Fermionic Chern-Simons theory for the Fractional Quantum Hall Effect in Bilayers

We generalize the fermion Chern-Simons theory for the Fractional Hall Effect (FQHE) which we developed before, to the case of bilayer systems. We study the complete dynamic response of these systems and predict the experimentally accessible optical properties. In general, for the so called $(m, m, n)$ states, we find that the spectrum of collective excitations has a gap, and the wave function has the Jastrow-Slater form, with the exponents determined by the coefficients $m$, and $n$. We also find that the $(m,m,m)$ states, {\it i.~e.~}, those states whose filling fraction is $1\over m$, have a gapless mode which may be related with the spontaneous appearance of the interlayer coherence. Our results also indicate that the gapless mode makes a contribution to the wave function of the $(m,m,m)$ states analogous to the phonon contribution to the wave function of superfluid $\rm{He}_4$. We calculate the Hall conductance, and the charge and statistics of the quasiparticles. We also present an $SU(2)$ generalization of this theory relevant to spin unpolarized or partially polarized single layers.