Excitons in moir\'{e} superlattices with disordered electrons

Moir\'{e} superlattices in transition metal dichalcogenides (TMDs) heterobilayers exhibit various correlated insulating states driven by long-range Coulomb interactions, and these states crucially alter exciton resonances, particularly at fractional fillings. We revisit a theoretical framework to investigate the doping dependence of exciton spectra by extending hydrogenic exciton wavefunctions, systematically analyzing how the 1$s$, 2$s$, and 3$s$ Rydberg states respond to moir\'e-induced mixing of $s$- and $p$-type orbitals. Notably, while the 1$s$ state remains relatively robust against doping, higher Rydberg excitons show strong redshifts and oscillator-strength quenching near specific fractional fillings. We incorporate both defect-induced quasi-ordering and thermal fluctuations to capture realistic device conditions, employing a large supercell approach. By selectively randomizing a subset of electrons or utilizing classical Monte Carlo simulations, we present direct calculations of exciton spectra under varying defect densities and temperatures. Our results emphasize how even moderate disorder or finite temperature can partially or completely suppress characteristic moir\'{e} exciton physics. Especially, we show how the 2$s$ exciton states respond to the phase transition in correlated electron states. This comprehensive picture not only clarifies recent experimental observations but also provides a framework to guide the design of moir\'{e}-based optoelectronic devices.