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Holographic entanglement entropy : structure and applications from noncommutative field theories to energy… Rabideau, Charles 2016

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Holographic Entanglement EntropyStructure and Applications from Noncommutative Field Theories toEnergy ConditionsbyCharles RabideauBSc Honours Mathematics and Physics, McGill University, 2010MSc Physics, The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2016c© Charles Rabideau, 2016AbstractThe holographic Ryu-Takayanagi formula for entanglement entropy connects theentanglement of a field theory to the geometry of a dual gravitational theory in astraightforward and universal way.The first part of this thesis applies this formula to study the entanglement en-tropy in strongly coupled noncommutative field theories. It is found that the groundstate of these theories have substantial entanglement at the length scale of the non-commutativity. The entanglement entropy in a different perturbative regime is alsocomputed, where in contrast it is found that noncommutative interactions do not in-duce long range entanglement in the ground state to leading order in perturbationstheory.The second part of this thesis explores some general consequences of this holo-graphic formula for the entanglement entropy. Identities involving entanglemententropies are related to nontrivial geometric constraints on gravitational duals. Inparticular, the strong subadditivity of entanglement entropy is used to show thatdual three dimensional asymptotically anti-de Sitter gravitational states must obeyan averaged null energy condition. Finally, this holographic formula allows us atleast in principle to express the entanglement entropy of a region in a holographicfield theory in terms of the one-point functions in that theory. This is exploredin the context of a two dimensional conformal field theory where explicit calcula-tions are possible. Our results in this case allow us to extend a recent proposal thatthe entanglement entropy of states near the vacuum of conformal theories can beunderstood by propagation in an auxiliary de Sitter space.iiPrefaceThis thesis includes work that has been published. This preface explains my con-tributions to the research presented here.Chapter 1 is a review and includes original research only when summarisingthe results of the rest of this thesis.Chapter 2 is published as J. L. Karczmarek and C. Rabideau,“Holographic en-tanglement entropy in nonlocal theories,” JHEP 1310, 078 (2013). The researchtopic was proposed by J. L. Karczmarek. This work was the result of a closecollaboration between the authors throughout the entire work. I completed the nu-merical calculations and prepared the figures. The publication was prepared byJ. L. Karczmarek with contributions from notes I prepared.Chapter 3 is published as C. Rabideau, “Perturbative entanglement entropy innonlocal theories,” JHEP 1509, 180 (2015). I am the sole author of this work. Thiswork was completed with the advice and supervision of J. L. Karczmarek.Chapter 4 is published as N. Lashkari, C. Rabideau, P. Sabella-Garnier andM. Van Raamsdonk, “Inviolable energy conditions from entanglement inequali-ties,” JHEP 1506, 067 (2015). This work is based on the close collaboration ofthese authors under the supervision of M. Van Raamsdonk. I was most heavilyinvolved in the results of Section 3 and 4 and Appendix B. The publication wasprepared by N. Lashkari and M. Van Raamsdonk with contributions from notesprepared by all of the authors.Chapter 5 is published as M. J. S. Beach, J. Lee, C. Rabideau and M. VanRaamsdonk, “Entanglement entropy from one-point functions in holographic states,”JHEP 1606, 085 (2016). This work is based on the close collaboration of the au-thors under the supervision of M. Van Raamsdonk. I was most heavily involvediiiin the work presented in Sections 4 and 5 and the Appendix, which are based oncalculations completed by J. Lee and myself. All the co-authors participated in thewriting and editing of this publication.All published material used in this thesis is published in an open access journaland as such is available under the Creative Commons CC BY 4.0 license whichpermits reproduction with attribution.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Relative entropy . . . . . . . . . . . . . . . . . . . . . . 71.2 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Holographic entanglement entropy . . . . . . . . . . . . . 111.3 Noncommutative gauge theories . . . . . . . . . . . . . . . . . . 141.3.1 Summary of results for holographic entanglement entropy 141.3.2 Summary of results for perturbative entanglement entropy 151.4 The structure of holographic entanglement entropies . . . . . . . 161.4.1 Constraints on geometry from entanglement . . . . . . . . 161.4.2 Expanding holographic entanglement entropies in terms offield theory one-point functions . . . . . . . . . . . . . . 181.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19vI Entanglement Entropy in Noncommutative Field Theories . . . 202 Holographic Entanglement Entropies in Noncommutative Theories 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Theories considered and their gravity duals . . . . . . . . . . . . 252.2.1 NCSYM . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Dipole theory . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Entanglement entropy for the strip . . . . . . . . . . . . . . . . . 302.3.1 Review of results for AdS space . . . . . . . . . . . . . . 322.3.2 Dipole theory . . . . . . . . . . . . . . . . . . . . . . . . 332.3.3 NCSYM . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Entanglement entropy for the cylinder in NCSYM . . . . . . . . . 412.5 Mutual information in NCSYM . . . . . . . . . . . . . . . . . . . 452.6 Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Perturbative Entanglement Entropies in Noncommutative Theories 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 n-sheeted surfaces . . . . . . . . . . . . . . . . . . . . . 563.4 Free theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . . 583.4.2 Entanglement entropy in the free theory . . . . . . . . . . 603.5 First order in perturbation theory . . . . . . . . . . . . . . . . . . 623.5.1 Commutative theory . . . . . . . . . . . . . . . . . . . . 623.5.2 Noncommutative theory . . . . . . . . . . . . . . . . . . 663.5.3 Complex scalar . . . . . . . . . . . . . . . . . . . . . . . 743.5.4 Dipole theory . . . . . . . . . . . . . . . . . . . . . . . . 753.6 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78II The Structure of Holographic Entanglement Entropy . . . . . 804 Inviolable Energy Conditions . . . . . . . . . . . . . . . . . . . . . . 81vi4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.1 Entanglement inequalities . . . . . . . . . . . . . . . . . 864.2.2 Holographic formulae for entanglement entropy . . . . . . 934.2.3 Energy conditions . . . . . . . . . . . . . . . . . . . . . 934.3 Constraints on spacetimes dual to Lorentz-invariant 1+1D field the-ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3.1 An averaged null energy condition . . . . . . . . . . . . . 964.3.2 Non-monotonic scale factors . . . . . . . . . . . . . . . . 984.4 Constraints on spacetimes dual to states of 1+1D CFTs . . . . . . 994.4.1 Constraints from positivity and monotonicity of relativeentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4.2 Constraints from strong subadditivity . . . . . . . . . . . 1064.5 Constraints on spherically-symmetric asymptotically AdS spacetimes1094.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.6.1 Constraints on entanglement structure from geometry . . . 1115 Entanglement Entropy of Holographic States in Terms of One-pointFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.1 Relative entropy and quantum Fisher information . . . . . 1195.2.2 Canonical energy . . . . . . . . . . . . . . . . . . . . . . 1215.3 Second-order contribution to entanglement entropy . . . . . . . . 1245.3.1 Example: CFT2 stress tensor contribution . . . . . . . . . 1255.3.2 Example: scalar operator contribution . . . . . . . . . . . 1305.4 Stress tensor contribution: direct calculation for CFT2 . . . . . . . 1325.4.1 Conformal transformations of the vacuum state . . . . . . 1335.4.2 Entanglement entropy of excited states . . . . . . . . . . 1345.4.3 Perturbative expansion . . . . . . . . . . . . . . . . . . . 1365.4.4 Excited states around thermal background . . . . . . . . . 1395.5 Auxiliary de Sitter space interpretation . . . . . . . . . . . . . . . 141vii6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A Analysis of the Potential Divergences from the j > 1 Terms . . . . . 158B Modular Hamiltonian for an Interval in a Boosted Thermal State ofa 1+1D CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161C Variation in Geodesic Length under Endpoint Variation . . . . . . . 164D Rindler Reconstruction for Scalar Operators in CFT2 . . . . . . . . 167viiiList of FiguresFigure 1.1 Example of minimal surfaces used to compute entanglemententropies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.2 Holographic entanglement entropy with a disconnected region. 12Figure 1.3 Holographic entanglement entropy with multiple extremal sur-faces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.1 Point of deepest penetration as a function of strip width for thedipole theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.2 Area of the minimal surface as a function of the strip width forthe dipole theory. . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.3 Point of deepest penetration as a function of the strip width forthe noncommutative theory. . . . . . . . . . . . . . . . . . . 37Figure 2.4 Shape of three extremal area surfaces. . . . . . . . . . . . . . 38Figure 2.5 Area of the minimal surface as a function of strip width fornoncommutative theory. . . . . . . . . . . . . . . . . . . . . 40Figure 2.6 Extremal surfaces homologous to a cylinder in NCSYM. . . . 42Figure 2.7 Point of deepest penetration as a function of the cylinder’s ra-dius in the noncommutative theory. . . . . . . . . . . . . . . 44Figure 2.8 Area of the minimal surface homologous to a cylinder, as afunction of the cylinder’s radius l. . . . . . . . . . . . . . . . 45Figure 3.1 Translations on the n-sheeted surface. . . . . . . . . . . . . . 62Figure 3.2 Non-planar diagrams and non-commutivity. . . . . . . . . . . 68Figure 3.3 Non-planar diagrams for the complex scalar λφ 4 theory. . . . 75ixFigure 4.1 Ryu-Takayanagi formula as a map from the space of geome-tries to the space of mappings from subsets of the boundary toreal numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.2 Spacelike intervals for strong subadditivity. . . . . . . . . . . 87Figure 4.3 Relative entropy constraints on coefficients in the Fefferman-Graham expansion of the metric. . . . . . . . . . . . . . . . . 105Figure 5.1 Rindler wedge associated to a ball-shaped region on the bound-ary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 5.2 Feynman diagram which computes δ (2)S. . . . . . . . . . . . 143xAcknowledgmentsI would like to thank my supervisor Joanna Karczmarek; my collaborators MatthewBeach, Nima Lashkari, Jaehoon Lee, Philippe Sabella-Garnier and Mark Van Raams-donk; and all of the people I have had the great pleasure of discussing physics within my office at UBC and at conferences and summer schools.To my parents and family, thank you for your constant encouragement through-out this degree and your unconditional support throughout my life.I was supported by funding from the Natural Sciences and Engineering Re-search Council of Canada’s Alexander Graham Bell Canada Graduate Scholarship,the Izaak Walton Killam Memorial Fund for Advanced Studies, the University ofBritish Columbia and the Walter C. Sumner Foundation.My co-authors and I are grateful for helpful discussions with Raphael Bousso,Daniel Carney, Laurent Chaurette, Keshav Dasgupta, Ori Ganor, Thomas Hartman,Michal Heller, Juan Maldacena, Don Marolf, Shunji Matsuura, Robert Myers, Fer-nando Nogueira, Hirosi Ooguri, Ali Izadi Rad, Jared Stang and Brian Swingle.This thesis is dedicated to Green College and all the wonderful people I havemet there.xiChapter 1IntroductionThe standard model of particle physics has applied the techniques of quantum fieldtheory very successfully to describe all observed particles and their interactionswith one important caveat: it does not include the effects of gravity. In the classicallimit, Einstein’s theory of general relativity and Maxwell’s equations describe thebehaviour of matter subject to gravity and electromagnetism. At small scales inlaboratory settings on earth, gravitational interactions become weak and can beignored, while electromagnetism dominates. At these small scales in the quantumregime, Maxwell’s classical description breaks down. The study of this regimelead to the development of the standard model, however as gravity is very weakin this regime the question of how to include it in the framework of the standardmodel could be postponed while still matching the results of experiments in particleaccelerators to exquisite precision.Applying the techniques of perturbative quantum field theory to general relativ-ity leads to a nonrenormalisable theory, indicating that general relativity can onlybe understood as an effective field theory and does not describe the correct degreesof freedom to understand quantum gravity at short distance scales. String theoryprovides an ultraviolet completion which reduces to general relativity at long dis-tances, but is not understood in a full nonperturbative sense. A full understandingof quantum gravity is still lacking and this is especially apparent in the context ofgeometries far from flat space such as black holes.One of the symptoms of the problem of quantum gravity is the information loss1problem for black holes [1]. In classical general relativity no signal can escape theevent horizon of a black hole. However, the study of quantum fields in curvedspace near a black hole reveals that black holes emit thermal radiation. As thisradiation carries away energy, the black hole must shrink until it reaches a smallsize where the calculation breaks down. This poses a problem for any quantumtheory which proposes to describe this system: time evolution in quantum systemsis unitary and therefore reversible, yet the semi-classical description of the collapseof matter into a black hole which eventually evaporates away into thermal radiationis not unitary as the information about the configuration of the initial matter cannotbe contained in thermal radiation. Either the unitarity of quantum gravity or thevalidity of the semi-classical approximation near the event horizon of the blackhole must be abandoned.Another symptom is the holographic bound on entropy in gravitational systems[2, 3]. In quantum field theories, thermal entropies are extensive. However, gravitycontains black holes which are thermal systems with an entropy proportional tothe area of their event horizon. In fact, in a gravitational system, thermal entropyin a region is bounded by the area of the boundary of that region. This lead tothe holographic principle, which conjectures that a gravitational system in a regionshould be described by degrees of freedom on the boundary of that region.Taken together, these suggest that we do not understand how to correctly or-ganise the degrees of freedom of quantum gravity. However, a concrete exampleof the holographic principle has been found in the context of string theory wherequantum gravity in an asymptotically anti-de Sitter (AdS) geometry was found tobe dual to a conformal field theory living on its conformal boundary [4, 5]. Thisexplicit example of holography can be used to shed light on quantum gravity as thedual conformal field theory is a well defined quantum system.1Although in order to make contact with reality we might be most interested inquantum gravity with asymptotically flat or de Sitter boundary conditions, asymp-totically AdS boundary conditions provide a good starting point for understanding1The classical limit of the gravitational description is dual to a strong coupling limit in the confor-mal field theory, which although well defined in principle is often inaccessible to perturbative fieldtheory techniques. This presents an obstacle to our original goal of understanding quantum gravity,but it also opens up new opportunities for understanding strongly coupled field theories by studyingthe dual classical gravitational description.2some of the problems of quantum gravity because they simplify the problem ofdefining diffeomorphism invariant operators. Diffeomorphisms are a gauge sym-metry of general relativity and so the physical states and operators of quantumgravity should be diffeomorphism invariant, which precludes the existence of localoperators in quantum gravity. However, the conformal structure of the asymptoticboundary of a geometry is invariant under