Vortex dynamics in self-dual Chern-Simons-Higgs systems

Abstract

We consider vortex dynamics in self-dual Chern-Simons Higgs systems. We show that the naive Aharanov-Bohm phase is the inverse of the statistical phase expected from the vortex spin, and that the self-dual configurations of vortices are degenerate in energy but not in angular momentum. We also use the path integral formalism to derive the dual formulation of Chern-Simons Higgs systems in which vortices appear as charged particles. We argue that besides the electromagnetic interaction, there is an additional interaction between vortices, the so-called Magnus force, and that these forces can be put together into a single 'dual electromagnetic' interaction. This dual electromagnetic interaction leads to the right Aharanov-Bohm phase. We also derive and study the effective action for slowly moving vortices, which contains terms both linear and quadratic in the vortex velocity.

First and second order vortex dynamics

“…This looks strange but is expected from the spin statistics as we have explained in Ref. [5]. We review this briefly in the section where two vortex dynamics is discussed.…”

“…While the theory is intrinsically relativistic, the low energy effective Lagrangian found in Ref. [5] was only of first order in vortex velocity. This first order effective Lagrangian was shown to be consistent and describes fractional statistics of vortices.…”

“…We should note that the statistics of vortices with nonzero fractional spin is not given by the naive Aharonov-Bohm phase, rather it is given by the sum of the Aharonov-Bohm phase and the additional Berry phase due to the 'Magnus force' [5]. This has led to the explanation for the famous sign flip of vortex spin compared with the spin of elementary charged particles in the symmetric phase.…”

“…(For the slow motion of selfdual vortices on Poincare upper half plane, there exists a beautiful work by Strachan [4].) There has been an attempt to find a similar effective Lagrangian for vortices in pure Chern-Simons Higgs theories [5]. While the theory is intrinsically relativistic, the low energy effective Lagrangian found in Ref.…”

“…From the study of two vortex dynamics, we can expect nontrivial bound states with binding energy of order of elementary particle mass when the Chern-Simons coefficient κ is much larger than one [5]. The binding energy would be order of elementary particle mass.…”