Topological Entanglement Entropy of Black Hole Interiors

Home , BTZ black hole

Reports in Advances of Physical Sciences Vol. 4, No. 1 (2020) 1940001 (19 pages) #.c The Author(s) DOI: 10.1142/S2424942419400012

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 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 ¯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.

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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 entropy 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 associated 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 information 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 correspondence. 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 actually 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

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they become global symmetries. Gauge invariance is a di®eomorphism invariance, therefore local operators are not gauge-invariant as there are no preferred coordinates. 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,

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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 trBðÞ, 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. The von Neumann entropy SA ¼trðA log AÞ depicts the total entanglement between the regions A and B. The von Neumann entropy is su±cient to quantify the entanglement of a system with a small ¯nite number of degrees of freedom. However, the entanglement entropy for a system with an in¯nite number of degrees of freedom is not very well understood. Topological systems do not have local degrees of freedom at all, and their non- local degrees of freedom are only in topological, geometry-independent relation with each other. A quantum Hall system is a gapped state, whose non-local excitations only possess statistical interactions and it is therefore a topological system. The scale-independent part of the von Neumann entropy is topological, therefore it does not depend on the geometry of the partition. Entanglement is a form of quantum correlation between particles that displays unique properties like topological order when a large number of particles are entangled in a non-trivial way. Kitaev found an explicit connection between entanglement entropy and topological order in two dimensions. Entanglement takes part in several condensed matter processes, like quantum-critical points, quantum spin liquids or topologically ordered phases of matter. The entanglement entropy was successfully used in low-dimensional quantum many-body theory to study the nature of quantum-critical systems. The correlation functions in a topological system remain invariant at distances longer than the distance and energies lower than the energy scale. The formula for entanglement

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com entropy,

SA ¼trAA log A;A ¼ trBjihj; ð2Þ

shows how highly correlated a wave function ji is. Beyond the Ginzburg–Landau paradigm, many-body wave functions of quantum ground states are described by their pattern of entanglement rather than their

by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. symmetry breaking structure. Entanglement entropy becomes useful here, explaining the topological structure embedded in the ground state wave function. In condensed matter physics, the quantum phases of matter are described by their speci¯c pattern of entanglement encoded in many-body wave functions of ground states. The ground state degeneracy on a higher-genus surface and the braiding properties of quasiparticle excitations are important because the ground state of the underlying medium is strongly correlated.

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A topological phase in two dimensions is a translationally-invariant ground state of matter that does not break spontaneously the symmetries of the system. The excitation spectrum is gapped. The ground state on a manifold with a non-trivial topology (torus) has a degeneracy that cannot be changed by the action of a local perturbation. For low energies and long distances the wave function of a set of excitations does not depend on the positions of the excitations, therefore is non-local. The path integral or partition function of such a topological ¯eld theory is independent of the metric of the space, therefore topological. The best understood topological quantum ¯eld theory is the Chern–Simons gauge theory in (2 þ 1)-dimensions. Gravity in (2 þ 1)-dimensions contains no propagating local degrees of freedom and in that sense its gauge constraints can be solved exactly. In (2 þ 1)-dimensions, General Relativity can be rewritten as a trivial Chern–Simons gauge theory that is able to embed the infrared properties of topologically ordered quantum-critical systems. The topological interactions of the ground state wave function are included. The ground state degeneracy of the system will depend on the global spacetime topology. The non-local behavior of the excitations of a topological phase depends on the braiding properties of the worldlines which also depend on the wave functions. The interior of black holes can be in this way modeled using patterns of long-range quantum entanglement, directly correlated to the braiding operations from non-Abelian statistics. Topological entanglement entropy will quantify the long-range quantum order. The entanglement entropy will be directly dependent on the degrees of freedom. A topological phase shows emergent universal properties like anyon quasiparticle excitations, dependent on the non-local topological state of the many-body system, exhibiting exotic braiding exchange statistics and remaining robust to any local perturbations. A topological state of matter has a speci¯c topological invariant with di®erent values in dif- ferent states, which can be computed from the ground state wave function. The excitations with non-Abelian statistics are described by multi-dimensional representa- tions of the braid group in multi-dimensional Hilbert spaces. Topological insulators,

