On fermions in a plane coupled to the nonlinear sigma model

A gas of fermions in two spatial dimensions, placed in a uniform magnetic field, and coupled to the O(3) nonlinear σ model, is studied. A new gauge field is introduced via a Hubbard-Stratonovich transformation, and, upon integration over the fermion fields, the effective action is computed to first order in derivatives of the new field and of the electromagnetic field. We also discuss a model which has only a mixed Chern-Simons term in the effective action. It involves two types of fermions with opposite electric charge, but the same coupling to the σ model.