diffeomorphisms and so this boundaryprovides a natural setting for diffeomorphism invariant observables. In asymp-totically AdS spaces, the conformal boundary is timelike and the observables ofquantum gravity can be matched to those of a conformal field theory.This thesis will focus on a particular diffeomorphism invariant quantity, thearea of the minimal surface anchored on a region of the boundary. The holographicduality relates this quantity to entanglement entropy in the dual conformal fieldtheory through the Ryu-Takayanagi (RT) formula [6, 7, 8] and its covariant gener-alisation given by Huberny, Rangamani and Takayanagi [9].From topological condensed matter systems, to black holes, to phase transitionsand the emergence of spacetime in holography, the study of entanglement entropyhas proven fruitful across many fields of physics [10, 11, 12, 8, 13]. Entanglementis one of the principal features distinguishing the quantum from the classical andentanglement entropy has proven to be an important tool for quantifying entangle-ment between two systems.A useful property of entanglement entropy is that its definition is independentof the details of the theory in question giving us an tool to study universal prop-erties of field theories and to compare the entanglement structure of very differentfield theories. This universality is reflected in the RT formula which relates theentanglement entropy directly to the geometry of the gravity dual.Part I of this thesis uses this holographic relation to compute the entangle-ment entropy in noncommutative field theories using known gravitational dualsand contrasts these results to those found in a different perturbative regime of thesetheories.One of the original motivations for studying noncommutative field theories wasto help regulate divergences in local quantum field theories. Much as introducinga commutation relation between positions and momenta regulated the ultravioletcatastrophe in black body radiation by introducing a minimal scale in phase space,3a commutation relation between coordinates can introduce a minimal length scale.Since this involves quantising coordinates, it was hoped that this might be related tothe quantisation of spacetime required to understand quantum gravity. In addition,noncommutativity appears naturally in string theory in the context of D-branes.Part II studies general properties of this holographic relation, with the goal ofbetter understanding the holographic map and ultimately quantum gravity. Chapter4 relates universal properties of entanglement entropy to constraints on the classicallimit of dual holographic geometries in the form of gravitational energy inequali-ties. This tells us that although the effective theory given by general relativity candescribe any geometry given appropriate matter, the ultraviolet completion givenby string theory can only accommodate geometries obeying some constraints.Chapter 5 uses properties of classical geometries to express entanglement en-tropy in terms of one-point functions, exhibiting constraints on quantum stateswhich have a classical gravitational description. The results of this chapter alsoextend a surprising connection between entanglement entropy in conformal fieldtheories and a field propagating in an auxiliary de Sitter (dS) space.1.1 EntanglementGiven two Hilbert spaces HA and HB, we can construct a tensor product spaceH =HA⊗HB 2. A state in the tensor product space is called a product state ifit can be written as the tensor product of states in the constituent Hilbert spaces.|ψ〉 ∈H is a product state if |ψ〉 = |ψA〉⊗ |ψB〉 for |ψi〉 ∈Hi. A state which isnot a product state is called an entangled state. Given an operator OA on HA anda product state |ψ〉 ∈H , 〈ψ|OA⊗ IB|ψ〉= 〈ψA|OA|ψA〉. The expectation value ofOA in a product state only depends on the state in the A-subspace. This is not thecase in an entangled state.A density matrix is an operator onH which encodes the state, but which canalso encode classical uncertainty like that found in statistical mechanics. Given an2Given a basis {|χa〉} forHA and {|ξb〉} forHB the tensor product space is the space spannedby {|χa〉}×{|ξb〉}.4ensemble of states |ψi〉 with classical probabilities pi, the density matrix isρ =∑ipi|ψi〉〈ψi|. (1.1)A density matrix describes a pure state if it can be written as ρ = |ψ〉〈ψ| for a singlestate, otherwise it describes a mixed state. The expectation value of an observableis tr(ρO). Note that ρ itself is not an observable as it depends on the state.Given a tensor product space and a basis {|χa〉} forHA and {|ξb〉} forHB, thepartial trace istrB(O) =∑b〈ξb|O|ξb〉, (1.2)which defines an operator on A.The reduced density matrix ρA = trBρ reproduces the expectation values ofoperators OA onHA,trρOA = trAρAOA. (1.3)The expectation value of local operators in a subspace can be reproduced witha state in that subspace at the cost of introducing classical uncertainty when theoriginal state was entangled.Quantifying this uncertainty using the von Neumann entropy of the reduceddensity matrix leads us to a natural way of quantifying the entanglement of a state.This is called the entanglement entropy3S(A) =− trA (ρA logρA) . (1.4)If ρ corresponds to a pure state on H =HA⊗HB, then the entanglemententropy computed using either subspace must be equal, S(A) = S(B). In this casethe classical uncertainty in the reduced density matrices on either subspace comesonly from the entanglement. If the state is mixed, this equality does not hold asthere is additional classical uncertainty which can be unevenly distributed between3A useful reference on the topic of quantum information is [14].5the subsystems.Of course this one number cannot fully describe the entanglement of a state.Other useful quantities include the Renyi entropiesSn =11−n log trA (ρnA) (1.5)and the entanglement negativity.Knowledge of all the Renyi entropies allows one to compute the spectrum ofthe reduced density matrix also known as the entanglement spectrum. In fact theRenyi entropies are often used to calculate the entanglement entropy sincelimn→1Sn = SA. (1.6)1.1.1 Field theoryThe Hilbert space of a field theory can be decomposed into a tensor product of thedegrees of freedom inside a region and those in its compliment. The entanglemententropy of a region in a field theory is that resulting from this decomposition. Thisentanglement entropy is defined for any state of the field theory, but unless oth-erwise specified we will usually be interested in the entanglement entropy in theground state of the theory in question.Since the von Neumann entropy of the reduced density matrix in a thermalstate4 will receive an extensive contribution related to the thermal entropy, somereserve the terminology of entanglement entropy for the vacuum state where theentanglement entropy can only be attributed to entanglement. This work will usethe term somewhat more loosely to refer to the von Neumann entropy of the re-duced density matrix in any state. The entanglement entropy in the ground state issometimes referred to as geometric entanglement entropy in the literature.When decomposing the Hilbert space of a gauge theory there are ambiguitiesrelated to the fact that the physical gauge invariant Hilbert space is not strictly local.The resolution of these ambiguities has been discussed in detail in the literature4A thermal state is a mixed state described by a density matrix where every state appears with aclassical probability given by the canonical ensemble, ρthermal = e−βH .6[15, 16, 17].In the ground states of local relativistic field theories, the entanglement entropyusually has an area law divergence.5 In d dimensions,S(A) =C|∂A|εd−1+ . . . , (d > 2) (1.7)S(A) =C logε+ . . . , (d = 2) (1.8)where ∂A is the boundary of A, |∂A| is the area of this boundary, C is a regulatordependent constant and ε is a UV regulator such as a lattice spacing. This arealaw divergence reflects the short range entanglement present in local quantum fieldtheories.In field theories, the entanglement entropy is often computed using the replicatrick. Computing the Renyi entropies defined in (1.5) requires evaluating the nthpower of the reduced density matrix, which can be computed by appropriatelysewing together along A the boundary conditions of n copies of the theory in apath integral. Analytically continuing the result for the nth Renyi entropy to maken continuous and taking the appropriate limit as n→ 1 allows us to compute theentanglement entropy. This procedure is described in detail in Section 3.3 where itis used in the context of perturbative quantum field theory.1.1.2 Relative entropyThe relative entropyS (ρ||σ) = tr(ρ logρ)− tr(ρ logσ) . (1.9)provides a measure of the distinguishability between two density matrices as S(ρ||σ)≥0 with S(ρ||σ) = 0 iff ρ = σ . If ρ and σ are two different pure states this measurewill diverge.Fixing a state σ , the modular Hamiltonian can be defined Hσ = logσ . The5See [18] for a review of area laws. Exceptions include Fermi surfaces [19].7relative entropy isS (ρ||σ) =∆E−∆S (1.10)∆S =Sρ −Sσ (1.11)∆E =〈Hσ 〉ρ −〈Hσ 〉σ . (1.12)The relative entropy is positive and monotonic. For any regions A and B suchthat A⊂ B,S (ρA||σA)≥0, (1.13)S (ρB||σB)≥S (ρA||σA) . (1.14)The relative entropy of a density matrix with itself vanishes. Since the relativeentropy is positive,ddλS(ρ+λδρ||ρ)∣∣∣∣λ=0= 0, (1.15)which leads to the first law for entanglement entropies: 6δS = δE, (1.16)where where δS and δE are the first order changes under the perturbation of thestate to the entanglement entropy and the expectation of the modular Hamiltonianrespectively.The monotonicity of relative entropy is equivalent to the strong sub-additivityof entanglement entropy (SSA), which says that for tripartite systems with H =HA⊗HB⊗HC, [20, 21]S(A∪B)+S(B∪C)≥ S(B)+S(A∪B∪C). (1.17)6This name is in analogy to the first law of thermodynamics.8This can be recast in terms of the mutual informationI(A,B)≡ S(A)+S(B)−S(A∪B), (1.18)so that strong sub-additivity readsI(A,B∪C)≥ I(A,B). (1.19)Restricting ourselves to a subset (B) of a subsystem (B∪C) cannot increase ourknowledge about the correlations between this subsystem and a reference subsys-tem (A).1.2 HolographyThis section will review gauge-gravity duality. The best established example is theAdS5×S5—N = 4,d = 4 Super-Yang Mills duality [4, 5]. A number of variationsof this example are also understood, which lead us to some entries in the dictionaryof the duality7 in its most general form.D-branes are extended dynamical objects in string theory, described in text-books on perturbative string theory, e.g. [22]. At small string coupling (gs),8 theirdegrees of freedom can be understood using string perturbation theory. These in-clude a gauge theory living on the world-volume of the brane. The rank of thegauge group of this theory counts the number of units of brane charge, so we thinkof the low energy excitations of a stack of N branes as including a U(N) gaugetheory. The Yang-Mills coupling of this U(N) gauge theory obeys g2Y M ∼ gs.A large stack of these branes has a complimentary description at large stringcoupling [5]. In this case, the stack of branes represents a large classical source andcan be described using classical gravity. We will focus on the case of D3-braneswhich fill 3 spatial directions. The curvature length scale of this classical solutionobeys R4 = 4pigsα ′2N, where α ′ is the dimensionful string tension parameter in7The gauge-gravity dictionary is the set of relationships that allows us to convert concepts be-tween the two sides of the duality.8String theory doesn’t have externally fixed dimensionless coupling constants. However, in stringperturbation theory amplitudes are expanded in the constant part of the expectation value of thedilaton field.9string theory which sets the length scale of fundamental strings. For this classicaldescription to be a good approximation, the curvature must be large compared tothe scale of both gravity and strings. In terms of string parameters, the string scaleis ls ∼√α ′ and the plank length is lp =√GN ∼√α ′√gs. We need thatR4l4s∼ gsN ∼ g2Y MN ∼ λ  1 (1.20)R4l4p∼ N 1, (1.21)where λ is the ’t-Hooft coupling. In the low-energy limit of this gravitationaldescription, the dynamics of a near horizon AdS5× S5 spacetime decouples fromthe rest. This is understood to be a complimentary strong coupling description ofthe low energy sector of this stack of branes, which at weak coupling is a U(N)gauge theory.This example related a specific spacetime to a specific conformal field theory(CFT). However, by considering different brane configurations, by varying the di-mensionality of the branes or by turning on other sugergravity fields, many gauge-gravity dualities can be found [23, 5, 24, 25].There is an important entry in the holographic dictionary which allows us toextend this duality. Deforming the action of a conformal theory by an operatorO with source J corresponds to adding a scalar field to the gravitational action[26]. J and 〈O〉 determine the boundary conditions for the scalar at the conformalboundary of the asymptotically AdS space. The mass of the scalar field is relatedto the conformal dimension of the operator O by R2m2 = ∆(∆−d).A general feature of the duality is that to describe a field theory defined ona manifold M the gravitational description will need a conformal boundary of M.This means that if the gravitational theory is defined on a manifold N, there mustbe a diffeomorphism which preserves the metric up to a local rescaling between∂N and M.9 This feature allows us to use local data in the field theory as boundaryconditions for the gravitational theory.9Given a region or a manifold M, ∂M denotes its boundary.10Figure 1.1: Two time slices of asymptotically AdS3 geometries in coordinateswhich bring the conformal boundary (in blue) into view. The minimalsurface (in red) homologous to a boundary region (in orange) is shown.On the left the geometry is empty AdS3 corresponding to the vacuumstate of a CFT2. On the right, the geometry is a BTZ black hole corre-sponding to a thermal state of a CFT2. The grey region is enclosed bythe horizon of the black hole.1.2.1 Holographic entanglement entropyThis thesis focuses on a particular entry in this holographic dictionary relating geo-metric entanglement entropy in a field theory to the area of boundary anchored min-imal surfaces in its gravitational description known as the Ryu-Takayanagi (RT)formula [6, 7, 9, 8, 27, 28].The entanglement entropy of a region A in a field theory in a particular state isgiven by the area of the minimal extremal area surface homologous10 to A on theconformal boundary of the dual gravitational description of that state.Let A˜ be an extremum of the area functional such that ∂ A˜ = ∂A, thenS(A) =|A˜|4GN, (1.22)where |A˜| is the area of A˜ and GN is Newton’s constant. Figure 1.1 presents a fewexamples of such minimal surfaces for asymptotically AdS3 spacetimes.10Two submanifolds A and B of dimension k of a manifold M of dimension d > k are homologousif ∂A = ∂B and there exists a submanifold C such that ∂C = A∪B. This essentially means that A issmoothly deformable into B.11Figure 1.2: When the region A (in orange) is disconnected, there are multiplehomologous extremal surfaces (in red). The extremal surface with theminimal area should be chosen.If there are multiple extremal surfaces, the one with minimal area should betaken. If the gravitational state has a black hole, it may be necessary to include thearea of an unconnected component of A˜ which wraps the horizon of the black holeso as to obey the homology constraint. In particular, there is no requirement for Aor A˜ to be connected. See Figure 1.2 for an example where A is disconnected andFigure 1.3 for an example where A˜ is disconnected even though A is not.For static geometries and regions defined on a fixed time slice of the field the-ory, the extremal surface is a minimal surface on the extension of that fixed timeslice into the bulk of the dual geometry.RegularisationThe entanglement entropy in field theories has a UV divergence as was discussed inSection 1.1.1. In holography this is reflected by the fact that the conformal bound-ary is at infinite distance, require any surface anchored there to have infinite area.In order to regularise the entanglement entropy, the surface is instead anchored to adistant cutoff surface. In this picture, the area law divergence typical of the entan-glement entropies in field theories arises from the part of the minimal surface nearthe conformal boundary. In asymptotically AdS spaces, minimal surfaces quicklydive into the interior of the space and the divergence in their area comes from athroat near the boundary of the region (∂A) where they must be anchored to the12Figure 1.3: Two local minima of the area functional (in red) homologous tothe same region (in orange) in an asymptotically AdS3 BTZ black holegeometry corresponding to a thermal state of a CFT2 (the conformalboundary is blue and the interior of the black hole horizon is grey). Theextremal surface with