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We propose an unconventional bulk-edge correspondence for two-dimensionalStiefel-Whitney insulators and Euler insulators, which are topologicalinsulators protected by the $PT$ symmetry. We find that, although the energyspectrum under the open boundary condition is generally gapped, theentanglement spectrum is gapless when the Stiefel-Whitney or Euler class isnonzero. The robustness of the gapless spectrum for Stiefel-Whitney insulatorcan be understood through an emergent anti-unitary particle-hole symmetry. Forthe Euler insulators, we propose a conjecture, which is supported by ournumerical calculation, that the Euler class is equal to the number of crossingin the entanglement spectrum, taking into account the degree of the crossings.We also discuss that these crossings of the entanglement spectrum are relatedto the gap closing points in the cutting procedure, which is the energyspectrum as the magnitude of the boundary hopping is varied.

Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the entanglement spectrum and cutting procedure