A Field theory for partially polarized quantum Hall states

We propose an effective field theory for partially polarized quantum Hall states. The density and polarization for the mean-field ground states are determined by couplings to two Chern-Simons gauge fields. In addition, there is a $\ensuremath{\sigma}$-model field, $\mathrm{m\ifmmode \hat{}\else \^{}\fi{}},$ which is necessary both to preserve the Chern-Simons gauge symmetry that determines the correlations in the ground state, and the global SU(2) invariance related to spin rotations. For states with nonzero polarization, the low-energy dynamics is that of a ferromagnet. In addition to spin waves, the spectrum contains topological solitons, or skyrmions, just as in the fully polarized case. The electric charge of the skyrmions is given by ${Q}_{\mathrm{el}}=\ensuremath{\nu}{\mathrm{PQ}}_{\mathrm{top}},$ where $\ensuremath{\nu}$ is the filling fraction, P the magnitude of the polarization, and ${Q}_{\mathrm{top}}$ the topological charge. For the special case of full polarization, the theory involves a single scalar field and a single Chern-Simons field in addition to the $\ensuremath{\sigma}$-model field, $\mathrm{m\ifmmode \hat{}\else \^{}\fi{}}.$ We also give a heuristic derivation of the model Lagrangians for both full and partial polarization, and show that in a mean-field picture, the field $\mathrm{m\ifmmode \hat{}\else \^{}\fi{}}$ is necessary in order to take into account the Berry phases originating from rotations of the electron spins.