the minimal area should be chosen.conformal boundary.Thermal statesIn thermal states, for regions A much larger than the thermal scale, the entangle-ment entropy receives an extensive contribution proportional to the thermal entropyS(A) ∝ |A|s, (1.23)where |A| is the volume of A and s is the entropy density in the thermal state.The gravitational dual to a thermal state is generally a black hole where thethermal scale is tied to the size of the black hole. The extensive contribution to theentanglement entropy comes from the minimal surface dwelling near the horizonof the black hole as can be seen in Figure 1.1.When A includes the entire boundary, the entanglement entropy is the thermo-dynamic entropy. In this case, the minimal surface homologous to the boundarycannot shrink to zero due to the presence of the black hole. Instead it wraps thehorizon, reproducing the well known Bekenstein-Hawking entropy of the black13holeS =Area4GN. (1.24)1.3 Noncommutative gauge theoriesNoncommutative spaces arise naturally in string theory in the context of D-branes.Noncommutative spaces first manifest themselves in that the coordinates of a stackof D-branes are given by a set of noncommuting matrices rather than a list ofnumbers. In addition, the gauge field living on the worldvolume of a brane in thepresence of a background NS-NS 2 form B field along the brane is a noncommu-tative field theory. In particular, the gauge theory on a D3-brane is Yang-Mills ona noncommutative plane. The gravity dual to this theory was found in [29, 30],which gives us the opportunity on one hand to learn about the application of theRT formula to new backgrounds11 and on the other hand to study the entangle-ment entropy of this theory [31]. Part I of this thesis is dedicated to studying theentanglement entropy of noncommutative theories.1.3.1 Summary of results for holographic entanglement entropyIn Chapter 2, the holographic RT formula is used to study entanglement entropiesin a class of nonlocal theories related to field theories on noncommutative spaces.This will draw on my work with Joanna Karczmarek [31].In a nonlocal theory, the behaviour of entanglement entropy could be expectedto deviate from an area law and this is precisely what was found using holographicmethods at strong coupling. In a simple nonlocal theory with a fixed scale ofnonlocality aL, a dipole deformation of N = 4 SYM, the entanglement entropyis extensive (proportional to the volume of A), for regions A of size up to aL. Atlength scales larger than aL, it follows an area law, with an effective number ofentangled degrees of freedom which is proportional to aL. This is consistent withall the degrees of freedom within a region A of size aL or smaller, and not only those11This gravity dual has a number of unusual properties, including not being asymptotically AdS,a nontrivial dilaton profile and a nonzero B field.14living close to the boundary of A, having quantum correlations with the outside of Adue to the nonlocal nature of the Hamiltonian. In contrast, in the noncommutativedeformation ofN = 4 SYM, which is known to exhibit UV/IR mixing and whosenonlocality length scale grows with the UV cutoff, the entanglement entropy isextensive for all regions as long as their size is fixed as the UV cutoff is takenaway.12Since our theories differ from N = 4 SYM in the UV, the holographic dualsare not asymptotically AdS spaces. Their non-asymptotically AdS geometry hasan interesting consequence. In previously studied examples of extensive behaviourof entanglement entropy (for example, in thermal states) this extensive behaviourwas due to the minimal surface ‘wrapping’ a surface in the IR region of the dual,such as a black hole horizon (see for example [33]). Here, however, the extensivityarises from the fact that the minimal surfaces stays close to the cutoff surface: thevolume law dependence of entanglement entropy is a UV phenomenon.1.3.2 Summary of results for perturbative entanglement entropyA natural question is whether the volume law behaviour found in the holographicanalysis of entanglement entropies is generic to nonlocal theories or if it is confinedto a strongly coupled, large N regime.In Chapter 3, based on [34], the role of interactions in this question is inves-tigated by considering field theories at small coupling and with one scalar degreeof freedom and nonlocal interactions. The leading divergence in entanglement en-tropy of large regions to leading order in perturbation theory is not found to beproportional to the length scale of the nonlocality, hence no evidence of a volumelaw is found. Instead, the leading divergence in both theories has the same formas the standard local λφ 4 theory which follows an area law. This result indicatesthat, perturbatively these nonlocal interactions are not generating sufficient entan-glement at distances of the nonlocality scale to change the leading divergence, atleast to first order in the coupling.12Entanglement entropy in the noncommutative theory was studied before in [32]. Here we extendand improve on those results.151.4 The structure of holographic entanglement entropiesPart II investigates general features of the RT formula. This has the double goal ofbetter understanding the two theories involved in the duality as well as clarifyingthe structure of the duality itself.1.4.1 Constraints on geometry from entanglementEntanglement entropy is a function on the subsets of a spacetime manifold. How-ever not all such functions can arise as the entanglements entropies of some state.There are a number of constraints that the entanglement entropy of any state mustobey. Similarly, not all such functions can arise as the areas of minimal surfaces insome geometry. Holographic entanglement entropy must satisfy both these sortsof constraints.Studying these constraints and their translations through the gauge-gravity dic-tionary can provide new constraints on quantum gravity theories dual to quantumfield theories. A strong form of gauge-gravity duality, where any quantum gravitytheory with asymptotically AdS boundary conditions is dual to some field theory,would lead us to interpret any constraints following from the axioms of quantummechanics as necessary conditions on any consistent theory of quantum gravity.Constraints on entanglement entropiesThe work in Chapter 4 concentrates on particular constraints on entanglement en-tropies following from the basic laws of quantum mechanics, which where dis-cussed in Section 1.1.2. See [35] for more details on these constraints and addi-tional references.The RT formula gives a geometric interpretation of these entanglement inequal-ities in terms of the areas of minimal surfaces. The goal is to transform theseconstraints on the areas of surfaces into more useful geometric constraints.Einstein EquationsThis approach was used to derive that the holographic duals of states near the vac-uum must obey the linearised Einstein equations near AdS [36, 37, 38].Using techniques developed by Wald and Iyer for proving the first law of black16hole thermodynamics [39, 40], an integral of the linearised Einstein tensor for aperturbation over a region bounded by a minimal surface can be related to thechange in the area and the energy associated to the asymptotic boost Killing vector.Through the gauge-gravity dictionary these are related to the entanglement en-tropy and the expectation of the modular Hamiltonian. In the case of black holethermodynamics a first law can be derived starting from the Einstein equations. Inthis case the logic can be reversed by using the first law of entanglement entropiesin (1.16) for spherical boundary regions of all sizes to derive the linearised Einsteinequations.Energy conditionsOnce we go beyond first order in perturbation theory, the field theory entanglementconstraints no longer have the form of an equality, but rather of an inequality.Inequalities that involve the Einstein tensor are referred to as energy or cur-vature conditions. Conditions of these form are usually assumed to hold for rea-sonable matter and are necessary to prove singularity theorems. Deriving suchinequalities from the tenants of quantum mechanics would put them on more solidfooting, so these provide a natural target.Indeed some progress towards relating these constraints to such energy condi-tions has been made in low spacetime dimensions [41, 35, 42].Chapter 4 shows that for 1+ 1-dimensional spacetimes which have transla-tional invariance, strong subadditivity can be related to an integrated null energycondition of the form ∫γdsTµνuµuν ≥ 0 (1.25)where γ is an arbitrary spatial geodesic and uµ is a null vector generating a light-sheet of γ defined such that translation by uµ produces an equal change in thespatial scale factor at all points.In addition, a local version of the weak energy condition in the field theorydirections of the dual geometry near the boundary follows from the positivity ofthe relative entropy. The near boundary expansion of Tµνuµuν must be positive forany timelike vector uµ with components only in the field theory directions, but not17in the holographic direction.This chapter based on my work with Nima Lashkari, Philippe Sabella-Garnierand Mark Van Raamsdonk [35].1.4.2 Expanding holographic entanglement entropies in terms offield theory one-point functionsIn classical gravitational states, boundary conditions at the conformal boundaryalong with the equations of motion determine the geometry. Using the holographicduality this geometry allows us to compute whatever we wish about the state ofa quantum theory with the boundary conditions at the conformal boundary as theonly input. These are dual to the expectation value or one-point functions of op-erators in the quantum theory and the sources or coupling constants in its action.In other words, a holographic state, that is a state which is well approximated by aclassical gravitational dual, is determined by its one-point functions. This can becontrasted to the fact that in a generic quantum state, knowledge of the expectationof an operator does not provide any information about the expectation value of thesquare of that operator.Chapter 5 explores methods to compute the entanglement entropy in an expan-sion of one-point functions of operators in the field theory using both holographicand field theory methods. In particular, we developed an iterative method to ex-press the entanglement entropy in a two-dimensional conformal field theory forstates dual to gravity with no additional matter in terms of the one-point functionof the stress tensor.In [43], it was realised that the leading order contribution to entanglement en-tropy in this expansion can be understood in terms of the propagation of a scalarfield in an auxiliary de Sitter space. We used our technique to compute the nextorder contribution and found that it could be understood by adding a simple inter-action term to this scalar field.This chapter is based on my work with Matt Beach, Jaehoon Lee and Mark VanRaamsdonk [44].181.5 OutlineThe following four chapters each contain my published work as described aboveand as detailed in the preface to this thesis. Each has its own introduction whichintroduces the concepts specific to that work and summarises the results of the restof that publication. The final chapter is a conclusion which will summarise theoriginal results contained in this thesis and discuss these results in the context ofexisting literature.19Part IEntanglement Entropy inNoncommutative Field Theories20Chapter 2Holographic EntanglementEntropies in NoncommutativeTheories2.1 IntroductionGeometric entanglement entropy as a tool to characterize physical properties ofquantum field theories has recently received a large amount of attention. One at-tractive feature of geometric entanglement entropy as an observable is that it is de-fined in the same way in any quantum field theory: it is simply the von Neumannentropy, −Tr(ρA logρA), associated with the density matrix ρA describing degreesof freedom living inside a region A. ρA arises when the portion of total Hilbertspace associated with degrees of freedom living outside of A is traced over. Uni-versality of entanglement entropy is reflected in the Ryu-Takayanagi holographicformula [6]S[A] =Vold(A˜)4G(d+2)N. (2.1)Here, we place A, a d-dimensional spacial region, on a spacelike slice of the bound-ary of the (d+2)-dimensional spacetime dual to the quantum field theory of interest.A˜ is a minimal area surface in the bulk of the holographic dual spacetime homolo-21gous to A. G(d+2)N is the (d+2)-dimensional Newton constant and the d-dimensionalvolume of A˜ is denoted with Vold(A˜). 1The Ryu-Takayanagi formula (2.1) is applicable to holographic duals wherethe dilaton and the volume of the internal sphere are both constant. However, dualsto the nonlocal theories we are interested in have neither, so the local gravitationalconstant G(d+2)N varies. Thus we must use a generalized version of formula (2.1),given by [7]S[A] =Vol(A¯)4G(10)N, with Vol(A¯) =∫d8σe−2φ√G(8)ind , (2.2)where G(10)N = 8pi6(α ′)4g2s is the (asymptotic) 10-dimensional Newton’s constantand φ is the local value of the fluctuation in dilaton field (so that the local valueof the 10-dimensional Newton’s constant is G(10)N e2φ ). Integration is now overa co-dimension two surface A¯ that wraps the compact internal manifold of theholographic dual.Because A¯ wraps the internal manifold, its boundary is the direct product ofthe boundary of A, ∂A, and the internal manifold. To obtain entanglement en-tropy, A¯ is chosen to to have minimal area (we will only work in static spacetimes).G(8)ind is the induced string frame metric on A¯. By considering the standard rela-tionship between local Newton’s constants in different dimensions: G(d+2),localN =G(10)N e2φ/V8−d , together with Vol(A¯) = V8−d VoldA˜, (2.1) can be easily recoveredfrom (2.2) for a scenario where the dilaton is a constant and the internal manifoldhas a constant volume V8−d (in string metric). The more general formula (2.2)has been used to study, for example, tachyon condensation [46] and confinement-deconfinement transition [47]. We will refer to the 8-dimensional Vol(A¯) as thearea of the minimal surface from now on.Generically, geometric entanglement entropy has a UV divergence, so it needsto be regulated with a UV cutoff. Holographically, this is accomplished the usualway by placing the region A on a surface which is removed from the boundaryof the holographic dual spacetime. Once the theory has been regulated with a1For an accessible introduction and some recent developments to holographic entropy, see forexample [8, 45].22cutoff, geometric entanglement entropy in the vacuum state can be thought to counteffective degrees of freedom inside A that have quantum correlations with degreesof freedom outside of A. In other words, it measures the the range of quantumcorrelations generated in the ground state by the interactions in the Hamiltonian.For a local theory, degrees of freedom with correlations across the boundary of Amust live near this boundary, which leads to the area law: entanglement entropyin local theories is generically proportional to the area of the boundary of A, |∂A|.While the area law has not been proven for a general interacting field theory, it isexpected to generically hold in local theories for the reason outlined above (see[18] for a review, focusing on lattice systems).In a nonlocal theory, behaviour of entanglement entropy could be expectedto deviate from the area law and this is precisely what we find using holographicmethods at strong coupling. In a simple nonlocal theory with a fixed scale of nonlo-cality aL, a dipole deformation ofN = 4 SYM, we find that entanglement entropyis extensive (proportional to the volume of A), for regions A of size up to aL. Atlength scales higher than aL, it follows an area law, with an effective number ofentangled degrees of freedom which is proportional to aL. This is consistent withall degrees of freedom within a region A of size aL or smaller, and not only thoseliving close to the boundary of A, having quantum correlations with the outside ofA due to the nonlocal nature of the Hamiltonian. In contrast, in the noncommuta-tive deformation of N = 4 SYM, which is known to exhibit UV/IR mixing andwhose nonlocality length scale grows with the UV cutoff, we find that entangle-ment entropy is extensive for all regions as long as their size is fixed as the UVcutoff is taken away to infinity.2Recent work [48] links behaviour of entanglement entropy to the ability of aquantum system to ‘scramble’ information. Whether a given physical theory is ca-pable of scrambling, and how fast it can scramble, has recently became of interestto the gravity community in the view of the so called fast scrambling conjecture[49]. It has been suggested that nonlocal theories might emulate the scramblingbehaviour of stretched black hole horizons. While the results of [48] do not applydirectly to quantum field theories, they are quite suggestive. Generally speaking,2Entanglement