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com superconductors, Weyl or Dirac semi-metals, integer and fractional quantum Hall states and spin liquids are examples of topological states of matter. In a gapped system where low energies are explained by a topological ¯eld theory, like ð2 þ 1Þ-dimensional condensed matter systems, the entanglement entropy provides important clues about the ground state. It is di±cult to calculate topological entropy in a holographic theory. In the absence of the Chern–Simons theory, for pure Yang–Mills in (2 þ 1)-dimensions the topological entropy vanishes. For non-trivial by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. results, Chern–Simons interaction is required. The gapless boundary modes of a (2 þ 1)-dimensional quantum theory are ef- fectively a (1 þ 1)-dimensional system and are conformal-invariant. The boundary modes and the bulk modes are in one-to-one correspondence therefore da also applies to the bulk states. Consider a disk A (genus 0) in (2 þ 1)-dimensions, with a smooth boundary of length L in the in¯nite plane. The interior of the black hole described by the region A

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has a disk topology of length L. The topological entanglement entropy is a universal characteristic of the many-body ground state and 0.In(2 þ 1)-dimensions, topological entanglement entropy is described by the modular matrix of a two- dimensional conformal ¯eld theory. Similarly, in a gapped 2D many-body system split into two regions A and B with smooth boundaries, the entanglement entropy of the ground state of A [ B should be proportional to the boundary between A and B. For a topologically ordered system, the ground state wave function of a topological phase, split into the regions A and B, has the entanglement entropy

1 SA ¼ L þ Stopo þ OðL Þ; ð3Þ

with L being the linear size of the region A and a non-universal constant that is short-distance dependent. The ¯rst term emerges from short-wavelength modes

laying on the boundary of the region A, Stop is the topological entanglement entropy and the ellipsis indicate terms vanishing in the limit L !1. The formula is generic for ground states with a gap to all excitations but also applies to gapless systems. The leading term is the \area law" because of its proportionality to the size of the boundary between A and B (in three dimensions it is an area). The term de¯nes the local entanglement near the boundary between the two systems and it is non- universal because of its strong dependency upon the Hamiltonian. The entanglement entropy becomes a measure of the correlation between the degrees of freedom of A and B regions. The topological entanglement entropy is de¯ned on the boundary between two regions A and B. In a topological theory, the boundary arises in order to cancel the chiral anomaly. As we integrate out the degrees of freedom beyond the boundary, using the subregion B, the topological entanglement entropy arises. The integration of the theory of B includes the boundary modes of B as well. In the same way, the integration of A's degrees of freedom includes the boundary that cancels the degrees on B. The mutual cancellation is a measure of the entanglement between the systems Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com A and B. The topological entanglement entropy arises here as evidence for topological order in quantum states. The scaling of the entanglement entropy for ground states of most gapped systems satis¯es the area law, as the leading term. The topological information is embedded in the subleading terms and is well hidden. The entanglement entropy provides the topological correction whenever the ground states are topologically ordered. For a topological system, the entanglement entropy

by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. which is computed in the ground states of topologically ordered states will follow the area law plus a universal correction that indicates the presence of long-range entanglement. Topological phases of matter exhibit long-range entanglement, as a probe for non-local order parameters such as ground state degeneracy or topological spin. In (2 þ 1)-dimensions, the topological order of the ground state or the topological entanglement entropy can be described by the constant term in the entanglement

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entropy of a disk in the limit of large radius. The topological entanglement entropy can be found by computing the entanglement entropy of a large disk (or a large ball for higher dimensions) and then extracting the constant term, independent of the disk circumference. The main properties of the topological order is the degeneracy in in¯nite volume limit of the ground state in a Riemann surface as a function of genus, the spectrum of the quasiparticle excitations and the structure of the gapless boundary excitations. The topological entanglement entropy is a measure of the topological order encoded in the wave function and not in the spectral properties of the Hamiltonian. In the case of an excitation gap, the correction term remains universal if the boundary between the two subregions is smooth and has a topological origin. For gapped systems, the leading term in the entanglement entropy is proportional to the surface area of the boundary. In two dimensions, the surface area is represented by a length. The coe±cient is non-universal because it depends on the UV cut-o®. Nevertheless, the topological entanglement entropy is separable from the length term. A non-zero topological entanglement entropy is an indication of the \long range entanglement" in the ground state. For highly entangled phases, the ground state cannot be smoothly deformed into a product state over a number of ¯nite regions and the entanglement entropy will show deviations from the area law. In this case, some excitations (or quasiparticles) will have non-local properties, exhibiting long-range interactions, as they cannot be individually produced by a set of local operators. The topological entanglement entropy describes the information inside a topological phase as a single number,

