential for transmutation of vortices’ quantumstatistics. In fact the statistical interactions come solely from the Schrodinger term. Finally we consider type II superconductors. Thanks to the normal current carriers inside a vortex core a defect of superfluid charge inside a vortex with respect to uniform background can be nonzero. It e-mail address: [email protected] address from October 1, 1995 Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, United Kingdom is shown that vortices in superconducting films are anyons. Vortices in superfluid helium films. In the local limit the Lagrangian density of the model for interacting hard-core bosons is [6] L = ih̄Ψ⋆∂t̃Ψ − h̄ 2m ∇̃Ψ∇̃Ψ+ μΨΨ− 1 2 λ(ΨΨ) , (1) where Ψ is the condensate wave-function defined on the plane. We assume quantisation of the model by path integrals does make sense. It is convenient to rescale the fields and coordinates in the above formula