entropy in the noncommutative theory was studied before in [32]. Here we extendand improve on those results.23they imply that local (lattice) theories, generally exhibiting area law for entan-glement entropy at low energies, do not scramble information at these low ener-gies, while theories with volume law entanglement entropy do. As we summarizedabove, we demonstrate here, in the two nonlocal theories we consider, that entan-glement entropy follows a volume law in the vacuum state. There is no reasonwhy entanglement entropy would cease to be extensive in an excited energy state;if anything, high energy states are more likely to have extensive entanglement en-tropy than low-lying states such as the vacuum state [50, 51]. Thus, the results of[48] would suggest that our nonlocal theories are capable of scrambling informa-tion. Combined with such results as those in [52], which shows that timescales forthermalization in nonlocal theories are accelerated compared to local theories, ourwork points towards these nonlocal theories at strong coupling being fast scram-blers.Since our theories differ from N = 4 SYM in the UV, the holographic dualswe use are not asymptotically AdS spaces. Their non-asymptotically AdS geom-etry has an interesting consequence. In previously studied examples of extensivebehaviour of entanglement entropy (for example, in thermal states) this extensivebehaviour was due to the minimal surface ‘wrapping’ a surface in the IR regionof the dual, such as a black hole horizon (see for example [33]). Here, however,the extensivity arises from the fact that the minimal surfaces stays close to thecutoff surface: the volume law dependence of entanglement entropy is a UV phe-nomenon.As we were finalizing this manustript, preprint [53] appeared, which also anal-izes entanglement entropy in the noncommutative SYM and which has some over-lap with our work.The reminder of the paper is organized as follows: in Section 2.2 we reviewnonlocal theories of interest and their gravity duals, in Section 2.3 we computeholographic entanglement entropy for a strip geometry, in Section 2.4 we computeholographic entanglement entropy in the noncommutative theory for a cylindergeometry, in Section 2.4 we briefly comment on mutual information in the non-commutative theory, and in Section 2.6 we offer further discussion of our results.242.2 Theories considered and their gravity dualsWe study the strong coupling limit of two different nonlocal deformations ofN =4 SYM in 3+1 dimensions: a noncommutative deformation and a dipole deforma-tion. Both of these can be realized as the effective low energy theory on D3-braneswith a background NSNS B-field. To obtain the noncommutative deformation,both indices of the B-field must be in the worldvolume of the D3-brane, whileto obtain the dipole theory, one of the indices must be in the worldvolume of theD3-brane while the other one must be in an orthogonal (spacial) direction.Since both of these theories are UV deformations of theN = 4 SYM, deep inthe bulk their holographic duals reduce to pure AdS:ds2R2= u2(−dt2+dx2+dy2+dz2)+ du2u2+dΩ25 (2.3)with a constant dilaton:e2φ = g2s . (2.4)In our coordinates, the boundary of AdS space, corresponding to UV of the fieldtheory, is at large u. It is in that region that the holographic duals in the next twosections will deviate from the above.2.2.1 NCSYMNoncommutative Yang-Mills theory is a generalization of ordinary Yang-Mills the-ory to a noncommutative spacetime, obtained by replacing the coordinates with anoncommutative algebra. We consider a simple set up where the x and y coor-dinates are replaced by the Heisenberg algebra, for which [x,y] = iθ and whichcorresponds to a noncommutative x− y plane.One way to define this noncommutative deformation of N = 4 SYM is toreplace all multiplication in the Lagrangian with a noncommutative star product:( f ?g)(x,y) = ei2θ(∂∂ξ1∂∂ζ2− ∂∂ζ1∂∂ξ2)f (x+ξ1,y+ζ1)g(x+ξ2,y+ζ2) |ξ1=ζ1=ξ2=ζ2=0(2.5)At low energy, this corresponds to deforming ordinary SYM theory by a gaugeinvariant operator of dimension six.25The holographic dual to this noncommutative SYM theory is given by the fol-lowing bulk data [29, 30]ds2R2= u2(−dt2+ f (u)(dx2+dy2)+dz2)+ du2u2+dΩ25 ,e2φ = g2s f (u) ,Bxy = −1− f (u)θ =−R2α ′a2θu4 f (u) ,f (u) =11+(aθu)4, (2.6)where Bxy is the only nonzero component of the NS-NS form background. Notethat x,y,z have units of length, while u has units of length inverse, or energy. aθ =(λ )1/4√θ is the weak coupling length scale of noncommutativity√θ scaled bya power of the ’t Hooft coupling λ and can be thought of as the length scale ofnoncommutativity at strong coupling.In the infrared limit, u a−1θ , f (u) ≈ 1 and the holographic dual appears toapproach pure AdS space (2.3), while the UV region at large u is strongly de-formed from pure AdS, so the holographic dual is not asymptotically AdS. Let εdenote the UV cutoff and uε = ε−1 the corresponding energy cutoff. For ε  a−1θ(uε  a−1θ ), the deformed UV region of the dual spacetime is removed: noncom-mutativity has been renormalized away. However, when uε > a−1θ , the non-AdSgeometry near the boundary can influence the holographic computations of anyfield theory quantities, including those with large length scales. This opens thepossibility of UV/IR mixing, defined as sensitivity of IR quantities to the exactvalue of the UV cutoff. Noncommutative theories are known to have UV/IR mix-ing [54]. The simplest way to understand the mechanism behind the UV/IR mix-ing in noncommutative theories is to consider fields with momentum py in they-direction in (2.5): f (x,y) = e−ipfy y fˆ (x), g(x,y) = e−ipgyygˆ(x). Then f ? g(x,y) =e−i(pgy+pfy )y fˆ (x−θ pgy/2)gˆ(x+θ p fy/2): the interaction in the x-direction is nonlocalon a length scale θ py. We will see that this momentum (or energy) dependence ofthe scale of nonlocality is reflected in holographic entanglement entropy.Finally, we need to understand the geometry of the boundary. The metric onthe boundary of the gravitational spacetime (2.6) is singular since f → 0 there.26However, this is not the metric applicable to the boundary field theory, as openstring degrees of freedom see the so-called open string metric. This is the effectivemetric which enters open-string correlation functions in the presence of a NS-NSpotential B, first derived in [55]3 and given byGi j = gi j−(Bg−1B)i j , (2.7)where gi j is the closed string metric. Substituting our holographic data at a fixedu, we obtain the open string metric, Gi j = R2u2(δi j). Removing an AdS conformalfactor, we see that the boundary field theory lives on a space with a conformallyinvariant metric ds2 = −dt2 + dx2 + dy2 + dz2. This is the metric we will use tocompute distances on the field theory side of the holographic correspondence.2.2.2 Dipole theoryAnother theory we will consider is the simplest dipole deformation ofN = 4 SYM[57, 58, 59]. A dipole theory is one in which multiplication has been replaced bythe following noncommutative product:( f ?˜g)(~x) = f(~x−~L f2)g(~x+~Lg2), (2.8)where~L f and~Lg are the dipole vectors assigned to fields f and g respectively. Atlow energy, this corresponds to a deformation by a vector operator of dimension 5.To retain associativity of the new product, we must assign a dipole vector~L f +~Lgto f ?˜g. A simple way to achieve it is to associate with each field f a globallyconserved charge Q f and to let ~L f =~LQ f . This can also be easily extended tomultiple globally conserved charges. We will take ~L = Lxˆ for some fixed lengthscale L, so that our theory is nonlocal only in the x-direction. As we saw in theprevious section, noncommutative theory can be thought of as a dipole theory withthe charges being momenta in a direction transverse to the dipole direction.43For an interpretation of the open string metric in the context of the AdS-CFT duality, see forexample [56].4This is not entirely accurate, as a field with transverse momentum p induces a dipole momentθ p in all the fields it interacts with instead of in itself, but this detail will not be relevant to our27Dipole SYM is a simpler nonlocal theory than the NCSYM. Since the scale ofthe noncommutativity is fixed, the theory does not exhibit UV/IR mixing. We willsee a clear signature of that in the entanglement entropy.The holographic dual to a dipole deformation of N = 4 SYM theory wherethe scalar and fermion fields in N = 4 SYM are assigned dipole lengths basedon global R-symmetry charges was found, using Melvin twists, in [25]. For thesimplest case, where supersymmetry is broken completely and where all the scalarfields have the same dipole lengths, the holographic dual is given by the followingbulk data:ds2R2= u2(−dt2+ f (u)(dx2)+dy2+dz2)+ du2u2+metric on a deformed S5 ,e2φ = g2s f (u) ,Bxψ = −1− f (u)L˜ =−R2α ′aLu2 f (u) , (2.9)f (u) =11+(aLu)2.Similar to aθ , aL = λ 1/2L˜, L˜ = L/(2pi) is the length scale of nonlocality at strongcoupling. The usual S5 of the gravity dual to a 3+1-dimensional theory is deformedin the following way: Express S5 as S1 fibration over CP2 (the Hopf fibration).Then the radius of the fiber acquires a u-dependent factor and is given by R f (u).The volume of the CP2 is constant and given by R4pi2/2. Thus the compact man-ifold at radial position u has a volume given by R5pi3 f (u). ψ is the global angular1-form of the Hopf fibration. For details, see [25].As we did for the noncommutative theory in the previous section, we need tounderstand what metric to use for distances in the boundary dipole theory. Unfor-tunately, it does not seem possible to give an argument similar to the one in [55]to find an ‘open string metric’ for the D-branes whose low-energy theory gives usthe dipole theory, since (in contrast to the noncommutative case) the dipole the-ory requires a nonzero NSNS field H and not just the nonzero potential B.5 Thereasoning.5A constant potential B which has only one of its indices in the worldvolume of a D-brane canbe gauged away completely. It is therefore important that the other index is in a direction of a circlewith varying radius, resulting in a nonzero H. In the holographic dual we consider, this circle is the28essence of the argument in [55] is that the only effect of the potential B is to changethe boundary conditions for open string worldsheet theory. Thus, the boundary-boundary correlator is modified in a simple way that is equivalent to modifying themetric. To understand the open string metric for the dipole set up we need a differ-ent way to make the NSNS field B ‘disappear’. We can accomplish this following[25] and using T-duality.First, let’s see what happens when we compactify the x direction in (2.6) on acircle of radius Rx and T-dualize using the prescription in [60]. The T-dual metricis(Ru)2(−dt2+(dy− (θ/α ′)dx˜)2+dz2)+ 1(Ru)2(dx˜)2+du2u2+dΩ25 , (2.10)where x˜ is the T-dual coordinate to x. In the T-dual, B is zero. It has been traded forthe twist around the x˜ circle: we identify (x˜,y) with (x˜+2piR˜x,y+2piR˜xθ), RxR˜x ∼α ′. Conformal invariance in the t− y− z directions has been restored by T-duality,and we recover the open string metric (2.7) in those directions.6 At the same time,the twist encodes the nonlocal structure of the theory. To see this recall that in thenoncommutative theory, fields with momentum px in the x-direction appear to havedipole lengths θ px. Taking x on a circle of radius Rx, p = n/Rx, with n an integer.When T-dualized, the corresponding open string mode has winding number n inthe x˜ direction. Given the twist, the ends of this open string are separated by ∆y =2piR˜x(θ/α ′)n. Substituting n=Rx p we get ∆y∼ θ p: the twist reproduces nonlocalbehaviour of the noncommutative theory when the distances are measured in theconformally invariant (or open string) metric.Returning to the dipole theory, we perform T-duality in the direction of theHopf fiber to obtain(Ru)2(−dt2+(dx− L˜dψ˜)2+dy2+dz2)+ (α ′)2R2(dψ˜)2+du2u2+d(CP2) . (2.11)Again, the NSNS potential Bψx has been replaced by a twist. However, due to theHopf fiber.6This is not a coincidence; the equation for the T-dual metric [60] and the equation for the openstring metric (2.7) are functionally the same.29twist of the Hopf fibration, in the T-dual there is a new NSNS potential compo-nent, Bxb where b lies in the direction of the CP2, resulting in a nontrivial NSNSfield Hxbu. Since ψ was a Dirichlet direction before T-duality, the interpretation isslightly different than it was in the noncommutative case. After T-duality, we havea twisted compactification identifying (ψ˜,x) with (ψ˜ + 2pi,x+ 2piL˜). The properdistance between (ψ˜,x) and (ψ˜,x+2piL˜) is therefore α ′/R, which is small on theboundary in the large u limit. This is a sign of the nonlocality at the dipole lengthL = 2piL˜. More relevant to us at this point is that, just like for the noncommutativetheory, conformal invariance in the t− x− y− z direction has been restored in theT-dual metric. It seems reasonable then to use the metric −dt2 + dx2 + dy2 + dz2to compute distances on in the boundary theory. For more details about this argu-ment, as well as a string worldsheet argument about the origin of dipole theories,see [25, 61].2.3 Entanglement entropy for the stripWe will start by studying entanglement entropy for degrees of freedom living onan infinitely long strip7 −l/2 < x < l/2, −W/2 < y,z < W/2, W → ∞. In thisgeometry, entanglement entropy follows the area law if it is independent of the stripwidth l. As we discussed in the Introduction, the relevant minimal surface is eight-dimensional; it wraps the compact (possibly deformed) sphere of the gravity dualand is homologous to the strip on the boundary in the non-compact dimensions. Itsarea is given byVol(A¯) = pi3R8W 2∫ l/2−l/2dx (u(x))3√1+(u′(x))2f (u)(u(x))4, (2.12)where function u(x) specifies the embedding of the bulk minimal area surface. Theabove formula for the area in terms of u(x), with the appropriate form for f (u),is applicable to all bulk metrics we are interested in: while the noncommutativetheory dual has more directions warped by a factor f (u) than the dipole one, in thedipole theory there is another factor of f (u) accounting for the deformation of the7 In dimensions three and higher it would be perhaps more accurate to call this region a ‘slab’rather than a ‘strip’; nevertheless, we will use established terminology.30sphere on which the entangling surface is wrapped.Following previous work, we can think of the problem of finding u(x) corre-sponding to a minimal area surface as a dynamics problem in one dimension: xplays the role of time, u(x) is the position and u′(x) the velocity. Since the La-grangianL (u,u′) = u3√1+(u′)2f (u)u4(2.13)does not depend explicitly on the ‘time’ x, there is a conserved Hamiltonian,H = u′∂L (u,u′)∂u′−L (u,u′) =− u3√1+ (u′)2f (u)u4. (2.14)The Hamiltonian H is equal to −u3∗, where u∗ is the smallest value of u(x) on theentangling surface. This point of deepest penetration of the minimal surface intothe bulk occurs at x = 0 by symmetry. u′(x) vanishes there.To implement the UV cutoff, the differential equation in (2.14) is to be solvedwith a boundary conditionu(x =±l/2) = uε = 1ε . (2.15)For some functions f (u), (2.14) can be integrated explicitly. The answer isa hypergeometric function for f (u) = 1 or f (u) = 1/(aθu)4, and an elementaryfunction for f (u)= 1/(aLu)2. For f (u)= 1/(1+(aLu)2) or f (u)= 1/(1+(aθu)4),(2.14) can only be studied using series expansions in different limits.To compute the area of the minimal surface, it is useful to solve (2.14) for u′(x)as a function of u and substitute the result into (2.12). We obtainVol(A¯) = 2pi3R8W 2∫ uεu∗duu′u6(−H) = 2pi3R8W 2∫ uεu∗duu4u3∗√u6∗f (u)(u6−u6∗).(2.16)To obtain the area of the minimal surface in terms of l from this equation, given uε ,it is necessary to find the relationship between u∗ and l.312.3.1 Review of results for AdS spaceFor pure AdS, with f (u) = 1, we can remove the boundary of AdS all the wayto infinity, uε → ∞. Then, by integrating (2.14), we obtain a simple relationshipbetween u∗ and the width of the strip l:lu∗ =Γ(2/3)Γ(5/6)√pi≈ 0.8624 . (2.17)This relationship has a nice interpretation: holographic entanglement entropy for astructure of size l is given by the minimal surface that probes AdS bulk from theUV cutoff down to energy scales of order l−1. Modes with wavelength longer thanl do not contribute to the entanglement entropy.To compute the leading order (for uε→∞) behaviour of the area of the minimalsurface, we can we can use (2.16). Since u∗ depends only on l and not on uε (i.e.,it remains finite in the uε → ∞ limit), the leading contribution to the area comesfrom large values of u. We can thus approximateVol(A¯) = 2pi3R8W 2∫ uεduu = pi3R8W 2ε2. (2.18)A more precise result for the entanglement entropy density is obtained from a next-to-leading order computation. It gives a universal term which is finite and indepen-dent of the cutoff:SW 2=R34G(5)N[1ε2−(2Γ(23)Γ(56))3pi3/21l2+ (terms that vanish for ε → 0)].