Stopo log D; ð4Þ

where D is the total quantum dimension of the system. The total quantum dimension D describes the entanglement properties of the ground state. For a topological phase in a topological quantum ¯eld theory described Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com by a speci¯c unitary modular tensor category C, the total quantum dimension is given by the sum over all the types of quasiparticles in the theory, rXffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D¼ d a; ð5Þ a2C

where da is the quantum dimension of the anyon with topological charge a. In order by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. to ¯nd D, da needs to be known.

The quantity Stopo is su±cient to fully describe the topological order up to chirality. For an anyon model, if N is the number of anyon types in a theory, the fusion rules imply that N D2. If the system is not purely topological, the length-dependent terms of the entanglement entropy are modi¯ed whereas the universal contribution remains the same.

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For an arbitrary compact orientable surface, the entanglement entropy between the two regions becomes ! XN X ðkÞ 1 SA ¼ Lk log Dþ p c log dc þð~AÞþOðL k Þ; ð6Þ k¼1 c

with k ¼ 1; ...; N being the connected components of the partition boundary, Lk the length of the kth connected partition boundary component with the topological ðkÞ charge c, ~A being the anyonic reduced density matrix for A, p c the probability of the state ~A, dc the quantum dimension of c and ð~Þ the anyonic entropy of the anyonic state ~. The study of entanglement entropy properties started in the context of understanding the area law in black hole thermodynamics in terms of quantum information theory. In general, in a quantum ¯eld theory in d space dimensions, the entanglement entropy of a region of linear size L will be proportional to the size of the boundary ðL=aÞd1 or the area of that region, containing a non-universal coe±cient which is dependent on the choice of the UV cut-o® a. In a generic (1 þ 1)-dimensional quantum ¯eld theory the non-universal area law reduces to a universal lnðL=aÞ size dependence with a universal factor equal to c=3, where c is the central charge. In (2 þ 1)-dimensions, the Bekenstein–Hawking entropy of a black hole depends on the length of the horizon, instead of the area, LengthðÞ SBH ¼ ; ð7Þ 4GN showing a similar form to the non-universal length term, if the coe±cient were a universal term. If the Bekenstein–Hawking formula de¯nes the geometric entanglement entropy, the black hole is in a near-maximally entangled state. The study of the concept of entanglement entropy in condensed matter systems is also related to an increased interest in the behavior of systems near quantum-critical points from a quantum ¯eld theory perspective and understanding quantum phase transitions for use in quantum computing. Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com In a (2 þ 1)-dimensional system in a gapped topological phase, the ¯rst term is independent of the size or geometric shape of the region A. The separation may be done if the region A is considered as a sum of three or more subsystems, with the resulting entanglement entropies in a linear combination that cancels the terms associated to the boundary, leaving only the universal term and leading to the novel concept and fundamental characteristic of the topological ¯eld theory, the quantum

by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. dimension of the excitations. By dividing the system into a number of subsystems and calculating the total entropy by using a linear combination of the entanglement entropies of each subregion, the length term is eliminated, leaving only the universal term, the topological entanglement entropy. As the correlation functions are always topologically invariant at distances longer than the correlation length between them and energies lower than the energy of the

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system, the ground state degeneracy on the higher-genus surface becomes important. The topological entanglement is therefore originated in the ground state wave function. A topological phase is described by a long-distance e®ective theory with topological properties. If a system has an energy gap, the quantum order is topological and it will be described by a topological quantum ¯eld theory in the infrared. Topological order has properties like the degeneracy in in¯nite volume limit of the ground state on a Riemann surface as a function of the genus, the spectrum of quasiparticle excitations and the structure of the gapless boundary excitations.

