CORRELATED MANY BODY COLD ATOMS AND EXTENDED FALICOV-KIMBALL MODELS

Correlated many body systems remain at the center of research in condensed matter theory for their rich properties and possible phase transitions. In the recent years, trapped atomic gases provide a unique playground to simulate many-body quantum physics in a very controlled way. The theoretical study on the properties of these cold atom system thus become important. In this thesis, we address the theoretical studies on two cold atom models: an impurity model and a bosonic model with k-space Berry curvature. Explicitly, we first study in Chap.2 an impurity model interacting with a large bath of cold atoms. This model is mapped into a spin-boson model using either the Bogoliubov theory (for bosonic atoms) or bosonization (for fermionic atoms) and the corresponding dissipative phase transitions are found, in consistent with previous theories. In Chap.3, we study the properties of cold bosons in a two-dimensional optical lattice system where bose-condensation occurs at a momentum point k with non-zero k-space Berry curvature. We show that the boson system carries non-universal equilibrium angular momentum and edge current at low temperatures. On the other hand, exact solutions of some special quantum interacting particle models are important resources for understanding the physics of strongly correlated systems. In Chap.4, we generalize a solvable Falicov-Kimball model[3] in two general classes: spin dependent hopping class and the Majorana Falicov-Kimball class. We explore the general criteria for exact solution in these two classes for arbitrary interaction strength. Corresponding to the two classes, we explicitly study two models: (1) a spin-dependent Haldane Hubbard model, in which we found an interaction driven topological phase transition; and (2) a p-wave BCS Hubbard model in a bipartite lattice, in which we found chargeless solitonic excitation and interaction dependent spin excitation.