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.
Symmetry and localization in periodic crystals: Triviality of Bloch bundles with a fermionic time- reversal symmetry.Acta Applicandae Mathematicae, 137(1):185–203, December 2014
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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.