  We propose an unconventional bulk-edge correspondence for two-dimensional
Stiefel-Whitney insulators and Euler insulators, which are topological
insulators protected by the $PT$ symmetry. We find that, although the energy
spectrum under the open boundary condition is generally gapped, the
entanglement spectrum is gapless when the Stiefel-Whitney or Euler class is
nonzero. The robustness of the gapless spectrum for Stiefel-Whitney insulator
can be understood through an emergent anti-unitary particle-hole symmetry. For
the Euler insulators, we propose a conjecture, which is supported by our
numerical calculation, that the Euler class is equal to the number of crossing
in the entanglement spectrum, taking into account the degree of the crossings.
We also discuss that these crossings of the entanglement spectrum are related
to the gap closing points in the cutting procedure, which is the energy
spectrum as the magnitude of the boundary hopping is varied.
