A variant of bulk-boundary correspondence in topological phases in AZ+I classification

Topological insulators usually have gapless boundary states due to the well-known bulk-boundary correspondence. For this to hold true there is an implicit necessary condition: the symmetry of the bulk must be maintained locally even near the boundary. This condition is violated for symmetries that involve spatial inversion operations, such as parity-time-reversal (PT) symmetry. For two-dimensional PT-symmetric topological insulators, bulk-edge correspondence in the usual sense does not generally hold for the reasons mentioned above. In the previous work, we showed that by using entanglement spectra instead of ordinary edge spectra, a variant of "bulk boundary correspondence" holds even for PT symmetric topological insulators [1].



In this presentation, we reveal the extension of such "bulk-boundary correspondence" to topological phases of the AZ+I classification, which is an extension of the AZ classification by replacing time reversal T and particle-hole symmetry C with PT and PC, respectively. We focus on the fact that the entanglement Hamiltonian always has an anti-symmetry Ξ. By combining the PT and PC operations with Ξ, we show that symmetries of the AZ+I classification are recovered in the entanglement Hamiltonian. As a result, the entanglement spectrum for topological insulators in the AZ+I classification becomes gapless, and a variant of “bulk-boundary correspondence” holds.



[1] R. Takahashi and T. Ozawa, Phys. Rev. B 108, 075129 (2023).