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Topological Entanglement Entropy of Black Hole Interiors Reports in Advances of Physical Sciences Vol. 4, No. 1 (2020) 1940001 (19 pages) #.c The Author(s) DOI: 10.1142/S2424942419400012 Topological Entanglement Entropy of Black Hole Interiors Eric Howard Centre for Quantum Engineering Department of Physics and Astronomy Macquarie University, Sydney, NSW, Australia Centre for Quantum Dynamics Gri±th University, Brisbane, QLD, Australia [email protected]; eric.howard@griffith.edu.au Received 13 October 2019 Accepted 19 January 2020 Published 19 February 2020 Recent theoretical progress shows that (2 þ 1) black hole solution manifests long-range topo- logical quantum entanglement similar to exotic non-Abelian excitations with fractional quan- tum 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 ¯nd several common properties between BTZ black holes and the Quantum Hall e®ect in (2 þ 1)-dimensional bulk/boundary theories. We calculate the topological entanglement entropy of a (2 þ 1) 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 ¯nd 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 ¯ndings are discussed. Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com Keywords: Black hole interior; entanglement entropy; Bekenstein–Hawking entropy; BTZ black hole; area law. 1. Introduction Since its discovery in 1992, the BTZ black hole solution1 has been used to model by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. realistic black holes, leading to the discovery that higher-dimensional black hole thermodynamics may be studied in terms of BTZ physics. Searching for a quantum theory of gravity remains a longstanding open problem in theoretical physics. However, the goal of ¯nding such a consistent theory still This is an Open Access article published by World Scienti¯c Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited. 1940001-1 E. Howard remains elusive. The strong connection between thermodynamics and black hole physics has been found successful in searching for a complete quantum gravity theory, leading to new ideas and vital developments in this ¯eld. Speci¯cally, without adding any matter degrees of freedom, quantum gravity alone cannot account for the microscopic entropy of black holes. The works of Bekenstein, Hawking and others showed that black holes are endowed with thermodynamic properties such as en- tropy and temperature, leading to the Bekenstein–Hawking area law. The formula caused a signi¯cant impact in theoretical physics, leading to the discovery of the holographic nature of quantum gravity. The Bekenstein–Hawking entropy SBH, Area of horizon SBH ¼ ; ð1Þ 4GN describes the gravitational entropy through the degeneracy of quantum ¯eld theory in microscopic description. In the near-horizon limit, AdS–CFT correspondence, a (d þ 1)-dimensional conformal ¯eld theory (CFTdþ1) is found equivalent to a gravity theory on (d þ 2)-dimensional AdSdþ2 space, living on the boundary. The origin of entropy would be hidden behind the quantum microstates associ- ated with the black hole horizon. The Bekenstein–Hawking formula (1) also matches the microscopic entropy of strings, D-branes and their excitations. Current progress about the microscopic origin of black hole entropy, renormalization, black hole in- formation paradox, holographic principle, emergent gravity and the connection to entanglement has its deep origins in black hole thermodynamics. The bulk–boundary correspondence and the holographic properties are a result of the AdS–CFT corre- spondence. Signi¯cant progress has been recently made in providing a holographic approach for particular ideas from condensed matter physics. We investigate the interface between high-energy physics and condensed matter physics with signi¯- cance for holographic principle. It is accepted that the holographic principle is ac- tually a key ingredient to unifying quantum and gravity into a consistent framework. Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com If gravity has indeed a holographic nature, the radial direction is strongly connected to a scale of renormalization group (RG) °ow of the dual theory. Conveniently, if classical spacetime is found to fundamentally emerge from an underlying quantum layer, a signi¯cant and robust insight into the quantum nature of spacetime would be provided. If entanglement is indeed the fabric of spacetime, general relativity can be pictured as a long-range theory encoding the quantum information °ow of the underlying microscopic degrees of freedom. Going further, by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. tools from many-body physics like entanglement renormalization and holographic gauge/gravity duality or other ideas from quantum information theory trying to describe quantum-critical points were brought into the picture. Some quantum ¯eld theories without gravity were found to be dual to theories of quantum gravity in a higher-dimensional bulk geometry. Gravity has a topological nature because any observable is coordinate-invariant. In gravity, local di®eomorphisms are gauge symmetries and when reaching in¯nity 1940001-2 Topological Entanglement Entropy of Black Hole Interiors they become global symmetries. Gauge invariance is a di®eomorphism invariance, therefore local operators are not gauge-invariant as there are no preferred coordi- nates. In (2 þ 1)-dimensions, the metric is independent of any matter distribution and locally invariant. A di®eomorphism-invariant theory of a manifold is described by a quantum ¯eld theory on the boundary, rather than using the interior local degrees of freedom. In this context, gravity could be emergent from long-distance connections of the ground state of a topologically ordered highly entangled ¯eld theory which is close to a quantum-critical point. The long-range correlations behind the emergence of the di®eomorphism symmetries in the gravitational theory are manifested in the degeneracy of the ground state as a function of genus. The holographic principle brings a geometrical interpretation of several concepts in strongly coupled ¯eld theories and a fresh perspective on particular systems for which the perturbative theory remains ine®ective. An insightful connection between holography and entanglement renormalization led to a full holographic picture of the spacetime, where the ¯eld theory observables are described by information encoded geometrically. The holographic principle should contain the complete information about a local coarse-grained description of the dual theory. In this sense, the renormalization group °ow and the energy scale may be the true underlying cause of the emergent holographic dimension. The correspondence between the RG °ow in the ¯eld theory and the radial dimension backed up by the holographic principle still remains unknown. If we take into account the long-range nature of gravitational interaction and the asymptotic nature of the spacetime metric, black hole information should involve long-range correlations that are coded in the structure of spacetime. The UV–IR correspondence implies that a greater depth in the bulk would correspond to a coarse graining structure in the ¯eld theory with a UV cut-o® dual in the bulk to a large radial cut-o®. The deep bulk geometry is described by non-local observables at progressively larger distances on the boundary and dual to the infrared energy scales. There is a strong connection between the non-local observables and the bulk depth, Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com with deep consequences for understanding the causal structure of the space-time. In order to explore the holographic origin of gravity, a further information-theoretic interpretation of the ¯eld theory on the boundary in the dual-¯eld theory would be necessary. 2. The Topological Entanglement Entropy by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. The concept of geometric entanglement entropy links the microscopic structure of spacetime to entanglement. Ryu and Takayanagi found a holographic derivation of the entanglement entropy in quantum (conformal) ¯eld theories from AdS–CFT correspondence. We are here interested in understanding the entropy of the interior of a black hole employing (2 þ 1) AdS spacetime. Let us divide a region of space into two quantum subsystems A and B, where A is the spacetime region inside a black hole, and B quanti¯es the space outside of A, 1940001-3 E. Howard where an observer is located, outside the horizon. A and B are partitions of a larger system that is assumed to be in a pure quantum state. The quantum state of the subsystem A is de¯ned by the reduced density matrix A, which is obtained by tracing out the information contained in the subsystem B that describes the rest of the system. The density matrix of A is A ¼ trBðÉÉÞ, with as the ground state of the quantum system. The trace is taken over all states of the complement B. SB is the entropy of an observer who is only accessible to the region B but does not receive any communication from A. The information is lost as it becomes traced out (or smeared out) on the outside.
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