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We study and present the results of Berry connection for the topologicalstates in quantum matter. The Berry connection plays a central role in thegeometric phase and topological phenomenon in quantum many-body system. Wepresent the necessary and sufficient conditions to characterize the topologicalnature of the system through the complex analysis. We also present thedifferent topological aspects of the system in the momentum space.

A Study of Berry Connection and Complex Analysis for Topological Characterization