| Summary: | Since the discovery of the fractional quantum Hall effect in 1982, a number of formalisms have been proposed to construct the topological phases. However, it is difficult to see if these constructions lead to the same fractional quantum Hall states. Characterizing and labeling the orders in these states had been a long-standing problem in condensed matter physics. This problem was addressed by Wen and Zee by the introduction of topological indices, [p, q, S] that characterize fractional quantum Hall states. In 2019, Yang proposed an intuitive way of constructing model states with these topological indices using classical constraints known as the Local Exclusion Condition (LEC). This formalism has shown a remarkable success in reproducing the fractional quantum Hall states in single-component quantum fluids for both Abelian and non-Abelian phases, without requiring model Hamiltonians or wavefunctions. Its generality makes it an ideal
tool to discover new topological phases. In recent years, many new two-dimensional materials with additional degrees of freedom, such as bilayer graphene, have garnered interest from researchers due to the possibilities of new topological phases that have no analogues in the single-component systems. This motivates us to generalize the LEC formalism to bilayer quantum Hall systems.
This thesis presents the progress in generalizing the LEC formalism to multi-component states. Our results show that the LEC formalism indeed can be generalized to some quantum Hall bilayers. Additionally, we propose a new formalism called the non-local exclusion conditions (NLEC) for the Halperin (332) states, where the exclusion conditions are applied to circular droplets opposite to each other. This leads to the conjecture that there is non-local interaction in the Halperin (332) state, the nature of which remains to be understood. Using this generalized LEC and NLEC, we hope to discover new topological phases that have no analogues in the single-component quantum Hall systems. In particular, we are interested in the non-Abelian phases which are potential building blocks for topological quantum computers.
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