Effective Lagrangian for a system of nonrelativistic fermions in 2+1 dimensions coupled to an electromagnetic field: Application to anyonic superconductors

Abstract

We study here, in two spatial dimensions, the effective Lagrangian of nonrelativistic charged fermions in an electromagnetic field. The fermionic integration is performed yielding a one-loop effective action that is evaluated using the inhomogeneity (derivative) expansion technique. The effective Lagrangian involves a Chern-Simons (CS) -like term with a coefficient that is a ``staircase'' function of B, the magnetic field. We then discuss the application of this effective Lagrangian to a system of anyons, showing that the cancellation of the induced CS term against a CS term included in the beginning to change the fermions to anyons, is favored energetically, together with the expulsion of B from the system of anyons. The cancellation implies the existence of a massless mode. This, together with the fact that B=0, implies superconductivity.