The Lorentz Force Exerted by the Aharonov—Bohm Flux Line

  • Andrei Shelankov
  • A. F. Ioffe
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 132)


Vortices in superfluids share certain similarities with the magnetic flux lines introduced to quantum mechanics by Aharonov and Bohm [1–3]. The Aharonov-Bohm (AB) line is a source of the vector potential A defined via the circulation ∳ dlA = Φ, the same for any contour around the line. The magnetic field B = rot A is zero at every point except the line, and the total magnetic flux is fixed to Φ. The superfluid vortex circulation k is analogous to the magnetic flux Φ, and lines of the super-current around the vortex resembles the distribution of the vector potential A around the AB-line. Furthermore, the wave equation for a phonon scattered by a vortex [4] can be presented in a form identical to the Schrödinger equation for a charge in the field of the AB-line. Of course, the analogy should not be extended too far. Unlike the purely gauge vector potential, rot A — O, which can be chosen almost arbitrary due to the gauge invariance, the superflow is an observable field and the region around the vortex is not force-free. Nevertheless, the analogy has proven to be rather useful in the vortex theory, in particular, for the understanding of the origin of the Iordanskii force [5] (see also References [4]). One may say that the analog to the Iordanskii force is the transverse force exerted by the AB-line, which is a quantum version of the classical Lorentz force.


Incident Wave Vector Potential Lorentz Force Flux Line Transverse Force 
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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Andrei Shelankov
  • A. F. Ioffe

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