Theory of photoluminescence from a magnetic-field-induced two-dimensional quantum Wigner crystal

We develop a theory of photoluminescence using a time-dependent Hartree-Fock approximation that is appropriate for the two-dimensional Wigner crystal in a strong magnetic field. The cases of localized and itinerant holes are both studied. It is found that the photoluminescence spectrum is a weighted measure of the single-particle density of states of the electron system, which for an undisturbed electron lattice has the intricate structure of the Hofstadter butterfly. It is shown that for the case of a localized hole, a strong interaction of the hole with the electron lattice tends to wipe out this structure. In such cases, a single final state is strongly favored in the recombination process, producing a single line in the spectrum. For the case of an itinerant hole, which could be generated in a wide quantum-well system, we find that electron-hole interactions do not significantly alter the density of states of the Wigner crystal, opening the possibility of observing the Hofstadter gap spectrum in the electron density of states directly. At experimentally relevant filling fractions, these gaps are found to be extremely small, due to exchange effects. However, it is found that the hole, which interacts with the periodic potential of the electron crystal, has a Hofstadter spectrum with much larger gaps. It is shown that a finite-temperature experiment would allow direct probing of this gap structure through photoluminescence.