Theory of Photoluminescence of the $ν=1$ Quantum Hall State: Excitons, Spin-Waves and Spin-Textures

We study the theory of intrinsic photoluminescence of two-dimensional electron systems in the vicinity of the $\nu=1$ quantum Hall state. We focus predominantly on the recombination of a band of initial ``excitonic states'' that are the low-lying energy states of our model at $\nu=1$. It is shown that the recombination of excitonic states can account for recent observations of the polarization-resolved spectra of a high-mobility GaAs quantum well. The asymmetric broadening of the spectral line in the $\sigma_-$ polarization is explained to be the result of the ``shake-up'' of spin-waves upon radiative recombination of excitonic states. We derive line shapes for the recombination of excitonic states in the presence of long-range disorder that compare favourably with the experimental observations. We also discuss the stabilities and recombination spectra of other (``charged'') initial states of our model. An additional high-energy line observed in experiment is shown to be consistent with the recombination of a positively-charged state. The recombination spectrum of a negatively-charged initial state, predicted by our model but not observed in the present experiments, is shown to provide a direct measure of the formation energy of the smallest ``charged spin-texture'' of the $\nu=1$ state.