Exciton interacting with the phonons of an electronic Wigner crystal
With the advent of atomically thin and tunable van der Waals materials, a two-dimensional electronic Wigner crystal has recently been observed. The smoking gun signal was the appearance of an umklapp branch in optical exciton spectroscopy coming from the periodic potential generated by the Wigner crystal assumed to be static. Vibrations of the Wigner crystal however leads to a gapless phonon spectrum, which may affect the exciton spectrum. To explore this, we develop a field theoretical description of an exciton interacting with electrons forming a Wigner crystal including the coupling to the phonons. We show that importance of the exciton-phonon coupling scales with the exciton-electron interaction strength relative to the typical phonon energy squared. The motion of the exciton leads to two kinds of scattering processes, where the exciton emits a phonon either staying within the same Bloch band (intraband scattering) or changing its band (interband scattering). Using a non-perturbative self-consistent Born approximation, we demonstrate that these scattering processes lead to the formation of quasiparticles (polarons) consisting of the exciton in Bloch states dressed by Wigner crystal phonons. The energy shift and damping of these polarons depend on the electron density in a non-trivial way since it affects both the exciton-phonon interaction strength, as well as the phonon and exciton spectra. In particular, the damping is strongly affected by whether the polaron energy is inside the gapless phonon scattering continuum or not. Using these results, we finally analyse their effects on the observed spectral properties of the exciton.