Interactions and topology in two-dimensional semiconductor and semimetal

This dissertation presents our study of interaction and topology effects in two-dimensional semiconductor and semimetal. In the first part, we explore the ground state thermodynamic phases in various systems using a mean-field theory including long-range Coulomb interaction. In particular, we have discovered two distinct phases with different number of exciton condensate flavors in a bilayer system of two-dimensional semiconductors with spin/valley degree of freedoms. The exciton condensate phase is characterized by spontaneous inter-layer phase coherence and counterflow superfluidity. We have also studied the phase diagram of a model quantum spin Hall system as a function of band inversion and band-coupling strength, demonstrating that when band hybridization is weak, an interaction-induced nematic insulator state emerges over a wide range of band inversion. We also develop a fully microscopic theory of equilibrium polariton condensates that treats the two-dimensional quantum well band states explicitly and goes beyond the commonly used model in which bare excitons are treated as Bose particles that are coupled via flip-flop interactions with cavity photons. In the second part, we present our study of collective excitations in an exciton condensate using a time-dependent mean-field theory. By constructing a fluctuation Lagrangian based on a variational wave-function capturing exciton density and phase fluctuations, we are able to get the collective mode spectra and demonstrate the stability of exciton condensate phase. Moreover, we apply a linear response theory to identify a number of collective modes with a strong electron-hole pairing amplitude(Higgs-like) component. Next, we generalize our time-dependent mean-field theory to finite temperature to address the long debate question that whether a CDW phase of a semimetal is due to Coulomb interaction or electron-phonon interaction. Our theory includes both intraband and interband excitations and direct and exchange interactions, and allows for the formation of exciton condensates at low temperatures.