Probing fractional quantum Hall effect by photoluminescence

The recent discovery of fractional quantum anomalous Hall states, fractional quantum Hall states realized without an external magnetic field, in twisted transition-metal dichalcogenide (TMD) bilayers represents a significant development in condensed-matter physics. These states were first observed via photoluminescence (PL) spectroscopy. Surprisingly, a general theoretical understanding of PL is not available even for the conventional fractional quantum Hall states. Our study of a fully spin-polarized system brings out the following picture. At the Jain filling factors $\ensuremath{\nu}=n/(2n\ifmmode\pm\else\textpm\fi{}1)$, when an electron is excited from the valence to the conduction band, the photoexcited electron and the hole lower their energies by turning into composite-fermion (CF) particles and holes. These are charge $\ensuremath{\mp}e/(2n\ifmmode\pm\else\textpm\fi{}1)$ anyons and, in general, form a bound CF exciton with energy below the band gap. Away from $\ensuremath{\nu}=n/(2n\ifmmode\pm\else\textpm\fi{}1)$, the CF exciton either binds with or annihilates an already present anyon (i.e., a CF in a partially filled level) to form a trion or a single CF quasiparticle, further lowering the energy. When the interaction between two electrons is independent of whether they occupy the conduction or the valence band, as in an ideal two-dimensional system, the resulting SU(2) symmetry implies that the energy of the emitted photon must equal the band gap, independent of correlations. In this case, the bound CF excitons and CF trions are dark, indicating an absence of photoemission at zero temperature, except for the exciton of the $\ensuremath{\nu}=\frac{1}{3}$ ($\ensuremath{\nu}=\frac{2}{3}$) state in the valence (conduction) band, but we show that the temperature dependence of the PL peak intensity can be used to measure their binding energies. We also study, using exact diagonalization methods, the effect of an SU(2)-symmetry-breaking term; we find that as the strength of this term is increased, the dominant peaks of the SU(2)-symmetric theory evolve continuously, while additional, low-intensity peaks appear. We discuss implications for PL experiments in semiconductor quantum wells and twisted TMD bilayers.