| Summary: | 碩士 === 國立臺灣師範大學 === 物理學系 === 106 === The edge state is known to be a characteristic of a topological material. In two-dimensional topological systems, one can use the \emph{Chern number} to describe the topological property of the systems. However, the Chern number fails to discern the topology for two special cases of two-band systems: (a) when the parameter space is restricted to a plane, and (b) when the system is a semimetal. One should find another way instead to characterize the nontrivial topology.
In this thesis, the SSH model is extended from one dimension to two dimensions by four different ways. None of them can be described by the Chern number. However, by applying the dimensional reduction, the systems are reduced to one dimension and are equivalent to the generalized SSH model, whose topological nontriviality is characterized by the \emph{winding number}. Since the open boundary conditions are preserved under the dimensional reduction, the edge effect should be described by the reduced Hamiltonian. Therefore, we find the quasi-bulk-boundary correspondence to connect the edge states of the two-dimensional systems and the winding number of the reduced Hamiltonian. Moreover, if the edges of SSH chains are preserved under the extension in the thesis, the edge states are also preserved.
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