We study low-dimensional quantum systems with analytical and computational methods. Firstly, the one-dimensional extended $t$-$V$ model of fermions with interactions of a finite range is investigated. The model exhibits a phase transition between liquid and insulating regimes. We use various analytical approaches to generalise previous theoretical studies. We devise a strong coupling expansion to go beyond first-order perturbation theory. The method is insensitive to the presence or the lack of integrability of the system. We extract the ground state energy and critical parameters of the model near the Mott insulating commensurate density. We also study the possible charge-density-wave phases that exist when the model is at the critical density. Secondly, we investigate Mott-Wannier complexes of two (excitons), three (trions) and four (biexcitons) charge carriers in two-dimensional semiconductors. Our study also includes impurity-bound complexes. We provide a classification of trions and biexcitons in transition-metal dichalcogenides, which incorporates the difference of spin polarisation between molybdenum- and tungsten-based materials. Using the diffusion Monte Carlo method, which is statistically exact for these systems, we extract binding energies of the complexes for a complete set of parameters of the model. Our results are compared with theoretical and experimental work on transition-metal dichalcogenides. An agreement is found for excitonic and trionic results, but we also observe a large discrepancy in the theoretical biexcitonic binding energies as compared to the experimental values. Possible reasons for this are outlined. We also calculate contact pair densities, which in the future can be used in the determination of the contact interaction.
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