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Now we look at the emergence of surface states. First, we need to define a word "topological equivalence". We say two insulator is topologically equivalent if their band structure can be continuously deformed into each without closing the energy gap and with conservation of symmetries. From Fig.2, the upper space is a trivial vacuum with Chern number of n=0, and the lower is a quantum Hall state with Chern number of n=1. Since these two systems have different Chern number, there is no way that one can maintain the topological invariant across the interface. The end result is that there exists gapless surface edge states at the surface conducting the current like shown in Fig.2(a).