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We propose characterization of the three-dimensional topological insulator byusing the Chern number for the entanglement Hamiltonian (entanglement Chernnumber). Here we take the extensive spin partition of the system, that pullsout the quantum entanglement between up spin and down spin of the many-bodyground state. In three dimensions, the topological insulator phase is describedby the section entanglement Chern number, which is the entanglement Chernnumber for a periodic plane in the Brillouin zone. The section entanglementChern number serves as an interpolation of the $Z_2$ invariants defined ontime-reversal invariant planes. We find that the change of the sectionentanglement Chern number protects the Weyl point of the entanglementHamiltonian and the parity of the number of Weyl points distinguishes thestrong topological insulator phase from the weak topological insulator phase.

Entanglement Chern number for three-dimensional topological insulators: Characterization by Weyl points of entanglement Hamiltonians