For a single excitation of charge a, the topological entanglement entropy Stopo will be de¯ned as a Stopo ¼ logðda=DÞ ¼ logðS 0 Þ; ð8Þ a where S 0 is the matrix element of the modular S-matrix of the (1 þ 1)-dimensional CFT that describes the boundary excitations of the topologically ordered medium. The system manifests quantum Hall states with di®erent ¯lling fractions preserving same symmetries. Let HaðNÞ be the Hilbert space of a (2 þ 1)-dimensional topological theory of N quasiparticles (linearly-independent states) of type a. In large-N limit, the quantum dimension da of the excitation of type a for the Hilbert space grows exponentially and the number of independent states with N local excitations of charge a becomes N proportional to d a ,

N dimHaðNÞ/ðdaÞ ; ð9Þ

where da is the quantum dimension of the excitation a. For any D > 1 the system will manifest topological order. For non-Abelian statistics, the condition da > 1 must be true, where the states mix with each other under braiding. A di®erent order will possess di®erent quantum correlations between the microscopic degrees of freedom, while it still cannot be distinguished by any other local Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com order parameter. Landau's symmetry-breaking and local order parameter theory becomes obsolete, making necessary the use of quantum numbers. The topological entanglement entropy survives in long-distance limit, L !1, and it is calculated using the low-energy topological ¯eld theory describing the braiding properties of the anyonic quasiparticle excitations. Topological entanglement entropy describes the topological order within the wave function, but not

by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. within the spectral properties of the Hamiltonian. Its origin and physical interpretation are still not clear. The Bekenstein–Hawking formula is proportional to the length and has no topological characteristics. While the Bekenstein–Hawking entropy contribution is universal, the length term is not. The topological properties of Bekenstein–Hawking entropy are intriguing but can be understood in the context of interpreting the black hole as a localized defect of a topologically ordered system. On large scales, locality of

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a property that behaves almost like classical position should naturally emerge from the theory.

3. The BTZ Black Hole The BTZ black hole is a solution of Einstein's equations with negative cosmological constant in (2 þ 1)-dimensions, and is de¯ned in Schwarzschild coordinates by 2 2 2 2 r r R ds ¼ S d2 AdS dr2 r2d2: ð10Þ BTZ R 2 r2 r 2 AdS S

The geometry becomes asymptotic for large r in global AdS3 global spacetime, with radius of curvature RAdS and AdS boundary at r ¼1. The horizon is located at r ¼ rS (Schwarzschild radius). If we rescale the coordinates, we have

RAdS rS r ! r; ! ; rS; ð11Þ rS RAdS

and use RAdS 1 unit. The BTZ metric has now a horizon located at r ¼ 1, dr2 ds 2 ¼ðr2 1Þd 2 r2d2: ð Þ BTZ r2 1 12 A pure (2 þ 1)-dimensional theory does not take into account gravitational °uctuations and backreaction. The theory would require UV completion because of its non-renormalizability in higher dimensions when coupled to propagating matter.

In AdSglobal spacetime, a BTZ black hole has a holographic dual described by a conformal ¯eld theory with the temperature dual to the Hawking temperature. The

BTZ solution connects two asymptotically AdSglobal boundary regions. The Kruskal extension of BTZ dual is 4dudv 1 uv 2 ds2 ¼ d2 ðjuvj < 1Þ: ð13Þ ð1 þ uvÞ2 1 þ uv

The holographic dual is described by two dynamically decoupled CFTs in a state Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com of \thermo¯eld" entanglement, where the wave function is de¯ned as 1 X eEn n n: ð Þ BTZ Z 14 n The entangled state is dual to the Hartle–Hawking vacuum for BTZ solution. In holographic context, entanglement of the dual CFT is important for understanding the emergence of bulk spacetime. by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. The AdS–Schwarzschild black hole solution has a dual to a CFT at ¯nite temperature. For a pure Hartle–Hawking state in a pair of two CFTs, CFT1 and CFT2, we have 1 X j i¼ eEn=2jni jni ; ð Þ Z 1 2 15 n P E where En are the energy eigenvalues of the CFT and Z ¼ ne n .

