Derivative Expansion of the Effective Action and Vacuum Instability for QED in 2+1 Dimensions

We investigate the effective action of 2+1 dimensional charged spin 1/2 fermions and spin 0 bosons in the presence of a $U(1)$ gauge field. We evaluate terms in an expansion up to second order in derivatives of the field strength, but exactly in the mass parameter and in the magnitude of the nonvanishing constant field strength. We find that in a strong uniform magnetic field background, space-derivative terms lower the energy, and there arises an instability toward inhomogeneous magnetic fields.