(2.19)In terms of field theory variables, we haveR34G(5)N=N22pi, (2.20)so that the divergent part of the entanglement entropy is proportional to N2ε−2,with a numerical coefficient which is specific to strongly coupled N = 4 SYM.The entanglement entropy is therefore of this generic form (applicable to 3+1 di-32mensions):S = Neff|∂A|ε2= NeffW 2ε2(2.21)with the number of effective on-shell degrees of freedom Neff proportional to N2.Formula (2.21) supports the following simple picture of entanglement entropyin theory with a local UV fixed point: A quantum field theory in 3+1 dimensionswith a UV cutoff ε−1 can be thought of as having on the order of one degree offreedom per cell of volume ε3. The divergent part of the geometric entanglemententropy computed the vacuum state of such a theory is a measure of the effectivenumber of degrees of freedom inside of a region A that have quantum correlationswith degrees of freedom outside of A. In a local theory, only ‘adjacent’ degreesof freedom are coupled via the Hamiltonian and the simple intuition is that there-fore quantum correlations between degrees of freedom inside of A and outside ofA happen only across the boundary ∂A. Thus, the dominant part of the entangle-ment entropy comes from degrees of freedom which live within a distance ε of theboundary of A, with entanglement entropy proportional to the volume of this ‘skin’region, equal to ε|∂A|. Dividing this volume by the volume of one cell, ε3, gives(2.21).2.3.2 Dipole theoryHaving briefly reviewed holographic entanglement entropy on a strip in unde-formed SYM, we will now study it in the dipole theory.In Figure 2.1, we show the relationship between l and u∗ for the dipole theory.We see that it approaches the AdS result (2.17) for large strip widths l and that itdeviates significantly from it for strips whose width is on the order of and smallerthan aL. For narrow strips, the entangling surface does not penetrate the bulk verydeeply into the IR region. To study these, we assume that u∗  a−1L and writef (u) ≈ (aLu)−2. Here we get a pleasant surprise: the exact shape of the minimalsurface can be obtained in terms of elementary functionsu(x) =u∗cos(3x/aL)1/3for x/aL ∈ [−pi/6,pi/6] . (2.22)The relationship between the penetration depth of the minimal surface and the33Figure 2.1: Point of deepest penetration u∗ as a function of the strip width lfor a minimal area surface in the gravity dual to the dipole theory (solidred line). The blue dotted line shows the result for pure AdS, given by(2.17), while the black dashed line shows the narrow strip approxima-tion, (2.23). In this figure, aLuε = 30.width of the strip isu∗ = uε (cos(3l/2aL))1/3 . (2.23)This equation is valid as long as u∗ a−1L , which, in the limit where uε is large, istrue for all strip widths l up to l = (pi/3)aL. Notice that, in contrast to pure AdS,the point of deepest penetration u∗ depends on the UV cutoff. Thus, if one worksat the limit of infinite cutoff, these minimal area surfaces will be missed.The area of the minimal surface under the approximation f (u)≈ (aLu)−2 isVol(A¯) = pi3R8[W 2aLε32sin(3l/2aL)3]≈ pi3R8 W2lε3, (2.24)where the final approximation is for a small strip width l aL. For narrow strips,entanglement entropy is extensive, proportional to the width of the strip. The firstpart of (2.24) gives the corrections to the volume scaling, controlled by the powers34Figure 2.2: The area of the minimal surface as a function of the strip widthl for for the dipole theory (solid red line). The blue dotted line showsresult (2.24), valid for narrow strips l < (pi/3)aL. In this figure, aLuε =30.of the ratio l/aL.For surfaces with large l (compared to aL), we can use the same approximationas in (2.18), with f (u)≈ (aLu)−2:Vol(A¯) = 2pi3R8W 2aL∫ uεduu2 = pi3R82W 2aL3ε3(2.25)We see that this area, which is independent of the width, is the same as thearea obtained from (2.24) at the extremal value of l, l = aLpi/3. The situation isillustrated in Figure 2.2.To summarize, we obtained the following result for the entanglement entropydensity in the dipole theory:SW 2=N22pi2aL3ε3G(l/aL) , where G(z) = sin(3z/2) for z < pi/3 ,1 for z > pi/3 . (2.26)35Entanglement entropy is extensive for very narrow strips, depends on the width ofthe strip in a nonlinear fashion for widths up to the nonlocality scale and smoothlygoes over to a non-extensive (area law) behaviour for wide strips. For wide strips,while the entanglement entropy follows an area law, it has a different form thanit would for a a generic local theory (given by (2.21)). To explain this, applyreasoning similar to that below (2.21) to a theory with a fixed scale of nonlocalityaL. By definition, the Hamiltonian of such a theory couples together degrees offreedom as far apart at aL, thus, for a large region, the dominant part of geometricentanglement entropy should be proportional to the volume of a set of points nomore than aL away from the boundary of A. This volume, for a large enoughregion, can be approximated by aL|∂A|, leading to S∝ aL|∂A|/ε3, which is exactlywhat we see in (2.26) for a strip with l > (pi/3)aL.Applying our reasoning to the narrow strip, we see that, for l < aL, all degreesof freedom inside the strip should are directly interacting with, and therefore entan-gled with, degrees of freedom outside of the strip. For a very narrow strip, degreesof freedom inside it will mostly be entangled with the degrees of freedom outside,and entanglement entropy should be proportional to l, which is exactly what wesee. As the strip gets wider, some of the degrees of freedom inside the strip be-come entangled with each other, decreasing the entanglement with the outside andimplying a sub-linear growth to the entanglement entropy as a function of l, againin agreement with (2.26).The exact way in which S deviates from S ∝ l can be viewed as a way to probethe distribution of quantum correlations in the ground state of this nonlocal theory.It would be interesting to consider this further.Finally, notice that above the nonlocality length scale aL, the shape of the min-imal surface is not greatly affected by the exact value of the cutoff; this is a signthat the dipole theory does not have UV/IR mixing. We will see a very differentbehaviour for the noncommutative theory.2.3.3 NCSYMFor entanglement entropy of a strip in the noncommutative theory, the situation ismore complicated. As is shown in Figure 2.3, there are as many as three extremal36Figure 2.3: Point of deepest penetration u∗ as a function of the strip widthl for extremal area surfaces in the gravity dual to the noncommutativetheory (solid red line). The blue dotted line shows the result for pureAdS, given by (2.17), while the black dashed line shows the result of(2.27). In this figure, aθuε = 30.area surfaces for a given width l of the strip. At large strip widths there is onlyone surface, for which the relationship between l and u∗ approaches that of pureAdS, given by (2.17). At small widths, similarly to the dipole theory, there is asurface which stays close to the cutoff surface.8 At intermediate l, there are threeextremal surfaces, whose shape is shown in Figure 2.4. As we will see, the middleof the three surfaces is always unphysical (its area is never smaller than the othertwo). As the width is increased from zero, at some critical width lc there is a phasetransition as the area of the surface on the top-most branch becomes larger than thearea of the surface on the bottom-most branch in Figure 2.3.We start by studying top-most branch, which contains surfaces anchored on8In [32], this surface was approximated by one placed exactly at the cutoff, at constant u.37Figure 2.4: Shape of three extremal area surfaces, given as u(x), all anchoredon the same boundary strip.narrow strips. To study these, we find u(x) as a series expansion for small x. Thisallows us to write the relationship between l, u∗ and uε for small l:uε −u∗ = 38u3∗1+(aθu∗)4l2+O((l/aθ )4) . (2.27)The integral in (2.12) can also be expanded and evaluated for small l. Finally,substituting u∗ from the expression above into the area integral, we can obtain thearea for small l:Vol(A¯) = pi3R8W 2[lε3− 38l3a4θε(1+(ε/aθ )4)+O((l/aθ )4)]. (2.28)We have kept the sub-leading terms in ε for completeness — expression (2.28), asgiven, is correct even for large ε as long as l is small.From (2.27) we see that as we increase uε keeping l fixed, (uε − u∗) ∝ l2/u∗,so that uε − u∗ approaches zero: the minimal surface approaches the boundarysurface.38This result turns out to hold even for large (but fixed) strip width l in the largeuε limit. In this limit, we approximate f (u) ≈ (aθu)4. This allows us to obtain land the area as a function of u∗ and uε in terms of hypergeometric functions. Wesee that l/uε is a function of the ratio u∗/uε only. As uε approaches infinity withl fixed, this ratio goes to 1, showing that the entire minimal surface stays close tothe boundary and that our approximation f (u)≈ (aθu)4 is self-consistent even forlarge l, as long as l is held fixed when the UV cutoff is removed. The followingrelationship holds under this approximation:∫ l/2−l/2dxL =lu3∗+uε√u6ε −u6∗4. (2.29)Thus, the leading order UV divergence of the area of the minimal surface at anyfixed width l isVol(A¯) = pi3R8W 2lε3. (2.30)Having understood the top-most branch of the plot in Figure 2.3 , correspond-ing to surfaces that stay close to the boundary, we now move to the bottom-mostone. These surfaces penetrate deeply into the bulk and their shape is not affectedby the cutoff point. We can therefore use the same method as before for obtainingtheir area:Vol(A¯) = 2pi3R8W 2a2θ∫ uεu3du = pi3R8W 2a2θ2ε4. (2.31)Since there are multiple extremal surfaces anchored on a strip, we need to findout which of them have the smallest area at a given l. At very small l there is onlyone surface (see Figure 2.3), thus, by continuity, for l less than some critical lengthlc, the surface of the smallest area corresponds to the top-most branch of Figure2.3. Its area is given by (2.30). At lc there is a first order phase transition.9 Abovelc, the surface with the smallest area is on the bottom-most branch of Figure 2.3and its area is given by (2.31). To compute lc, we set (2.30) and (2.31) equal andobtain that lc = a2θuε/2.Since the critical length increases with uε , if we hold l fixed and take the limit9This is similar to [62] and to [47], as well as to the recent paper [63]. Entanglement entropy iscontinuous across the transition, but its derivative is not.39Figure 2.5: Area of the minimal surface as a function of strip width l fornoncommutative theory. Top: Plots with aθuε = 10, 30, 50 and 70are shown. Area is scaled by a power of the cutoff to allow functionsfor different cutoffs to be plotted on the same set of axis. Dashed linecorresponds to the leading term in (2.28), Vol(A¯)/u3ε ∝ l. The rangeof validity of this approximate expression increases with increasing uε .Bottom: Detail of the fish-tail phase transition is shown. The greendotted line and the blue dashed line correspond to (2.31) and (2.28)respectively. aθuε = 30.uε → ∞, lc will diverge to infinity as well and (2.30) will hold for any l.Our analysis implies that in the limit ε → 0, the entanglement entropy densityfor a strip of a fixed length l isSW 2=N2pi[lε3− 38l3a4θε+ terms vanishing for ε → 0], (2.32)which, to the leading order, is the same answer as for the dipole theory in thenarrow strip limit ( (2.26), l aL).To understand the physics behind this result, we recall that in the noncommu-tative theory a mode with momentum py in the y-direction can be thought of asa dipole field with an effective dipole length θ py in the x-direction. The high-momentum modes which dominate the divergent part of entanglement entropy allhave large effective dipole moments. Therefore the entanglement entropy is that ofa nonlocal theory with a large effective scale of nonlocality. This is precisely what40we see.In the complementary limit, fixing a (large) UV cutoff first and then consider-ing wide strips, l > lc, (2.31) shows that entanglement entropy density is equal toSW 2=N22pia2θ2ε4. (2.33)We see that the area law applies and the number of degrees of freedom whichare near enough to the boundary of the region to be entangled with the outsideis proportional to a2θ/ε2. This is equal to the scale of noncommutativity at theUV cutoff (a2θuε= a2θ/ε) divided by the cutoff length scale ε , consistent with ourprevious discussions.In the next section, we will compute the entanglement entropy in the noncom-mutative theory for another geometry: a cylinder whose circular cross-section is inthe two noncommutative directions x and y and which is extended infinitely in thecommutative direction z. We will obtain a result for the entanglement entropy thatis similar to the one in this section, while the geometry of the entangling surfaceswill be very different.2.4 Entanglement entropy for the cylinder in NCSYMConsider a region on the boundary extended in the z direction (−W/2 < z <W/2,W → ∞) and satisfying x2 + y2 < l2 in the x and y directions. The area functionalfor a surface homologous to this cylindrical region, assuming rotational symmetryin the x− y plane and translational symmetry in the z direction, isVol(A¯) = 2pi4R8W∫ l0dr r (u(r))3√1+(u′(r))2f (u)(u(r))4, (2.34)where r =√x2+ y2 and the surface is specified by a function u(r).Since r appears explicitly in the Lagrangian density, the equation of motioncannot be integrated explicitly even once. We will therefore have to rely more onnumerical computation.Figure 2.6 shows shapes of extremal surfaces anchored on a disk in the bound-ary noncommutative theory. As is easy to check analytically, all these surfaces41Figure 2.6: Extremal surfaces homologous to a cylinder in NCSYM, pre-sented as u(r). On the left, the straight dashed line is the asymptoticbehaviour given by aθu =√3r/aθ . On the right, surfaces with l suffi-ciently smaller or larger than lc = a2θuε/√3 to reach the cutoff beforethey had time to approach the this asymptote are shown as well.asymptote at large r and u to a single ‘cone’ given by aθu =√3r/aθ . A linearanalysis about this asymptotic solution givesaθu(r)≈√3r/aθ + t cos(√72ln(r/aθ )+ϕ), (2.35)where t and ϕ are free parameters, with t small. In principle, a relationship betweent and ϕ could be derived given that u′(0) = 0, but it cannot be obtained withinperturbation theory. It is interesting and perhaps surprising that the fluctuationsabout the asymptote are oscillatory in r. This behaviour, which can be seen inFigure 2.6, agrees very well with more detailed numerical analysis.From Figure 2.6 we see that surfaces with u∗ relatively close to a−1θ approachthe asymptote u =√3r/a2θ before reaching the cutoff, while those with large u∗(aθu∗ 1) or small u∗ (aθu∗ 1) do not. At a fixed cutoff, then, we have threeclasses of surfaces: shallowly probing surfaces, aθu∗  1, with l smaller thanand bounded away from lc := a2θuε/√3, deeply probing surfaces, aθu∗ 1, with llarger than than and bounded away from lc and the intermediate category, for whichl is approximately equal to lc. In the first and second category, there is a unique42extremal surface at a given radius l, while for radii close to lc the situation is morecomplicated, due to the oscillatory nature of the near-asymptotic solutions shownin (2.35). Since the cutoff radius lc increases with uε (similar to the behaviour inthe strip geometry), the entanglement surface for a region with any radius l belongsto the first category for a sufficiently high UV cutoff.First, let us consider the surfaces with small l/aθ . These can be studied byexpanding in l/aθ . We get the following two results:uε −u∗ = 34u3∗1+(aθu∗)4l2+O((l/aθ )4) , (2.36)Vol(A¯) = 2pi4R8W[l22ε3− 932l4a4θε(1+(ε/aθ )4)+O((l/aθ )4)]. (2.37)The l/aθ expansion for the area of the minimal surface has a structure which is sim-ilar to the one we obtained for the strip in the noncommutative theory: organizingthe expansion in powers of l, the term of order ln has as its leading ε dependence1/ε5−n (with n even). Assuming that this analytic structure is valid for finite l/aθ ,we obtain that in the limit ε → 0, the entanglement entropy density for a cylinderat a fixed radius l isSW=N2pi[pil2ε3− 932l4a4θε+ terms vanishing for ε → 0]. (2.38)Qualitatively, this is the same answer as we obtained for the strip: entanglemententropy is extensive, proportional to the volume of the considered region. Noticethat neither expression has a non-zero universal (independent of ε part).At finite (and large) cutoff, we can consider large radius cylinders. For l suf-ficiently larger than lc we see from Figure 2.6 that u∗aθ  1 and the entanglingsurface seems close in shape to that in pure AdS (as it approaches the boundary atapproximately the right angle, based on numerical evidence). Thus, u∗ ∝ l−1 and43Figure 2.7: Point of deepest penetration u∗ as a function of the cylinder’s ra-dius l for the minimal surface homologous to a cylinder in the noncom-mutative theory. The black dashed line corresponds to (2.36). Linearscale on the left, log-log scale on the right; aθuε = 30 for both plots.the area is approximatelyVol(A¯) = 