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The thermal density matrix is obtained by tracing out one of the Hilbert space of the second CFT: 1 X ¼ j ih j¼ eEn jni hnj : ð Þ 1 Tr2 Z 1 1 16 n The AdS boundary correlators of local CFT operators \living" on the AdS boundary, @AdS, describe a non-perturbative and UV-complete di®eomorphism-invariant theory of quantum gravity as a generalization of the S-matrix for asymptotic Min- kowski spacetime. The bulk theory cannot be pure gravity. In (2 þ 1)-dimensions, gravity will describe the interaction at long distances between massive localized excitations, without explaining the excitation spectrum in the bulk. Everything inside the bulk space is described by the degrees of freedom residing on its boundary. The infrared–ultraviolet connection (Susskind and Witten) describes that the infrared long-distance e®ects in the bulk are strongly connected to the ultraviolet short-distance e®ects on the boundary. The non-locality of the interactions leads to the IR–UV mixing. The UV–IR mixing connects the in¯nitely large scale in the bulk and in¯nitesimally short scale in the ¯eld theory. When evaluating the on-shell action, the in¯nite volume of the bulk spacetime produces a divergence that can be eliminated by cutting o® the spacetime near to the boundary in the dual-¯eld theory. The cut-o® represents a short-distance regulator of the UV divergence. The AdS–CFT correspondence would provide a geometric view for the renormalization group °ow of a system with gravity dual. In this picture, the extra radial coordinate of a spacetime with asymptotically AdS geometry in the bulk geometry corresponds to the energy scale on the boundary, whereas the bulk radial °ow is regarded as the renormalization group °ow parameter of the boundary ¯eld theory. In the scheme of the holographic renormalization group the radial coordinate parameterizes the renormalization group °ow of the dual ¯eld theory, as the evo- lution of ¯elds along the bulk radial direction de¯nes the renormalization group °ow

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com of the coupling constants on the boundary. The close relation between the radial °ow in AdS space and the renormalization group °ow in the dual ¯eld theory is important in understanding the IR–UV mixing. A UV cut-o® in the ¯eld theory is dual in the bulk to a large radial cut-o®. The Ryu– Takayanagi formula uses the entanglement entropy of large boundary intervals to relate the deep geometry in the bulk to non-local observables across large distances on the boundary. by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. The radial coordinate becomes the natural holographic direction, which in the case of a black hole generates a loop of minimal length or event horizon hiding the bulk degrees of freedom from the rest of the spacetime. The event horizon of a minimal length ðÞ is a boundary feature of topology whereas its geometry is identi¯ed by the black hole quantum state, completely speci¯ed by its mass M and spin J. At progressively larger spatial separation intervals, the non-local observables on the horizon probe deeper into the bulk of the spacetime.

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4. The Bulk and Boundary Entropies High-energy physics processes are located near the boundary of the QFT whereas low- energy dynamics reside deep in the bulk. The area law de¯ning the black hole entropy con¯ned by horizon is a direct consequence of the existence of the radial direction in the bulk. The entanglement between the inside and outside of the event horizon may be relevant for understanding the entropy of the black hole. One of the most essential aspects of holographic renormalization is that the bulk theory possesses an extra spatial dimension relative to the dual-boundary CFT. In holographic condensed matter physics, the radial dimension geometrizes the renormalization group. A spatial dimension as a two-dimensional membrane in three-dimensional space emerges as a direct consequence of a basic intrinsic property of the event horizon which is the membrane paradigm at null in¯nity, governed by an in¯nite set of symmetries and conservation laws. In this perspective, the in¯nite number of con- served charges is de¯ned by the stress-energy tensor and the asymptotic conditions for the metric. In Chern–Simons theory, the bulk information (the fractional charge and the statistics of quasiparticle excitations) is mapped to the chiral conformal ¯eld theory ona(1 þ 1)-dimensional boundary of the (2 þ 1)-dimensional system. The topological entanglement entropy is de¯ned as the boundary entropy in the conformal ¯eld theory. From an AdS–CFT perspective, the correspondence is realized as a

