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.

EDIT: 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). Is there good reason to be confident that the bulk-boundary correspondence will continue to hold in these cases?

  • $\begingroup$ If I understand it correctly, for quasicrystal (boundary condition) it may not hold. arxiv.org/abs/1109.5983 $\endgroup$
    – Z.Sun
    Commented Aug 21, 2012 at 7:25
  • $\begingroup$ Did you mean a different preprint than the first one referenced? 0901.2696 seems to be "Strong Morita Equivalence of Inverse Semigroups," which could be relevant to TIs, but prima facie seems not to be. $\endgroup$ Commented Aug 23, 2012 at 17:47

1 Answer 1


There is no proof of bulk-boundary correspondence for topological phases in general. In fact, topological phases like toric code model does not have gapless excitations on the boundary.

For non-interacting fermion systems protected by internal symmetries (as in the "periodic table" classification), bulk-boundary correspondence holds. For non-interacting fermion systems protected by a spatial symmetry, gapless surface states also exist on those crystal surfaces preserving the symmetry.

In some sense, the existence of some kind of boundary states is all there is about topology in non-interacting fermions.

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    $\begingroup$ Dear Prof. Fu, Welcome to Physics Stackexchange! Regarding the non-interacting case, isn't the bulk to boundary correspondence essentially very generally proved in the work of Teo and Kane (among other approaches)? And would it be true to generalize the characterization of these states in your last sentence, to "existence of gapless modes in various types of topological defects (domain walls, points, hegdehogs, ...)", and not necessarily only on boundaries? $\endgroup$
    – Heidar
    Commented Aug 23, 2012 at 14:22
  • $\begingroup$ The reference I have in mind is arxiv.org/abs/1006.0690 . $\endgroup$
    – Heidar
    Commented Aug 23, 2012 at 14:36

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