2pi4R8Wa2θ∫dr r u3 u′(r) = 2pi4R8Wa2θ l∫ uεduu3 = pi3R82pilWa2θ4ε4,(2.39)where we have used f (u) ≈ (aθu)−4 and approximated r ≈ l in the region nearthe boundary. Resulting entanglement entropy has the same interpretation as theone in (2.33), with the area of the strip’s boundary, W 2 replaced by the area of theboundary of the cylinder, 2pilW .Having understood the minimal surface in the large l and small l limits, we nowturn to l near the cutoff radius lc = a2θuε/√3, which corresponds to u∗aθ close to1. Figure 2.7 shows the dependence of u∗ on l over the entire range for a finitecutoff. We notice that near lc, there are multiple values of u∗ at a given l: just likein the case of the strip, there is a range of radii l for which there exist multipleextremal surfaces anchored on the same cylinder. This is related the oscillatingnature of the asymptotic solution (2.35). Since taking a large cutoff limit removesthe radius lc, at which phase transition take place, to infinity, we will not attempta detailed study of the properties of the phase transition, which is complicated bythe oscillatory nature of the minimal surfaces near the critical radius.44Figure 2.8: Area of the minimal surface homologous to a cylinder, as a func-tion of the cylinder’s radius l, with both axis shown in logarithmic scale.aθuε = 30. The green dotted line and the blue dashed line correspondto (2.39) and (2.37) respectively.It is interesting to notice that, apart from the details of the phase transition, theentanglement entropy for the cylinder has the same qualitative behaviour as it doesfor the strip, ever though the geometry of the minimal surfaces is very different.2.5 Mutual information in NCSYMTo strengthen our discussion of UV/IR mixing in noncommutative SYM theory, itwould be interesting to study the behaviour of an observable that (in the commu-tative theory) is finite in the large UV cutoff limit. One such observable is mutualinformation.Consider two disjoint regions A and B. Mutual information is defined byI(A,B) := S(A)+ S(B)− S(A∪B). Subadditivity implies that mutual informationis a non-negative quantity. For local theories, holographic mutual information isUV finite, since contributions from the near-boundary parts of the minimal sur-faces cancels. It is known to exhibit a phase transition [64]: if the regions A and45B have width l and the distance between them is x, I(A,B) is nonzero for x lessthan some xc and zero for x greater than xc, with xc/l of order 1. The origin of thisphase transition is shown in Figure 1.2: for large x/l, the minimal area surface hasthe the topology shown on the right of Figure 1.2, while for small x/l, it has thetopology shown on the left of Figure 1.2. Behaviour of mutual information and theexistence or disappearance of this phase transition can be used to find characteristiclength scales, see for example [65] and [66]. For NCSYM we find that the mutualinformation phase transition is absent for length scales less than lc. The fact that lcdepends on the UV cutoff is then a signature of the UV/IR mixing.To study the details of this signature, let regions A and B be strips of widthlA and lB respectively, separated by a distance x. Then, if lA, lB and x are heldfixed as the cutoff uε is taken to infinity, entanglement entropies associated withstrips of width x, lA, lB and lA+ lB+ x are all extensive. Therefore, S(lA)+S(lB)<S(lA + lB + x)+ S(x), i.e. the surface on the right in Figure 1.2 has a smaller areathan that on the left in Figure 1.2. This implies that I(A,B) = 0 for any x andthere is no phase transition. On the other hand, if lA and lB are both larger thanlc, then S(lA) ≈ S(lB) ≈ S(lA + lB + x) because to leading order the entanglemententropies do not depend on the width of the strip. Mutual information is positive(and divergent, since entanglement entropy in the noncommutative theory doesnot have a UV-finite piece) as long as x is small enough and undergoes a phasetransition as x is increased just like it does for a local field theory.It would be interesting to study the behaviour of mutual information near thephase transition in detail. We leave this to future work.2.6 Final commentsA key ingredient in our analysis was keeping the cutoff finite, if large. Only whenthe entangling region A is placed on a cutoff surface at finite u = uε can the correctminimal area minimal surfaces be found. This is especially true in the noncommu-tative theory, where UV/IR mixing implies that infrared physics is affected by theprecise value of the cutoff.We have already discussed the origins of the dependence of the entanglemententropy on the size (volume or area) of the region A, on the cutoff length ε and on46the intrinsic length scales aL and aθ built into our nonlocal theories. The numericalcoefficients we obtain are of physical significance: In the volume law regime, thecoefficient measures whether degrees of freedom inside of A are are entangledwith the outside of A or with each other. Therefore, this coefficient controls themaximum size of the region over which the theory thermalizes [48]. A similarstatement can be made about the coefficient in the area law regime.While the open string metric gives distances in the nonlocal boundary fieldtheory, it is the closed string metric that determines the causal structure of thetheory. In a local field theory, knowledge of the density matrix ρA in the region A isenough to compute all observables within the domain of dependence of A. Whilewe don’t know exactly which portion of the total holographic dual spacetime isdual to ρA itself [67, 68, 69, 70], the answer must involve the bulk (closed string)metric and its causal structure. Applying this argument to our nonlocal theories,we see that it is the bulk metric that determines the extent of the holographic dualto the density matrix ρA. For example, this holographic dual might be boundedby the minimal surface. Then, the intersection between the AdS boundary andthe lightsheets originating from the minimal surface might be interpreted as theboundary of the “domain of dependence” of the region A in a nonlocal theory.We would expect that knowledge of the density matrix ρA would be sufficient todetermine all observables within this “domain of dependence”. This new “domainof dependence” is determined causally not by the open string metric but by the bulkclosed string metric at a fixed cutoff. This closed string metric is not isotropic,in fact, it has a very large “speed of light” in the nonlocal directions, comparedwith the open string metric. Field theory computations show that nonlocal fieldtheories have large propagation speeds , see for example the behaviour discussed in[54], or the observations that the propagation speed in the noncommutative theoryis effectively infinite [71, 72]. As a result, in a nonlocal theory the “domain ofdependence” should have a very small time-like extent. This is consistent withit being bound by lightsheets which originate on a minimal surface which doespenetrate the bulk very far, a feature we have observed.A related feature of our minimal surfaces is that they are not necessarily or-thogonal to the boundary at a finite cutoff. Therefore, for example, the two pro-posals given in [9] for a covariant version of holographic entanglement entropy are47not necessarily equivalent, raising an interesting question about time-dependentnonlocal theories. Similarly, arguments for strong subadditivity of covariant holo-graphic entanglement entropy in time dependent spacetimes, in [73], do not applyeither (however, the simple argument for static spacetimes, in [74], does apply,and therefore the entanglement entropies computed in this paper do satisfy strongsubadditivity).Since our computations were done using holography, they are reliable in thestrong coupling limit. It would be interesting to see whether the same results ap-ply at weak coupling, with the appropriate nonlocal scale, aθ or aL, replaced by itsweak coupling counterpart,√θ or L respectively. This might not necessarily be thecase: for example, the enhancement to the rate of dissipation provided by noncom-mutativity at strong coupling is not seen at weak coupling [52]. The analysis in [52]points towards strong coupling being necessary for scrambling in noncommutativetheory, and, if the results in [48] can be extended to this situation, strong couplingbeing necessary for extensive entanglement entropy. It would be interesting to set-tle this question by a direct computation of geometric entanglement entropy in aweakly coupled noncommutative theory. Unfortunatelly, it will not be possible tolearn anything from free noncommutative theories as these are equivalent to theircommutative counterparts.A simple example of a nonlocal field theory with volume scaling of its en-tanglement entropy was given in [75]. In that work, it was proposed that volumescaling was a necessary feature of entanglement entropy in a hypothetical field the-ory dual to flat space. In contrast to this hypothetical theory, our nonlocal theoriesdo not have vanishing correlation functions.Finally, it would be interesting to study other extremal surfaces in holographicduals to nonlocal theories, following the work for local theories [76], as well as toextend our results to finite temperature.48Chapter 3Perturbative EntanglementEntropies in NoncommutativeTheories3.1 IntroductionLocal field theories generally exhibit what is know as an area law behaviour, wherethe leading divergence in the entanglement entropy of a spatial region is propor-tional to the area of the boundary of that region. That is, S ∼ |∂A|Λd−2, where Sis the entanglement entropy, |∂A| the area of the boundary of the region and Λ isthe momentum scale of the UV regulator of the theory, for example the inverse ofa lattice spacing.1 However, recent holographic studies of strongly coupled non-local theories have found a volume law behaviour instead [32, 75, 53, 31, 77].That is, for a nonlocality scale l, S ∼ |A|Λd−1 for regions much smaller than l andS∼ l|∂A|Λd−1 for regions much larger than l [31], as discussed in Chapter 2. Notethat entanglement entropy of large regions is sufficient to differentiate this type ofvolume law from an area law, as the entanglement entropy is proportional to thelength scale of the nonlocality times an additional factor of the UV regulator. To1See for example [18] for a review of area laws in entanglement entropy.49summarise,area law : S∼ |∂A|Λd−2, (3.1)volume law : S∼ |A|Λd−1, (small regions) (3.2)S∼ l|∂A|Λd−1. (large regions) (3.3)These results can be understood intuitively by assuming that all the degrees offreedom within the range of the nonlocality are equally entangled with each other.Then, for regions much smaller than l, all the degrees of freedom inside the region,not only those near the boundary, are entangled with degrees of freedom outside.For regions much larger than l, all the degrees of freedom within a distance l of theboundary are entangled with those outside. In both cases, the number of degreesof freedom strongly entangled across the boundary is proportional to Λd−1 ratherthan the Λd−2 expected from an area law.A natural question is whether this behaviour is generic to nonlocal theoriesor if it is confined to a strongly coupled, large N regime. One approach is tostudy entanglement entropy for a free scalar field on the fuzzy sphere [78, 79,80, 81]. This turns out to be proportional to the area2 for small polar caps [80, 81].However, two issues arise which question whether this should be characterised as avolume law. First, the dependence of the entanglement entropy on the UV regulatordoes not match the volume law described above. Second, the entanglement entropydoes not scale like the number of degrees of freedom contained in the polar cap,as the degrees of freedom are not uniformly distributed across the sphere. Insteadit scales as the number of degrees of freedom near the boundary [78, 79]. Anotherlimitation of this theory is that the nonlocality scale is tied to the size of the sphereso it is not possible to study regions much larger than the nonlocality scale.Another approach is to study a free field theory on a lattice with a nonlocalkinetic term, in which case a volume law was found [82].This paper investigates the role of interactions in this question by consideringtwo theories with nonlocal interactions: scalar λφ 4 theory on the noncommutativeplane and λφ 4 theory with a dipole type nonlocal modification with fixed nonlo-2The fuzzy sphere is a two-dimensional surface, thus |A| is an area and |∂A| is a circumference.50cality scale. The leading divergence in entanglement entropy of large regions iscalculated to leading order in perturbation theory and is not found to be propor-tional to the length scale of the nonlocality, hence no evidence of a volume law isfound. Instead, the leading divergence in both theories has the same form as thestandard local λφ 4 theory which follows an area law. This result indicates that, per-turbatively these nonlocal interactions are not generating sufficient entanglement atdistances of the nonlocality scale to change the leading divergence, at least to firstorder in the coupling.The free theory with λ = 0 for both of these nonlocal theories is equivalent tothe regular commutative λφ 4 theory. There is no modification of the entanglemententropy at this order. Perturbation theory can be used to study the nonlocal theoriesat small λ .The entanglement entropy is calculated using the replica trick and the formulaS = −∂n [lnZn−n lnZ1]n=1, where Zn is the partition function of the field theorydefined on an n-sheeted space [83, 84, 85]. This partition function can be reducedto computing vacuum bubble diagrams and the O(λ ) contribution in perturbationtheory comes from bubble diagrams with one vertex and two loops. Consistent withthe results of previous investigations of perturbative noncommutative theories [54],the planar diagrams in the nonlocal theories give the standard commutative result,which is S∼G1(0)∫dx∂n=1Gn(x)∼ A⊥Λ2 ln(Λ/m), where A⊥ is the (infinite) areaof the boundary of our region, Λ our UV regulator, m our IR regulator and Gn isthe Green’s function on the n-sheeted space used in the replica trick [85]. Thiscontribution follows an area law, as S ∝ A⊥Λ2 up to logarithmic corrections.The nonlocality only affects the nonplanar diagram. This diagram contributes aterm of the form S∼G1(0,∆x)∫dx∂n=1Gn(x,x+∆x)∼ A⊥(∆x)2 ln f (Λ,m,∆x), wherenow ∆x corresponds to a translation from the nonlocality.In the dipole theory, ∆x is proportional to the fixed dipole length. Thus thenonplanar diagram has only a logarithmic IR divergence and is subleading com-pared to the planar diagrams. In the noncommutative theory the translation alongthe noncommuative plane is proportional to the momentum in the other noncom-mutative direction, so this contribution must be integrated over this momentum.If we don’t impose an IR regulator, the momentum controlling the translation isallowed to vanish and G(0,∆x)→ G1(0)∼ Λ2. This gives a contribution that is of51the same order as the planar diagrams. However, if we impose an IR regulator, ∆xhas a minimal value and this divergence can be reinterpreted as an IR divergence.This is familiar from the UV/IR connection described for example in [54].Our results for the O(λ ) contribution to the entanglement entropy, S1, arereal scalar : S1 =2λSplanar+λSnonplanar (3.4)complex scalar : S1 =(2λ0+λ1)Splanar+λ1Snonplanar, (3.5)where Splanar and Snonplanar denote the contributions from planar and nonplanar di-agrams respectively.The leading divergences from these diagrams in each of the theories consideredareSplanar =− A⊥Λ221032pi3lnΛ24m2(3.6)Commutative theory : Snonplanar =Splanar (3.7)Noncommutative plane : Snonplanar =− A⊥Λ22932pi3− ln(Θ2m2Λ24)1− Θ2m2Λ24+ subleading(3.8)Dipole theory : Snonplanar is subleading, (3.9)where Λ is our UV regulator, m is our IR regulator, A⊥ is the area of the boundary,Θ is the noncommutativity parameter of the plane and a is the nonlocality scale ofthe dipole theory. The details of the expansion in mΛ used to extract these leadingdivergences are discussed in Section 3.5.2.In both cases, the contribution from these nonplanar diagrams does not havethe right form to be interpreted as the sign of a volume law in the entanglemententropy and we must conclude that these nonlocal theories at least to first orderin perturbation theory obey an area law. This can be contrasted with the strongcoupling result which found clear signs of the volume law even for large regions[31]. Thus, the volume law must either only appear at higher orders in perturbationtheory or it must require strong coupling. Consistent with our analysis, previousinvestigations of perturbative dynamics of the noncommutative theory [54] have52shown that noncommutativity does not introduce any new perturbative UV diver-gences that cannot be reinterpreted as IR divergences. Thus, is it hard to see howthe higher degree of divergence required for a volume law can arise in perturbationtheory. We are lead to the conclusion that entanglement on distances of the non-locality scale and volume laws require strong coupling and are not accessible toperturbation theory.The remainder of the paper is organised as follows: Section 3.2 describes thetheories we study, Section 3.3 explains how the entanglement entropy can be com-puted perturbatively in these theories, Section 3.4 shows that the results for the freetheory are unchanged in these nonlocal theories, Section 3.5.1 computes the firstorder correction in the coupling to the entanglement entropy in a real scalar φ 4theory for a warm-up and for later reference. Section 3.5.2 extends the calculationto the real scalar on the noncommutative plane. Section 3.5.3 reproduces the re-sults for the previous two sections in the case of the complex scalar. Section 3.5.4computes the result for the complex scalar in the dipole theory. Finally, Section 3.6concludes with a discussion of these results.3.2 TheoriesThe theories used in this paper are scalar field theories on R1,3 where productsof fields are replaced with a possibly noncommutative product denoted ?. Threeexamples of this product will be used: the regular commutative one, the Moyalproduct associated with the noncommutative plane and the dipole product with afixed nonlocality scale. See [86] for a review of noncommutative field theory. TheEuclidean action isSE =∫ddx[−12∂φ ?