Do Chern Insulators (QAHE) have topological order (long-range quantum entanglement)?

Question

I know IQHE is a example having "invertible" topological order from Professor Wen's definition. And Topological Insulators is SRE because of necessary of underlying symmetry protection. After that, the Chern Insulator (QAHE) needs a underlying TRS-broken not a TRS protection like IQHE. Except the external magnetic field, it is almost as same as IQHE. Is it also a example having long-range quantum entanglement? More precisely speaking, what topological order does it have? Also "invertible" topological order?

Answer 1

Chern Insulator = QAHE = IQHE. Chern Insulator has "invertible" topological order and long-range entanglement as defined in https://arxiv.org/abs/1004.3835 . Chern Insulator does not need any symmetry, although one usually assume Chern Insulator has an U(1) symmetry.

Q: what topological order does IQHE have?

A: An "invertible" topological order characterized by a gravitational Chern-Simons term.