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arXiv:2104.00227

The topological order of the space

Jingbo B. Wang

Published Apr 1, 2021• 0 citations
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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.

DOI
10.48550/arxiv.2104.00227

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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.
Topological order (long-range entanglement) is a new type of order that beyond Landau's symmetry breaking theory. This concept plays important roles in modern condensed matter physics. 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 states. So the BTZ black hole has the same topological order with the fractional quantum Hall state. We show that the four dimensional black hole can also have topological order, by showing that the topological entanglement entropy of black holes is non-zero in loop quantum gravity. We also consider the Hamiltonian constraint in loop quantum gravity. By comparing with the BF theory, we found that the physical Hilbert space for loop quantum gravity is degenerate on non-trivial manifold, which indicate the existence of topological order 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.
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Topological Order of Quantum Gravity in AdS3AdS_3AdS3​ Spacetime
Topological order is a new type order that beyond Landau's symmetry breaking theory. It has some interesting properties, such as producing quasi-particles with fractional quantum numbers and fractional/Fermi statistics, robust gapless boundary modes and emergent gauge excitations. In this essay, we will show that the quantum gravity in AdS3AdS_3AdS3​ spacetime can also have topological orders. Actually the theory has all the three features that define the topological order. We conjecture that quantum gravity in four dimension can also have topological orders.
After the discovery of fraction quantum Hall states in the 1980s,[1] it became more and more clear that Landau symmetry breaking theory does not describe all possible quantum phases of matter. The new quantum phases of matter were called topologically ordered phases[2,3] (for gapped cases) or quantum ordered phases[4] (for gapless cases), which correspond to patterns of many-body entanglement.[5–7] One may wonder: besides quantum Hall systems, are there other systems that realize the new topological/quantum order?