The topological entanglement entropy of black holes in loop quantum gravity

Topological order (long-range entanglement) play important role in modern condensed matter physics. In this paper, we will show that the four dimensional black hole can also have topological order, by showing that the topological entanglement entropy is non-zero. The properties of the topological order show that the large diffeomorphisms will act as symmetry (not gauge) transformation on the physical states. More importantly, the long-range entanglement will make the Hawking radiation pure.