Entanglement Chern Number in Tree Dimensions

We have characterized some of topological phases by the entanglement Chern number (e-Ch), which is defined as the Chern number of the entanglement Hamiltonian.\footnote{ T. Fukui and Y. Hatsugai, JPSJ, {\bf 83}, 113705 (2014)} The partition of the system is not necessarily spatial but can be spin partition, which is the extensive partition. If a system respects the time reversal symmetry, the Chern number is trivial but the e-Ch can be non-zero. For instance, the e-Ch characterizes the quantum spin Hall phase of the Kane--Mele model and its phase diagram by the $Z_2$ topological number is successfully reproduced by the e-Ch.\footnote{H. Araki, T. Kariyado, T. Fukui and Y. Hatsugai, JPSJ, {\bf 85}, 043706 (2016)} For the Fu--Kane--Mele model,\footnote{L. Fu, C. L. Kane and E. J. Mele, PRL, {\bf 98}, 106803 (2007)} its weak phases are well described by the non trivial section e-Ch and the strong phase is characterized by the existence of the Weyl points of the entanglement Hamiltonian.\footnote{ H. Araki, T. Fukui and Y. Hatsugai, in preparation.}