A Field Theory for Partially Polarized Quantum Hall States

TL;DR

This paper introduces a new effective field theory for partially polarized quantum Hall states, incorporating gauge fields and a sigma-model to describe spin dynamics, skyrmions, and their topological properties.

Contribution

It develops a comprehensive field theory framework that captures the spin polarization, topological excitations, and gauge symmetries of partially polarized quantum Hall states, extending previous models.

Findings

The theory describes spin waves and skyrmions in partially polarized states.

Skyrmion charge relates to filling fraction, polarization, and topological charge.

In the fully polarized case, the model simplifies to a scalar and Chern-Simons field with a sigma-model field.

Abstract

We propose a new 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 $\sigma$-model field, $\mh$, 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 non zero 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 $Q_{el}=\nu P Q_{top}$, where $\nu$ is the filling fraction, $P$ the magnitude of the polarization, and $Q_{top}$ 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 $\sigma$-model field, $\mh$. 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 $\mh$ is necessary in order to take into account the Berry phases originating from rotations of the electron spins.