Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum

We have studied extensively the band crossing patterns of the bulk entanglement spectrum (BES) for various lattice Chern insulators. We find that only partitions with dual symmetry can have either stable nodal-lines or nodal-points in the BES when the system is in the topological phase of a nonzero Chern number. By deforming the Hamiltonian to lift the accidental symmetry, one can see that only nodal points are robust. They thus should bear certain topological characteristics of the BES. By studying the band crossing patterns in details we conclude that the topological characteristics of the BES are inherited from the topological order of the underlying Chern insulators and the former can have more refined topological structures. We then propose the conjecture that the sum of the vorticities in the BES in a properly chosen reduced Brillouin zone equals the Chern number of the underlying Chern insulator. This relation is beyond the usual classification scheme of topological insulators/superconductors.