讲座摘要：Topological materials which include Chern insulators, topological insulators and Weyl semimetals, are characterized by their topological invariants. Bulk-edge correspondence tells us that this topological number can be expressed as the number of chiral edge-localized states. Moreover, in a gapped continuum system, a definition of topological invariant(Hall conductivity) as the Chern-Simons level of its effective theory is commonly used as well.
In this talk, I will discuss my work with collaborators on the study on the various manifestations of topological invariants in condensed matter systems. First, topology can be represented by open boundary conditions in Weyl semimetals and topological insulator. Second, when boundary state carries a topological number, itself can host a localized on the boundary of itself. Third, for a generic gapped Hamiltonian, the equivalence of Chern-Simons level of its effective theory as topological invariant to its Chern number is proven to be true.