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## Main Question or Discussion Point

In the literature on topological insulators and superconductors the 'bulk-boundary correspondence' features quite heavily. One version of this conjecture says roughly: "At an interface between two materials belonging to the same symmetry class with bulk invariants n and m, precisely |n-m| gapless edge modes will appear". Are there any known counterexamples to this statement when the invariants are of the usual non-interacting Bloch band type? (specifically I have in mind the invariants appearing in the "periodic table" of T.I.s/T.S.Cs, see 0901.2696 and 0912.2157). As far as I know no comprehensive proof of the statement exists, although considerable supporting evidence has been found in a number of special cases.

As some extra motivation, suppose that there are new bulk invariants waiting to be found protected by symmetries falling outside the usual classification schemes (e.g. the recently proposed topological crystalline insulators protected by point group symmetries). Are there any known reasons to be confident that the bulk-boundary correspondence will continue to hold in these cases?

As some extra motivation, suppose that there are new bulk invariants waiting to be found protected by symmetries falling outside the usual classification schemes (e.g. the recently proposed topological crystalline insulators protected by point group symmetries). Are there any known reasons to be confident that the bulk-boundary correspondence will continue to hold in these cases?