AdS3–CFT2 correspondence. The boundary entropy de¯nes the ground state degeneracy due to the existence of the boundary and it is identi¯ed by the entanglement entropy emerging at the boundary. For (1 þ 1) systems, the dependence of the entanglement entropy on the length L can be used to separate critical and non-critical systems because in the ¯rst case the entropy diverges logarithmically. In a gapped system, the leading term scales the area of the boundary and the coe±cient is cut-o®-dependent. The presence of conformal invariance in the bulk constrains the low-energy behavior of the boundary. We can consider that a ð1 þ 1Þ-dimensional critical system is the boundary of the Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com ð2 þ 1Þ-dimensional chiral topological system. In a (2 þ 1)-dimensional gapped system the ¯rst leading term scales with the area but it is universal and independent of the size and shape of this area A. The event horizon of the black hole becomes the point of contact that makes possible the tunneling process between gapless excitations at the boundary. The singularity behind the event horizon can now be treated as a defect or impurity in a

by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. ð1 þ 1Þ-dimensional system. We can recall the entanglement entropy formula and treat it now as an impurity problem,

1 SA ¼ Simp ¼ L ln DþþOðL Þ: ð17Þ

Here L is the size of the one-dimensional space. The universal term in the entanglement entropy scales with the central charge of the conformal ¯eld theory. When calculating the entropy by initially taking the

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limits LT !1and T ! 0, the dimension D is not necessarily an integer. For D to be an integer, the formula needs to compute the entropy of the impurity. The entanglement entropy becomes a property of the ground state for zero-temperature limit and L in¯nite. If the temperature is decreased, the length term is eliminated if we take into

account the di®erence SUV SIR, with SUV as the entropy in the ultraviolet limit or the entropy in the absence of the impurity, and SIR as the entropy in the infrared limit or the entropy calculated at zero temperature. In the infrared limit, the boundary breaks into two regions and the topological entanglement entropy becomes

2 ln D. Therefore we have SUV SIR ¼ ln D where D > 1. The degrees of freedom of the impurity in ð2 þ 1Þ dimensional are coupled to the ð1 þ 1Þ degrees of freedom, making the ð1 þ 1Þ-dimensional critical system to be a boundary for the ð2 þ 1Þ-dimensional system in a topological phase. The boundary entropy is a topological entropy for the subregions of the split system. The partition function for the chiral theory is X Z ¼ NaaðqÞ; ð18Þ a

with Na as the number of duplications of the primary ¯eld a, the character aðqÞ de¯ned as H c=24 L aðqÞtrae ¼ q traq 0 ð19Þ and q e2=R with R being the system spatial length. The modular S-matrix can be written as X b að~qÞ¼ S abðqÞ; ð20Þ b where ~q e2R=. As the theory is invariant under modular transformations, the partition function also is given by X

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com Z ¼ N~ aað~qÞ: ð21Þ a

A(2 þ 1)-dimensional system gapped in the bulk and gapless on the boundary can be described by a rational conformal ¯eld theory. The boundary entropy can be de¯ned in terms of the chiral conformal ¯eld theory and contains a similar universal part lnðDÞ as the topological entanglement entropy. The universal part of the boundary entropy is by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

Sboundary ¼lnðDÞ; ð22Þ where D is not necessarily an integer. To understand the bulk entropy, we introduce the topological degeneracy of a state which is the number of ways one can fuse the quasiparticles to produce a fusion channel (possible topological charges) de¯ned in a non-local space shared by non- Abelian anyons, independent of their location, called the fusion space. The bulk term

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arises because of multiple possible fusion channels [for the toric code, the entropy is lnð2Þ]. The fusion space is generated by all di®erent ways the anyons can be fused over how they are fused and represents a collective non-local property of the anyons. Topological degeneracy implies topological order or long-range entanglement in the ground state. For an na number of quasiparticles labeled by a, a matrix T can be de¯ned as Y n T ¼ ðQaÞ a ð23Þ a

using the ðQaÞ matrix of fusion coe±cients. The largest eigenvalue of ðQaÞ represents the quantum dimension. The entanglement entropy provides the associated topological correction whenever the ground states are topologically ordered. An important feature of many ground states of strongly correlated many-body systems is that the entanglement entropy is not an extensive quantity. Unlike in the case of the thermodynamic entropy, which is extensive (meaningthatitscaleswiththesizeorextent of the system), the area law postulates a proportionality of the entanglement entropy to the boundary of the two systems in the ground state. The area law sets a constant upper bound on the entanglement entropy given by a ¯nite correlation length. At the same time, the bulk entanglement entropy is an extensive quantity and can be de¯ned in terms of the number of bulk quasiparticles and topological degeneracy, ! X Sbulk ¼ ln T0cdc : ð24Þ c

