The topological order of the space

TL;DR

This paper explores the concept of topological order in various physical systems, including black holes and quantum states, proposing that black holes may share topological properties with quantum Hall states and suggesting new research directions.

Contribution

It introduces the idea that black holes exhibit topological order similar to fractional quantum Hall states and proposes applying string-net condensation methods to loop quantum gravity.

Findings

BTZ black hole's topological entanglement entropy matches that of fractional quantum Hall states

Black holes in higher dimensions may also possess topological order

Proposes integrating string-net condensation techniques into loop quantum gravity

Abstract

Topological order is a new type order that beyond Landau's symmetry breaking theory. The topological entanglement entropy provides a universal quantum number to characterize the topological order in a system. The topological entanglement entropy of the BTZ black hole was calculated and found that it coincides with that for fractional quantum Hall state. So the BTZ black holes have the same topological order with the fractional quantum Hall state. We conjecture that black holes in higher dimensions also have topological orders. Next we want to study the topological order of ordinary spaces which can be described by spin network states in loop quantum gravity. We advise to bring in the methods and results in string-net condensation to loop quantum gravity to solve some difficult problems.