The Zak phase defines a Z2 topological invariant for certain 1D AZC symmetry classes but vanishes under quaternionic anti-unitary symmetries, providing only partial information about topological phases in generalized Kitaev chains.
On the k-theoretic classification of topological phases of matter.Annales Henri Poincaré, 17(4):757–794, 2016
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The Zak phase in topologically insulating chains: invariants and limitations
The Zak phase defines a Z2 topological invariant for certain 1D AZC symmetry classes but vanishes under quaternionic anti-unitary symmetries, providing only partial information about topological phases in generalized Kitaev chains.