0 0 If dc and Tc0 are rewritten in terms of the modular S-matrix and using dc ¼ S c =S 0 , we have X Sbulk ¼ na lnðdaÞ: ð25Þ a Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com The entropy contains separate contributions of lnðdaÞ from each quasiparticle of quantum dimension da. For a number of type-a quasiparticles in the bulk na, we can write the holographic partition function as X Z ¼ T0aaðqÞ; ð26Þ a by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. where T is the topological degeneracy matrix de¯ned in (23). Finding the path integral for a (2 þ 1)-dimensional Chern–Simons theory for a spacetime D S1,whereD is a disk of radius R and S1 acircleofradius, leads to the holographic partition function. The set of degeneracies T0a require the usage of a set of na Wilson loops of type a puncturing the disk and wrapping around S1.

1940001-14 Topological Entanglement Entropy of Black Hole Interiors

The thermodynamic entanglement entropy in R= !1limit results from the partition function (26), X 0 Z ¼ T0aS a 0ð~qÞþ : a The total entropy S is found via the expansion lnðZÞ¼fR þ S þ : b 0 If db ¼ S 0=S 0 is used in conjunction with (24) and (25), we have ! X 0 S ¼ ln T0aS a a ! X 0 ¼ ln T0ada þ lnðS 0 Þ X a ¼ na lnðdaÞlnðDÞ: ð27Þ a For R= !1limit

S ¼ Sbulk þ Sboundary: ð28Þ

Let SUV be the entropy in the UV limit and SIR the entropy in the IR limit. In this case,

SUV SIR ¼lnðDÞ ð2 lnðDÞÞ ¼ lnðDÞ: ð29Þ

If q1 ¼ expð2=R1Þ and q2 ¼ expð2=R2Þ,withR1 and R2 being the sizes of the two regions, the partition function in UV is Z0 ¼ 0ðqÞ while in the IR, the function takes the form

Z0;0 ¼ 0ðq1Þ0ðq2Þ:

The universal term in aðqÞ is independent of q and leads to separate entropies of

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com ðDÞ Z S ¼2 ðDÞ ln for eachP region. For 0;0 we have IR ln . In the IR, the bulk term of the entropy is ana lnðdaÞ and brings a contribution of lnðdaÞ. The partition function is Za;0 ¼ aðq1Þ0ðq2Þ or Z0;a ¼ 0ðq1Þaðq2Þ and therefore the entanglement

entropy becomes SIR ¼ lnðdaÞ2 lnðDÞ. If a quasiparticle of type a in UV has a holographic partition function aðqÞ, then

the entanglement entropy is SUV ¼ lnðdaÞlnðDÞ. At the point contact, the boundary entropy diminishes from 2 lnðDÞ to lnðDÞ. by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. If two regions are in contact, the quasiparticles tunnel between them, perturbing and rearranging the incident boundary degrees of freedom, resulting into a loss of entropy. The holographic partition function for the two regions is the product of their separate holographic partition functions. The boundary entropy for the two regions is 2 lnðDÞ. The contact between the regions acts as a perturbation and the two thermodynamic entropies associated with each boundary will evolve into one

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single entanglement entropy of a shared boundary in the UV limit. This entropy loss on the boundary is actually a ð2 þ 1Þ-dimensional entanglement entropy. The bulk entanglement entropy of the system arises from the uncertainty associated with the quantum state of each bulk quasiparticle. The bulk entanglement entropy is generated by the gapped bulk quasiparticle excitations in a given state but still remains a property of the boundary of the system as the (2 þ 1)-dimensional bulk entropy is encoded in the dimensional partition function (1 þ 1) system.

