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arXiv:2303.15760

The topological order in loop quantum gravity

Jingbo Wang

Published Mar 28, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2303.15760

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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.
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.
Recent theoretical progress shows that ([Formula: see text]) black hole solution manifests long-range topological quantum entanglement similar to exotic non-Abelian excitations with fractional quantum statistics. In topologically ordered systems, there is a deep connection between physics of the bulk and that at the boundaries. Boundary terms play an important role in explaining the black hole entropy in general. We find several common properties between BTZ black holes and the Quantum Hall effect in ([Formula: see text])-dimensional bulk/boundary theories. We calculate the topological entanglement entropy of a ([Formula: see text]) black hole and recover the Bekenstein–Hawking entropy, showing that black hole entropy and topological entanglement entropy are related. Using Chern–Simons and Liouville theories, we find that long-range entanglement describes the interior geometry of a black hole and identify it with the boundary entropy as the bond required by the connectivity of spacetime, gluing the short-range entanglement described by the area law. The IR bulk–UV boundary correspondence can be realized as a UV low-excitation theory on the bulk matching the IR long-range excitations on the boundary theory. Several aspects of the current findings are discussed.
Abstract Black hole entropy is one of the few windows into the quantum aspects of gravitation, and its study over the years has highlighted the holographic nature of gravity. At the non-perturbative level in quantum gravity, promising explanations are being explored in terms of the entanglement entropy between regions of space. In the context of loop quantum gravity, this translates into an analysis of the correlations between the regions of the spin network states defining the quantum state of the geometry of space. In this paper, we explore a class of states, motivated by results in condensed matter physics, satisfying an area law for entanglement entropy and having non-trivial correlations. We highlight that entanglement comes from holonomy operators acting on loops crossing the boundary of the region.
We study a proposed statistical explanation for the Bekenstein-Hawking entropy of a black hole in which entropy arises quantum-mechanically as a result of entanglement. Arguments for the identification of black hole entropy with entanglement entropy are reviewed in the framework of quantum field theory, emphasizing the role of renormalization and the need for a physical short-distance cutoff. \n \nOur main novel contribution is a calculation of entanglement entropy in loop quantum gravity. The kinematical Hilbert space and spin network states are introduced, and the entanglement entropy of these states is calculated using methods from quantum information theory. The entanglement entropy is compared with the density of states previously computed for isolated horizons in loop quantum gravity, and the two are found to agree up to a topological term. \n \nWe investigate a conjecture due to Sorkin that the entanglement entropy must be a monotonically increasing function of time under the assumption of causality. For a system described by a finite-dimensional Hilbert space, the conjecture is found to be trivial, and for a system described by an infinite-dimensional Hilbert space a counterexample is provided. \n \nFor quantum states with Euclidean symmetry, the area scaling of the entanglement entropy is shown to be equivalent to the strong additivity condition on the entropy. The strong additivity condition is naturally interpreted in information-theoretic terms as a continuous analog of the Markov property for a classical random variable. We explicitly construct states of a quantum field theory on the one-dimensional real line in which the area law is exactly satisfied.