Mass-imbalanced two-dimensional Bose-Fermi mixtures with boson-fermion pairing

Ultra-cold atomic Bose-Fermi mixtures constitute a versatile platform for studying quantum many-body phenomena, offering tunability over interparticle interactions, dimensionality, and component populations. Similar mixtures also occur in two-dimensional semiconductor nanostructures known as transition metal dichalcogenides, where excitons act as bosons interacting attractively with charge carriers (electrons or holes). Both examples motivate the theoretical study of Bose-Fermi mixtures with boson-fermion pairing and bosons and fermions with unequal masses. This thesis investigates two-dimensional mass-imbalanced Bose-Fermi mixtures at zero temperature, extending previous analyses of mass-balanced systems. Using a diagrammatic T-matrix approach that has been validated in three dimensions by a recent experiment as well as by Quantum Monte Carlo simulations, we numerically study fundamental thermodynamic quantities as functions of the boson-fermion attraction strength. (Semi-)analytical expressions in the weak- and strong-coupling regimes are also derived. Mass-imbalance effects are systematically analyzed, and numerical results are tested against the (semi-)analytical expressions. We further introduce formal improvements to the T-matrix method, including fermion-mediated boson-boson interactions in the anomalous boson self-energy and a partial self-consistency scheme in the fermion and boson self-energies. The inclusion of the additional anomalous term is essential to restore the correct weak-coupling expansion within the partially self-consistent T-matrix approach, suggesting the necessity of both improvements for an even more accurate modeling of the system.