5. Conclusions The holographic partition contains both bulk and boundary entanglement entropies. The topological entanglement entropy between the two separate regions becomes thermodynamic entropy of each region. The smoothness of the spacetime is achieved in the limit where each region of spacetime in contact with the neighboring region shares its degrees of freedom at the boundary. This fact changes the boundary entropy. We apply the quasiparticle tunneling dynamics (Moore–Read Pfa±an or Read–Rezayi states non-Abelian quantum Hall states) of a point contact between neighboring points on the boundary of a disk. The degrees of freedom live on the boundary and the information in the bulk is holographically encoded in the boundary (edge) modes. If spacetime geometry is directly related to the entanglement structure of the underlying degrees of freedom, the holographic principle may act as a constraint for this framework. The smoothness of the spacetime continuum is explained by the fact that the entropy has, apart from a bulk term describing the local degrees of freedom by the area law, a boundary term that splits into two and shares in a non-local way its degrees of freedom between any two regions that are in contact. A black hole state arises when the two regions (or the interior region and its environment) are maximally entangled. In this sense, we can also say that the entanglement entropy saturates the Bekenstein–Hawking or, for more general spacetimes, the Bousso bound. If gravity is holographic, the Bousso

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com bound should impose the necessary and su±cient conditions for the smoothness of the spacetime, leading to a UV-complete theory of quantum gravity on AdS. This entanglement saturation implies that the splitting of the boundary occurs only for non-maximal entanglement and while we consider the spacetime as a continuum manifold, we have to assume a maximal entanglement in the UV limit. Geometry and quantum entanglement are intimately connected here. If spacetime is built from entanglement, the bulk–boundary correspondence can help understanding by MACQUARIE UNIVERSITY on 03/08/21. Re-use and distribution is strictly not permitted, except for Open Access articles. the smoothness of the spacetime. Geometric space and the standard Einstein–Hilbert action emerge in the continuum limit from the holographic bound that holds the stitching of the spacetime in a maximally entangled state. The entropy formula touches two distinct problems: the emergence of locality from non-local degrees of freedom and the emergence of smooth spacetime. The degrees of freedom have a non-geometric nature and refer to disconnected components of a not necessarily geometric spacetime. In this \boundary" approach, the

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concept is separate from the coarse-grained MERA and cMERA emergent metric procedures using a unitary transformation or \disentangler", designed to remove the entanglement at a given scale. The continuum limit is achieved by gluing disconnected components of the spacetime that share the same boundary. If the entanglement between these spacetime regions is maximal, the two boundaries of each region become one unique boundary, in the same way as the two horizons of a black hole become one single horizon in the extremal case (extremely rotating or charged black holes). Spacetime emerges in a natural way from entanglement via holographic principle while maximal entanglement assures the continuity of the spacetime. This novel approach explains the nature of Raamsdonk's glue of spacetime via the holographic principle by viewing the non-local degrees of freedom of the boundary between disconnected regions of spacetime as \shared" between distinct regions. The fresh concept of a \shared boundary" can be used in conjunction with Raamsdonk's glue as the entanglement between pure states, as the idea behind the nature of this glue or independently as a starting point of formulating an emergent metric from renor- malizing discrete metric spaces and a new geometric renormalization method. As Raamsdonk's glue is the actual entanglement between pure states in the components of spacetime, such a metric can be constructed using discrete structure underlying the smooth spacetime. This paper concentrates only on understanding the nature and properties of the topological entanglement entropy and its role in explaining locality and smoothness of the spacetime. Further work is necessary to formulate an emergent metric using an RG °ow in the locally compact metric space and leading to the continuum limit of the classical spacetime. The conditions on the existence of the continuum limit and the stability under the RG °ow should be held by the holographic bound. Without gravity, the quantum ¯eld should feature a Landau pole at high energies, leading to the triviality problem and breaking down in UV limit. Turning a trivial theory into a non-trivial one should be here done at the cost of introducing the holographic bound

Rep. Adv. Phys. Sci. 2020.04. Downloaded from www.worldscientific.com constraint.

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