Topological Quantum Phase Transitions and Quench Dynamics

Topological phases of matter and the phase transitions between them have been the focus of much recent theoretical and experimental interest. In this thesis, we firstly study a type of topological phase transitions between quantum Hall states driven by pairing interactions through Feshbach resonances. Although quantum Hall effects were first discovered in electronic condensed matter systems, this type of phase transitions was proposed to ultra cold atom field for their ability of the control of interactions. In addition, we also investigate the quench dynamics of topological phase transitions based on specifically Haldane model and checkerboard model in the second part. In the study of topological phase transitions between fermionic integer quantum Hall (FIQH) and bosonic fractional quantum Hall (BFQH) phases, we first provide a general picture of this kind of quantum Hall phase transitions. Subsequently, we use exact diagonalization to study the quantum phases and phase transitions when a single species of fermionic atoms at Landau level filling factor vf = 1 in a rotating trap interact through a p-wave Feshbach resonance. We show that under weak pairing interaction, the system undergoes a second order quantum phase transition from vf = 1 fermionic integer quantum Hall (FIQH) state at positive detuning, to vb = ¼ bosonic fractional quantum Hall (BFQH) state at negative detuning. However, when the pairing interaction increases, a new phase between them emerges, corresponding to a fraction of fermionic atoms stay in a coherent superposition of bosonic molecule state and an unbound pair. The phase transition from FIQH phase to the new phase is of second order and that from the new phase to BFQH phase is of first order. Furthermore, we investigate the quantum phases and phase transition in a system made of two species of fermionic atoms that interact with each other via s-wave Feshbach resonance, and are subject to rotation or a synthetic gauge field that puts the fermions at Landau level filling factor vf = 2. We show that the system undergoes a continuous quantum phase transition from a vf = 2 fermionic integer quantum Hall state formed by atoms, to a vf = ½ bosonic fractional quantum Hall state formed by bosonic diatomic molecules. In the disk geometry we use, these two different topological phases are distinguished by their different gapless edge excitation spectra, and the quantum phase transition between them is signaled by the closing of the energy gap in the bulk. Comparisons will be made with field theoretical predictions, and the case of p-wave pairing. In the second part of this thesis, we study the dynamics of systems quenched through topological quantum phase transitions and investigate the behavior of the bulk and edge excitations with various quench rates. Specifically, we consider the Haldane model and checkerboard model in slow quench processes with distinct band-touching structures leading to topology changes. The generation of bulk excitations is found to obey the power-law relation Kibble-Zurek and Landau-Zener theories predict. However, an anti-Kibble-Zurek behavior is observed in the edge excitations. The mechanism of excitation generation on edge states is revealed, which explains the anti-Kibble-Zurek behavior.