∂φ(x)+12m2φ ?φ(x)+λ4!φ ?φ ?φ ?φ(x)]. (3.10)The entanglement entropy in these three theories is calculated to leading order inthe coupling λ . The mass is present to serve as an IR regulator and will be takento be small in the end.First, the standard commutative case, where ( f ?g)(x) = f (x)g(x), is reviewedand presented in our notation in Sections 3.4 through 3.5.1. The entanglement53entropy for this theory was studied in [85] and the approach contained therein willbe followed for each of the theories we consider.Second, in Section 3.5.2, the entanglement entropy of a field theory defined onthe noncommutative plane, where( f ?g)(x) = exp(i2Θµν∂∂ξ µ∂∂ζ ν)f (x+ξ )g(x+ζ )|ξ=ζ=0, (3.11)is studied. The noncommutativity is parametrised by the antisymmetric tensor Θ.This theory has been studied perturbatively in [54]. In this case especially, themass should be thought of as an IR regulator and taken to zero at the end of thecalculation in order to see full effects of the UV/IR mixing present in this theory.We specialise to the case commonly referred to as the noncommutative plane whereΘµν =Θ(δ 1µδ 2ν −δ 2µδ 1ν) for simplicity.Finally, the entanglement entropy of the a simpler nonlocal theory with a fixednonlocality scale along a particular axis, known as a dipole theory, is studied. Forthis product, a vector called a dipole must be assigned to every field. The noncom-mutative product is( f ?g)(xµ) = f (xµ +12Lµ(g))g(xµ − 12Lµ( f )), (3.12)where Lµ( f ) is the dipole assigned to the field f .These dipoles must obey various rules set out in [61]. In particular, the dipoleof the ?-product of two field must be the sum of their dipoles. As well, the dipoleof the complex conjugate of a field must be minus the dipole of the original field.This means that a real field must have a zero dipole and that a complex scalar mustbe used rather than the real scalar field theory discussed so far. The action for acomplex scalar isSE =∫ddx[−∂φ † ?∂φ(x)+m2φ † ?φ(x)+ λ04φ † ?φ ?φ † ?φ(x)+λ14φ † ?φ ?φ ?φ †(x)].(3.13)where there two φ 4 terms which are inequivalent due to our noncommutative prod-54uct [61].3The result from the real scalar theory will be extended to this complex scalartheory in Section 3.5.3, then the dipole theory will be studied in Section 3.5.4.Setting Lµ(φ) = aδ µ1, the terms in the action can be written in a more explicitform: ∫dx(φ † ?φ)(x) =∫dxφ †(x+12a)φ(x+12a) =∫dxφ †(x)φ(x),∫dx(φ † ?φ)? (φ † ?φ)(x) =∫dxφ †(x)φ(x)φ †(x)φ(x), (3.14)∫dx(φ † ?φ)? (φ ?φ †)(x) =∫dxφ †(x+12a)φ(x+12a)φ(x− 12a)φ †(x− 12a),where only the dependence on the first coordinate, labelled x, is highlighted as theother coordinates are unaffected by this deformation.In fact, renormalisability requires that we include in the action terms of theformλn∫dx(φ †φ)(x+12na)(φ †φ)(x− 12na) (3.15)for all n [61]. However, the contributions from these terms can be obtained bysimply substituting a→ na into the results for n = 1 and summing over n. Theresults in Section 3.5.4 are such that this sum is guaranteed to converge as long asthe λn don’t grow too quickly. As the inclusion of these terms would not affect ourconclusions, we will not consider them separately.3.3 Entanglement entropyThe standard technique of the replica trick is used to compute the entanglemententropy [83]. This technique was used in a perturbative context in [85], whoseapproach is followed here.Starting with ρA, the reduced density matrix of the ground state of the theory3These noncommutative products are constructed to ensure that integrals of products of fields areinvariant under cyclic permutations.55in question for a region A, the idea is to evaluateS =− Tr (ρA lnρA) =− ∂∂n ln Tr (ρnA)|n=1, (3.16)by calculating Tr ρnA for arbitrary n and analytically continuing. In this paper wewill concentrate on the simplest case where A is the half plane (A = {(x1,x2,x3) ∈R3|x1 > 0}).The main result that will be needed can be lifted directly from [83, 85]:ln Tr (ρnA) = lnZn−n lnZ1, (3.17)where Zn is the partition function of the theory on an n-sheeted surface with a cutalong the region A that connects the sheets. However, some details of this n-sheetedspace will be needed in the argument to follow, so the rest of this section will defineit more carefully.3.3.1 n-sheeted surfacesThe density matrix can be written as a path integral, (at finite inverse temperatureof β )〈φ2|ρ |φ1〉= (Z1)−1∫Dφφ(x,β )=φ2φ(x,0)=φ1 e−SE , (3.18)where Z1 is a normalisation factor to ensure that Tr ρ = 1. Then the reduceddensity matrix for a region A is obtained by periodically identifying the field in theEuclidean time direction along A¯, the complement of A, while leaving the boundarycondition along A untouched. To look at the ground state, β must be sent to infinity.We do this while keeping the cut along A near the origin.Then,Tr (ρnA) = (Z1)−n[∫Dφφ(x∈A,0−)=φ2φ(x∈A,0+)=φ1 e−SE][∫Dφφ(x∈A,0−)=φ3φ(x∈A,0+)=φ2 e−SE]. . .[∫Dφφ(x∈A,0−)=φ1φ(x∈A,0+)=φn e−SE].(3.19)This identification of boundary conditions can be replaced by defining the field56theory on an n-sheeted surface with a cut along A that takes you from one sheetto the next. Calling this n-sheeted surface(Rd \A)n, the projection onto the sheetpi :(Rd \A)n→ Rd \A and the indicator function telling you if you are on the kthsheet χk :(Rd \A)n → Z1, this means that Φ : (Rd \A)n → R can be defined asΦ(x) = ∑Nk=1 φk(pi(x))χk(x), so thatTr (ρnA) = (Z1)−n[∫DΦe−SE], (3.20)where SE for Φ has the same form as that for each φ , since the action for eachsheet is additive.With our simple region A, a half-plane, polar coordinates can be defined in thex-τ plane of Rd \A. Then the glueing required to create this n-sheeted surface issimply to identify θ = 2pi on one sheet to θ = 0 on the next. Thus polar coordinatescan be defined on(Rd \A)n where θ ∈ [0,2pin), such that each interval of length2pi corresponds to a sheet, i.e. pi(r,θ ,y,z) = (r,θ mod 2pi,y,z) and χk(r,θ ,y,z) =χ[2pi(k−1),2pik)(θ).This gives us the result from [83, 85] cited above, as Zn =∫DΦe−SE . Thispath integral over Φ is the path integral over the n-sheeted surface.3.4 Free theoryThe first step is to understand the free theories where λ = 0. The action for thefree noncommutative and dipole theories is the same for that of the commutativetheory, since the star product of 2 fields is the same as the regular product up to atotal derivative [54].For the noncommutative theory,∫d4x( f ?g)(x) =∫d4x∞∑n=0in2nΘµ1ν1 . . .Θµnνn ∂µ1 . . .∂µn f (x) ∂ν1 . . .∂νng(x)(3.21)=∫d4x[f (x)g(x)+∂µ1∞∑n=1Θµ1ν1 . . .Θµnνn ∂µ2 . . .∂µn f (x) ∂ν1 . . .∂νng(x)],so that the quadratic term in the action is the same as for the commutative case up57to a total derivative. As there are no boundaries, the only place this total derivativecould make for a finite contribution is at the conical singularity introduced at theorigin when considering the n-sheeted path integral.Around the origin this term contributes (note that the singularity is at the originof the x-τ plane and is not localised in the y-z directions),limr→0A⊥∞∑n=1∫rdθ Θrν1Θµ2ν2 . . .Θµnνn ∂µ2 . . .∂µnφ ∂ν1 . . .∂νnφ ∼ limr→0∑nr∂ nφ∂ n+1φ ,(3.22)where A⊥ is the area of the y-z plane. As long as ∂ nφ ∂ n+1φ is regular at the originthis term will not contribute to the action. This means that φ needs to be C∞ at theorigin, which is just the regular boundary condition imposed in the commutativecase.For the dipole theory, direct calculation of the ?-product of two fields can beseen to reduce to the commutative result in (3.14).Thus the free theory is the same for all three theories.3.4.1 Green’s functionsSince the free theories are the same, they have the same Green’s functions. ThisGreen’s function is straightforward in the polar coordinates introduced in Section3.3.1. Since the action for Φ living on the n-sheeted surface is the same as theaction for φ living on any particular sheet, the local equation that the Green’s func-tion must obey will be the same. The only difference is that θ must be periodicwith period 2pin rather than the usual period of 2pi . The Green’s function for thefield living on the n-sheeted surface is, from [85],Gn(x,x′) =12pin∫ dd⊥ p⊥(2pi)d⊥∞∑k=0ak∫ ∞0dqqJk/n(qr)Jk/n(qr′)q2+ p2⊥+m2cos(k(θ −θ ′)/n)eip⊥(x⊥−x′⊥),(3.23)where a0 = 1, ak 6=0 = 2, p⊥ = (py, pz) and x⊥ = (x2,x3). ⊥ refers to the directionsorthogonal to the cut introduced by the replica trick.58The Euler-Maclaurin formula,∞∑k=0akF(k) = 2[∫ ∞0dkF(k)]− 16F ′(0)−2∑j>1B2 j(2 j)!F(2 j−1)(0), (3.24)can be applied to this Green’s function to replace the sum over k,Gn(x,x′) =∫ ∞0dkpi∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqqJk(qr)Jk(qr′)q2+ p2⊥+m2cos(k(θ −θ ′))eip⊥(x⊥−x′⊥)− 112pin2∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥) (3.25)−∑j>1B2 jpin2 j(2 j)!∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq(∂ν)2 j−1[Jν(qr)Jν(qr′)cos(ν(θ −θ ′))]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥).It will be useful to define Gn(x,x′; p) asGn(x,x′; py) =12pin∫ d pz2pi∞∑k=0ak∫ ∞0dqqJk/n(qr)Jk/n(qr′)q2+ p2y + p2z +m2cos(k(θ −θ ′)/n)eipz(x3−x′3)+ipy(x2−x′2)(3.26)such thatGn(x,x′) =∫ d py2piGn(x,x′; py) (3.27)∂∂x2Gn(x,x′; p) =− ∂∂x′2Gn(x,x′; p) = ipGn(x,x′; p). (3.28)It is also useful to define fn(x,x′) and fn(x,x′; p) asfn(x,x′) = Gn(x,x′)−G1(x,x′) (3.29)=n2−112pin2∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥)+( j > 1)fn(x,x′; p) = Gn(x,x′; p)−G1(x,x′; p), (3.30)where G1 is the Green’s function on the 1-sheeted surface, that is just the regularGreen’s function.59Single sheeted limitThis Green’s function for the n-sheeted space must reduce to the regular Green’sfunction in the limit where n→ 1. Starting with our expression for the Green’sfunction in (3.23), defining ϕ = θ −θ ′ for convenience and setting n = 1,G1(x,x′) =12pi∫ dd⊥ p⊥(2pi)d⊥∞∑k=0ak∫ ∞0dqqJk(qr)Jk(qr′)q2+ p2⊥+m2cos(kϕ)eip⊥(x⊥−x′⊥). (3.31)(10.9.E2) in the DLMF [87] provides a useful integral representation of theBessel functions, which can be rewritten as, Jn(z) =∫ pi−pidγ2pi ei(zsinγ−nγ). Using thisrepresentation and the fact that J−k(z) = (−1)kJk(z), 4∞∑k=0akJk(qr)Jk(qr′)cos(kϕ) =∞∑k=−∞∫ pi−pidγdκ(2pi)2eiq(r sinγ+r′ sinκ)−ik(γ+κ)eikϕ=∫ pi−pidγ2pieiq[r sinγ+r′ sin(ϕ−γ)]. (3.32)Defining our position axes on the x0-x1 plane such that ~x = (0,r) implies that~x′ = (−r′ sinϕ,r′ cosϕ). Then defining~q = (qcosγ,qsinγ),~q · (~x−~x′) =q[r sinγ+ r′ sin(ϕ− γ)] (3.33)∞∑k=0akJk(qr)Jk(qr′)cos(kϕ) =∫ pi−pidγ2piei~q·(~x−~x′) (3.34)Finally, defining p = (~q, p⊥),G1(x,x′) =∫ dd p(2pi)deip(x−x′)p2+m2, (3.35)which is the usual Euclidean Green’s function.3.4.2 Entanglement entropy in the free theoryThe entanglement entropy when λ = 0 must be identical in the three theories as itwas shown above that the quadratic terms in the action are the same. This can4 (10.4.E1) in [87]60be seen more explicitly by using the approach from [85]. Starting from SA =−∂n [lnZn−n lnZ1]n=1, the part of the entanglement entropy which depends onthe mass can be related to the Green’s function by∂∂m2lnZn =−12∫nddx〈Φ2(x)〉n. (3.36)In the commutative case, 〈Φ2(x)〉n = Gn(x,x). In the non-commutative case,〈Φ?Φ(x)〉n =(exp[i2Θ(∂∂ξ1∂∂ζ2− ∂∂ξ2∂∂ζ1)]〈Φ(x+ξ )Φ(x+ζ )〉n)ξ=ζ=0=(exp[i2Θ(∂∂ξ1∂∂ζ2− ∂∂ξ2∂∂ζ1)]Gn(x+ξ ,x+ζ ))ξ=ζ=0(3.37)=∫ d py2pi(exp[12Θpy(∂∂ξ1+∂∂ζ1)]Gn(x+ξ ,x+ζ ; py))ξ=ζ=0=∫ d py2piGn(x+12Θpy ıˆ,x+12Θpy ıˆ; py).That the ?-product turns out to just translate the argument of the Green’s functionis an important theme of the calculation in this paper.The only difference for a complex scalar is that the mass term in the action isproportional toΦ† ?Φ instead ofΦ?Φ, however the expectation value of this leadsto the same Green’s function and the same result follows.The dipole theory is identical except that translations by Θ times the momen-tum in the y-direction are replaced by translations by a.Thus, still for the non-commutative case,∂∂m2lnZn =− 12∫nddx〈Φ?Φ(x)〉n=− 12∫nddx∫ d p2piGn(x+12Θpıˆ,x+12Θpıˆ; p) (3.38)=− 12∫nddx∫ d p2piGn(x,x; p) =−12∫nddxGn(x,x),recovering explicitly the result from the commutative case by shifting the integra-tion variable.61Figure 3.1: Translations on each of the sheets of the n-sheeted surface (on theleft) give a well defined map on the whole surface (shown for n = 2 inthe polar coordinates described in Section 3.3.1 on the right), except fora measure zero set near the singularity at the origin.However, this shift of the integration variable on the n-sheeted surface bearsfurther investigation. It is sketched in Figure 3.1.This shift is well defined except for the region which gets translated into or outof the origin. However, this region has measure zero and cannot affect the resultof the integral. As long as only a countable number of such shifts are done, thesepoints can be omitted from the integral without changing the result. Finally, theintegral over the whole n-sheeted surface can be written as a sum over the sheetsand the Jacobian of this shift on each sheet is 1, so the Jacobian of the whole shiftdoes not introduce any new factors into the integral. Thus shifting the variable ofintegration on this n-sheeted surface is allowed with no Jacobian, just as for theplane.3.5 First order in perturbation theory3.5.1 Commutative theoryWe will start by computing the first order correction to the entanglement entropyfor the commutative φ 4 theory. This was done previously in [85], but will be re-peated here with more explicit regulators that will allow a direct comparison to the62nonlocal cases. From [85],lnZn = ln∫Dφe−SE [φ ]= lnZn,0− λ4!∫nd4x〈Φ4(x)〉0+ ... (3.39)= lnZn,0− 3λ4!∫nd4x [Gn(x,x)]2+ ...,where∫n denotes integration over the n-sheeted surface and lnZn,k is the kth orderterm in a λ expansion of lnZn. Generally, adding subscript will denote the order ofa term in a λ expansion, e.g. X = X0+X1+X2+ . . .The entanglement entropy can be calculated using (3.16) and (3.17),ln Tr (ρnA)1 = lnZn,1−n lnZ1,1=− 3λ4!∫nd4x [Gn(x,x)]2+3nλ4!∫d4x [G1(x,x)]2 (3.40)=− 3λ4!∫nd4x[2G1(x,x) fn(x,x)+ f 2n (x,x)].Recalling from (3.29),fn(x,x′) =n2−112pin2∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥)+( j > 1).(3.41)The j > 1 terms don’t contribute [83], so they will be dropped in what follows.This is the same on each sheet, so the integral over the n-sheeted surface is n timesin integral on one sheet. Finally, f1(x,x′) = 0, so ∂n f 2n (x,x′)|n=1 = 0 andS1 =−∂n [ln Tr (ρnA)1]n=1 =6λ4!∫d4xG1(x,x)∂n [n fn(x,x)]n=1 (3.42)S1 =12λA⊥12pi ·4!∫rdrdφ∫ d4kd pyd pz(2pi)61k2+m2∫ ∞0dqq∂ν [Jν(qr)Jν(qr)]ν=0q2+ p2y + p2z +m2.(3.43)63Schwinger parameters are introduced to allow the denominators to be com-bined, using1A=∫ ∞0dαe−Aα . (3.44)This allows us to regulate the UV divergence in S1 by introducing a factor of e− 1αΛ2 ,as was done in previous perturbative studies of noncommutative theories [54]. Thisregulator is convenient in the noncommutative case and is used here so that theresults can be compared. Using (25) from p.146 in volume I of [88],∫ ∞0dte−pt−a4t =√apK1(√ap), (3.45)the effect of this regulator is∫ ∞0dαe−α p2− 1αΛ2 =2ΛpK1(2pΛ)pΛ→∞−−−→√2Λp3e−2pΛ ,pΛ→0−−−→ 1p2. (3.46)Thus it regulates the UV and leaves the IR unaffected. This can be seen simplyfrom the fact that e−1αΛ2 vanishes for α  Λ−2 and goes to one for α  Λ−2. Amass m regulates the IR by contributing a factor of e−αm2 , which has the oppositebehaviour.Introducing these Schwinger parameters and regulating,S1 =λA⊥3 ·23∫drd4kd pyd pz(2pi)6dq∫ ∞0dαdβqre−αk2−β [q2+p2y+p2z ]−αm2− 1αΛ2−βm2− 1βΛ2 ∂ν [Jν(qr)Jν(qr)]ν=0.(3.47)All the momenta integrals except q are Gaussian,S1 =λA⊥3 ·29pi3∫drdq∫ ∞0dαdβqrα2βe−βq2−αm2− 1αΛ2−βm2− 1βΛ2 ∂ν [Jν(qr)Jν(qr)]ν=0.(3.48)Using (10.22.E67) from the Digital Library of Mathematical Functions (DLMF)64[87],∫ ∞0te−p2t2Jν(at)Jν(bt)dt =12p2e− (a2+b2)4p2 Iν(ab2p2), (3.49)the q integral can be evaluated. This along with the fact that ∂ν Iν(z)|ν=0 =−K0(z)5givesS1 =− λA⊥3 ·210pi3∫dr∫ ∞0dαdβrα2β 2e− r22β −αm2− 1αΛ2−βm2− 1βΛ2 K0(r22β). (3.50)(21) on p. 131 of [88],∫ ∞0dte−atK0(ty) =arccos(ay )√y2−a2ay→1−−−→ 1y, (3.51)after substituting r2→ t and setting a = y = 12β , givesS1 =− λA⊥3 ·210pi3(∫ ∞0dαα2e−αm2− 1αΛ2)(∫ ∞0dββe−βm2− 1βΛ2). (3.52)Looking at the α integral first,∫ ∞0dαα2e−αm2− 1αΛ2 =∫ ∞0dαe−m2α − αΛ2=2mΛK1(2mΛ)mΛ→0−−−→ Λ2 (3.53)by substituting α → 1α in the first line and using (3.45) as well as in the second.This recovers the Λ2 divergence seen previously in this case [85].Using (29) from Volume 1, p. 146 of [88]∫ ∞0tν−1e−pt−a4t dt =2(a4p) ν2Kν(√ap) (3.54)5 (10.38.E4) in the DLMF [87].65the β integral gives,∫ ∞0dββe−βm2− 1βΛ2 =2K0(2mΛ)mΛ→0−−−→−2ln 2mΛ= lnΛ24m2, (3.55)as K0(z)→− lnz as z→ 0. This reproduced the logarithmic divergence seen pre-viously in this case [85] and makes explicit its form in our regularisation scheme.Combining, the first order in λ correction to the entanglement entropy in thecommutative theory isS1,Comm. =−3λ A⊥Λ232 ·210pi3 lnΛ24m2. (3.56)This is proportional to the area of the boundary of A, that is A⊥, and the leadingdivergence is of order Λ2, so this result fits with the area law picture discussed inthe introduction.3.5.2 Noncommutative theoryNext we will compute the first order correction to the entanglement entropy for thenoncommutative φ 4 theory. Similarly to the commutative theory,lnZn = ln∫Dφe−SE [φ ]= lnZn,0− λ4!∫nd4x〈Φ?Φ?Φ?Φ(x)〉0+ ... (3.57)Using the associativity of the ?-product, this can be written as∫nd4x〈Φ?Φ?Φ?Φ(x)〉0 =∫nd4x(exp[i2Θ(∂∂ξ1∂∂ζ2− ∂∂ξ2∂∂ζ1)])ξ=ζ=0(exp[i2Θ(∂∂η1∂∂ς2− ∂∂η2∂∂ς1)])η=ς=0(3.58)(exp[i2Θ(∂∂γ1∂∂κ2− ∂∂γ2∂∂κ1)])γ=κ=0〈Φ(x+ξ +η)Φ(x+ξ + ς)Φ(x+ζ + γ)Φ(x+ζ +κ)〉.66The usual Wick’s Theorem can be applied to calculate the four-point function,〈Φ(w)Φ(x)Φ(y)Φ(z)〉= Gn(w,x)Gn(y,z)+Gn(w,y)Gn(x,z)+Gn(w,z)Gn(x,y).(3.59)The key point is that while the conical singularity breaks the translational in-variance in the x0-x1 plane, it is preserved in the x2-direction. Thus the star productreduces to a translation in the x1-direction by an amount determined by the mo-mentum in the x2-direction. Defining Gn(w,z) =∫ d py2pi Gn(w,z; py) as in (3.26),exp(i2Θ∂∂w1∂∂ z2)Gn(w,z) =∫ d py2piexp(12pyΘ∂∂w1)Gn(w,z; py)=∫ d py2piGn(w+12pyΘıˆ,z; py), (3.60)this can be used to evaluate the 4-point function,∫nd4x <Φ?Φ?Φ?Φ(x)>0 (3.61)=∫nd4x∫ dkyd py(2pi)2[Gn(x+12Θky ıˆ,x+12Θky ıˆ;ky)Gn(x+12Θpy ıˆ,x+12Θpy ıˆ; py)+Gn(x+12Θky ıˆ,x+12Θ(ky+2py)ıˆ;ky)Gn(x+12Θ(2ky+ py)ıˆ,x+12Θpy ıˆ; py)+Gn(x+12Θky ıˆ,x+12Θky ıˆ;ky)Gn(x+12Θ(2ky+ py)ıˆ,x+12Θ(2ky+ py)ıˆ; py)].Then, by shifting the spatial integral,=∫nd4x∫ dkyd py(2pi)2[Gn(x,x;ky)Gn(x+12Θ(py− ky)ıˆ,x+ 12Θ(py− ky)ıˆ; py)+Gn(x− 12Θpy ıˆ,x+12Θpy ıˆ;ky)Gn(x+12Θky ıˆ,x− 12Θky ıˆ; py) (3.62)+Gn(x,x;ky)Gn(x+12Θ(ky+ py)ıˆ,x+12Θ(ky+ py)ıˆ; py)].In [54] it is seen that the effects of the non-commutativity manifest themselvesin the diagrams where lines cross each other. This is also present here, as Figure3.2 shows that it is only the second term that involves lines crossing. The other67Figure 3.2: Vacuum bubble diagrams at leading order in a real scalar λφ 4theory. The only vacuum bubble where lines cross is the second one.This is the only one which is affected by the non-commutativity, asdiscussed in [54].two terms are two self-coincident Green’s functions – the same result as was foundin the commutative case in Section 3.5.1 and [85]. The second term, which corre-sponds to the nonplanar diagram, is the only one which is different than what wasfound in the commutative case.The entanglement entropy can be calculated using (3.16),S1 =−∂n [lnZn,1−n lnZ1,1]n=1 (3.63)=2λ4!∂n(∫d4x[2G1(x,x)n fn(x,x)+∫ dkyd py(2pi)2G1(x,x+Θpy ıˆ;ky)n fn(x,x−Θky ıˆ; py)])n=1where the fact that the spatial integral can be shifted, that the momenta can berenamed, that G1(x,x; py) =G1(x+a,x+a; py), that fn(x,x′, py) = fn(x,x′;−py) aslong as x2 = x′2 and that f1 = 0 so that the terms with f2n can be ignored have all beenused. The j > 1 terms in fn have also been dropped again, which allows us here towrite the integral over the n-sheeted surface as n times the integral over a sheet. Inthe commutative case, it was clear that these j > 1 terms do not contribute [83]. InAppendix A it is argued that the leading divergence must be entirely contained inthe j = 1 term even in this noncommutative theory.New contribution from the nonplanar diagramThe first term in (3.63) is the contribution from the two planar diagrams. Thesegive the same result as in the commutative case, namely λA⊥Λ221032pi3 lnΛ24m2 from eachdiagram. However, the nonplanar diagram gives a new contribution to the entangle-ment entropy from the non-commutativity. The contribution from this nonplanar68diagram will be denoted Snonplanar,Snonplanar =2λ4!∫d4x∫ dkyd py(2pi)2G1(x,x+Θpy ıˆ;ky)∂n [n fn(x,x−Θky ıˆ; py)]n=1=4λA⊥12pi ·4!∫rdrdφ∫ d4kd pyd pz(2pi)6eiΘkx pyk2+m2∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2y + p2z +m2,(3.64)where r′2 = (~r−Θky ıˆ)2 = r2+(Θky)2−2Θrky cosφ and A⊥ is the area of the x2-x3plane that bounds the region for which the entanglement entropy is being calcu-lated.The next step is to introduce Schwinger parameters and to regulate this inte-gral in the same manner as the integrals for other perturbative calculations in thisnoncommutative theory were regulated in [54], as discussed in Section 3.5.1,Snonplanar =λA⊥2332pi∫drdφd4kd pyd pz(2pi)6dq∫ ∞0dαdβqre−αk2−β [q2+p2y+p2z ]− 1αΛ2−αm2− 1βΛ2−βm2eiΘkx py∂ν [Jν(qr)Jν(qr′)]ν=0.(3.65)The py, pz and k except for ky integrals are all Gaussian (recall that r′ is afunction of ky),Snonplanar =λA⊥2832pi 92∫drdφdkydq∫ ∞0dαdβqrα√β√4αβ +Θ2e−αk2y−βq2− 1αΛ2−αm2− 1βΛ2−βm2∂ν [Jν(qr)Jν(qr′)]ν=0. (3.66)In order to make explicit some of the symmetry between r and r′, ρ and ϕ canbe defined such that r = ρ sinϕ and ky = ρΘ cosϕ , with ρ ∈ [0,∞) and ϕ ∈ [0,pi].Then defining g(φ ,ϕ) =√1+ sin2ϕ cosφ , gives r′ = ρg(φ ,ϕ) in these variables.69Performing this change of variables,Snonplanar =λA⊥2832pi 92Θ∂ν |ν=0∫dρdϕdφdqdαdβqρ2 sinϕα√β√4αβ +Θ2e− αΘ2ρ2 cos2ϕ−βq2− 1αΛ2−αm2− 1βΛ2−βm2Jν(qρ sinϕ)Jν(qρg(φ ,ϕ)).(3.67)From the DLMF (10.22.E67) [87],∫ ∞0te−p2t2Jν(at)Jν(bt) =12p2e− (a2+b2)4p2 Iν(ab2p2)(3.68)so that,Snonplanar =λA⊥2932pi 92Θ∂ν |ν=0∫dρdϕdφdαdβρ2 sinϕαβ 32√4αβ +Θ2e− αΘ2ρ2 cos2ϕ−ρ2 sin2 ϕ+g2(ϕ,φ)4β − 1αΛ2−αm2− 1βΛ2−βm2Iν(ρ22βg(φ ,ϕ)sinϕ).(3.69)Now ρ and α can be rescaled to simplify this expression as ρ → 2√βρ andα → Θ24β α ,Snonplanar =λA⊥2632pi 92Θ2∂ν |ν=0∫dρdϕdφdαρ2 sinϕα√α+1(∫ ∞0dβe−β(m2+ 4Θ2Λ2α)− 1β(1Λ2+Θ2m2α4))e−αρ2 cos2ϕ−ρ2[sin2ϕ+g2(φ ,ϕ)]Iν(2ρ2g(φ ,ϕ)sinϕ). (3.70)(25) from p.146 in volume I of [88],∫ ∞0dte−pt−a4t =√apK1(√ap), (3.71)70allows the β integral to be evaluated,∫ ∞0dβe−β(m2+ 4Θ2Λ2α)− 1β(1Λ2+Θ2m2α4)=√√√√ 4Λ2 +Θ2m2αm2+ 4Θ2Λ2αK1(√(4Λ2+Θ2m2α)(m2+4Θ2Λ2α)),=Θ√αK1(4ΘΛ2√α+Θm2√α). (3.72)Using the identity ∂ν |ν=0Iν(z) =−K0(z),Snonplanar =− λA⊥2632pi 92Θ∫dρdϕdφdαρ2 sinϕ√α√α+1e−ρ2[α cos2ϕ+sin2ϕ+g2(φ ,ϕ)]K0(2ρ2g(φ ,ϕ)sinϕ)K1(4ΘΛ2√α+Θm2√α). (3.73)Taking a large Λ limit of this expression and expanding K1(x) ≈ 1x for x→ 0allows us to extract an overall quadratic divergence. However, more progress canstill be made by evaluating the ρ integral.Using in order (23) from p. 131 of [88] and (15.9.E19) of [87],∫ ∞0dρρ2e−Aρ2K0(Bρ2) =∫ ∞0dx√xe−AxK0(Bx)=12√pi[Γ(32)]2Γ(2)(A+B) 322F1(32,12;2;A−BA+B)(3.74)=pi 328√2B321√(AB)2−1P1− 12(AB),where P1− 12(x) is the appropriate branch of the associated Legendre function withnon-integer degree.71Defining z= α cos2ϕ+sin2ϕ+g2(ϕ,φ)2g(φ ,ϕ)sinϕ and recalling that g(φ ,ϕ)=√1+ sin2ϕ cosφ ,Snonplanar =− λA⊥21132pi3Θ∫ ∞0dαG(α)√α√α+1K1(4ΘΛ2√α+Θm2√α)and(3.75)G(α) =∫ pi0dϕ∫ 2pi0dφ1[g(φ ,ϕ)]32√sinϕP1− 12(z)√z2−1 , (3.76)where G(α) is dimensionless and finite for α ∈ (0,∞).At this point, the asymptotic behaviour of G(α) can be analysed numerically,as no analytic formula for this integral was found in the tables consulted. However,while analysing this asymptotic behaviour, we found that G(α) = 16√α+1 gives anexact match up to high numerical accuracy across the many orders of magnitudethat were checked.6Using this result for G(α),Snonplanar =− λA⊥2732pi3Θ∫ ∞0dα√α1α+1K1(4ΘΛ2√α+Θm2√α). (3.77)Note that this result is invariant under ΘΛ2 ↔ Θm2, another sign of the UV/IRconnection in non-commutative theories.This integral has two regulators, Λ and m. The only other dimensionful pa-rameter is Θ, so the only dimensionless products of these regulators are mΛ andΘmΛ. As is familiar from the UV/IR mixing in this theory, the limits Λ→ ∞ andm→ 0 do not commute. This can be resolved by taking mΛ → 0 while fixing ΘmΛ.Then taking the limit m→ 0 or Λ→ ∞ first corresponds to the limits ΘmΛ→ 0 orΘmΛ→ ∞ respectively. 76The only potential divergences in the integral for Snonplanar come from the regions of small andlarge α . If the reader is uncomfortable with this numeric argument, this functional form for G(α)could also be thought of more conservatively as a function with the right asymptotic behaviour toreproduce the correct divergences in this integral.7This discussion applies even if we want to think of m as a physical mass, as the ratio mΛ will stillvanish if m is fixed while Λ→ ∞. This case corresponds to ΘmΛ→ ∞.72Introducing γ =√α ,Snonplanar =− λA⊥2632pi3Θ∫ ∞0dγ1γ2+1K1(2mΛ[2ΘmΛγ+ΘmΛγ2])mΛ→0−−−→− λA⊥Λ2832pi3Θm∫ ∞0dγ1γ2+112ΘmΛγ +ΘmΛγ2(3.78)=− λA⊥Λ22932pi3− ln(Θ2m2Λ24)1− Θ2m2Λ24=− λA⊥2732pi3Θ2m2− ln( 4Θ2m2Λ2 )1− 4Θ2m2Λ2,where the last line uses (2) from Volume 2 p.216 of [88].This result illustrates the UV/IR connection in non-commutative theories. If theIR regulator is removed first (ΘmΛ 1), Snonplanar∼A⊥Λ2 – a quadratic UV diver-gence. However if the UV regulator is removed first (ΘmΛ 1), Snonplanar ∼ A⊥Θ2m2 ,allowing the same divergence to be interpreted as an IR divergence. In addition,whether Θ2m2Λ24 is taken to be large or small there is a logarithmic divergence asis found in the commutative case. However, here there is the additional option ofkeeping both regulators, that is keeping 12ΘmΛ finite, which eliminates the loga-rithmic divergence seen in the commutative case.8 In particular, there is a naturalchoice of IR regulator9, m = 2ΘΛ whereSnonplanar =− λA⊥2732pi3Θ2m2 =−λA⊥Λ22932pi3. (3.79)From a mathematical point of view, this UV/IR connection can be seen tooriginate from the translation of the arguments of the Green’s function. In thecommutative theory, Snonplanar ∼∫n dxGn(x,x) fn(x,x) where as in the noncommu-tative theory, the non-planar diagram made a contribution of the form Snonplanar ∼∫n dxGn(x,x+Θp) fn(x,x+Θp). If an IR regulator is imposed, this momentumcannot vanish and regulates the integral. This can be seen more clearly in thedipole theory (analysed in Section 3.5.4) where the fixed translation regulates theUV divergence of the integral.8Note that if a Θ→ 0 limit is taken, this option is no longer available and the commutative resultis recovered, although the exact form of the logarithmic divergence depends on how the Θ limit istaken.9See Section 6 of [54]73It is important to note that contributions from the j > 1 terms in (3.25) weredropped at the start of this section and are not present in (3.78) or elsewhere in theseresults. However, as is discussed in Appendix A, these do not affect the leadingdivergence in Snonplanar or the conclusion that there is no volume law.In contrast to strong coupling results, which saw signs of a volume law for theentanglement entropy even with large regions, this perturbative calculation is onlysees an area law. The leading divergence in Snonplanar is quadratic and proportionalto the area of the boundary of the region, A⊥, in line with the area law discussed inthe introduction.3.5.3 Complex scalarThe difference when considering a complex scalar is the Wick contraction in (3.39)and (3.59) for the commutative and the noncommutative theory respectively. Forthe real scalarλ 〈φ(w)φ(x)φ(y)φ(z)〉= λ (Gn(w,x)Gn(y,z)+Gn(w,y)Gn(x,z)+Gn(w,z)Gn(x,y)) ,(3.80)whereas for the complex scalar this must be replaced withλ0〈φ †(w)φ(x)φ †(y)φ(z)〉+λ1〈φ †(w)φ(x)φ(y)φ †(z)〉 (3.81)= λ0 (Gn(w,x)Gn(y,z)+Gn(w,z)Gn(x,y))+λ1 (Gn(w,x)Gn(z,y)+Gn(w,y)Gn(z,x)) .In the commutative theory, the fields in the 4-point function are all insertedat the same point, that is w = x = y = z. Taking into account the difference inthe normalisation of the φ 4 term in the action, the only change is to replace anoverall factor of 3λ4! by2(λ0+λ1)4 . This has no effect on the intermediate steps of thecalculation and can just be carried through straight to the final result:S1,Comm.→− (λ0+λ1)A⊥Λ23 ·28pi3 lnΛ24m2. (3.82)For the noncommutative theory, it is a simple matter of writing out the ?-products explicitly and following through similar transformations of the integra-74Figure 3.3: Vacuum bubble diagrams at leading order in the noncommutativecomplex scalar λφ 4 theory. The two on the left come from the λ0φ † ?φ ?φ †?φ term in the action whereas the two on the right from the λ1φ †?φ ?φ ?φ † term.tion variables as in the previous section. This procedure gives 2λ0 +λ1 times thecommutative result plus λ1 times the result for the nonplanar diagram already en-countered for the real scalar. This result can be obtained directly by looking at the4 diagrams in Figure 3.3 and realising that only the term proportional to λ1 gives anonplanar diagram.Thus the result for the noncommutative theory with a complex scalar isS1,NC→(2λ0+λ1) A⊥Λ23 ·29pi3 lnΛ24m2−λ1 A⊥Λ23 ·28pi3− ln(Θ2m2Λ24)1− Θ2m2Λ24(3.83)3.5.4 Dipole theoryFor the dipole theory, the explicit form of the interaction terms was written out in(3.14). Thus,lnZn = ln∫Dφe−SE [φ ] (3.84)= lnZn,0−∫nd4x〈λ04Φ†(x)Φ(x)Φ†(x)Φ(x)+λ14Φ†(x+12a)Φ(x+12a)Φ(x− 12a)Φ†(x− 12a)〉0+ . . .Applying Wick’s Theorem, using the facts that G1(x,x) = G1(x+a,x+a) and75fn(x+a,x) = fn(x,x+a) (when ignoring the j > 1 terms) and shifting the integral,lnZn,1 =− 14∫nd4x [2λ0Gn(x,x)Gn(x,x) (3.85)+ λ1(Gn(x+12a,x+12a)Gn(x− 12a,x−12a)+Gn(x+12a,x− 12a)Gn(x− 12a,x+12a))]S1 =−∂n [lnZn,1−n lnZ1,1]n=1 (3.86)=24∂n(∫d4x[(2λ0+λ1)G1(x,x)n fn(x,x)+λ1G1(x,x+a)n fn(x,x−a)])n=1.Again this is as expected from the diagrammatic approach. Only the singlenonplanar diagram gives a new contribution and the 3 planar diagrams give contri-butions identical to those in the commutative theory.Focusing on the contribution from the nonplanar diagram, the explicit forms ofG1 and fn giveSnonplanar =4λA⊥12pi ·4∫rdrdφ∫ d4kd pyd pz(2pi)6eikxak2+m2∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2y + p2z +m2,(3.87)where now r′2 = (~r−aıˆ)2 = r2+a2−2racosφ .Introducing Schwinger parameters and regulating,Snonplanar =λA⊥223pi∫drdφd4kd pyd pz(2pi)6dq∫ ∞0dαdβqre−αk2−β [q2+p2y+p2z ]− 1αΛ2−αm2− 1βΛ2−βm2eikxa∂ν [Jν(qr)Jν(qr′)]ν=0.(3.88)In this case, all the momenta integrals except q are Gaussian,Snonplanar =λA⊥283pi4∫drdφdqdαdβqrα2βe− a24α− 1αΛ2−αm2−βq2− 1βΛ2−βm2∂ν [Jν(qr)Jν(qr′)]ν=0.(3.89)76The α integral can be factored out to give, using (3.45),∫ ∞0dαe−1α(a24 +1Λ2)−αm2α2=∫ ∞0dαe−α(a24 +1Λ2)−m2α =2m√a24 +1Λ2K1(2m√a24+1Λ2)Λ→∞−−−→4maK1 (ma) (3.90)m→0−−−→ 4a2This factor came from evaluating G1(0,aıˆ) which goes as ∼ 1a2 as expected. Thefixed nonlocality scale has regulated the UV divergence in this case. In the dipoletheory the distance of the translation is fixed, as opposed to the non-commutativecase where the translation is proportional to the momentum in the y-direction whichcan vanish in the IR.Using (3.49),Snonplanar =− λA⊥273pi4a2∫ ∞0dβ[∫ ∞0dr∫ 2pi0dφrβ 2e−r2+r′24β K0(rr′2β)]e− 1βΛ2−βm2.(3.91)Rescaling r→ ar and β→ a2β to make them dimensionless (r′→ a√r2+1−2r cosφunder this) and defining H(β ) as the part of the previous equation enclosed inbrackets,Snonplanar =− λA⊥273pi4a2∫ ∞0dβH(β )e−1βa2Λ2−βa2m2. (3.92)H(β ) is dimensionless and finite for β ∈ (0,∞). The integrand is exponentiallysuppressed for small β and numerical evaluation of the r and φ integrals confirmthat H(β ) β→0−−−→ 0. The other potential source of a divergence is at large β andnumerical integration finds that H(β ) β→∞−−−→ 2piβ leading to a logarithmic divergenceat large β that must be regulated by e−βa2m2 ,∫ ∞ dββe−βa2m2 =− ln(a2m2)+O(m0), (3.93)77to leading order in the small m limit.Thus Snonplanar has only an IR divergence in the dipole theory. The leadingdivergence in the j = 1 term isSnonplanar =− λA⊥3 ·26pi3a2[− ln(a2m2)] , (3.94)however there will be contributions to this order from the j > 1 terms which weredropped. The the conclusion of this analysis is that the nonplanar diagram does notcontribute to the leading divergence of entanglement entropy at this order as it issubleading to the contribution from the planar diagram.The nonlocality introduced in the dipole theory does not affect the area law, asthe total entanglement entropy at this order in perturbation theory is dominated bythe planar diagrams which matched the result from the commutative theory. Eventhe subleading terms we have analysed do not follow any sort of volume law as theyare not proportional to the lengthscale of the nonlocality. The only effect of thenonlocality is to regulate the UV divergence otherwise present. Similar behaviourwas observed in [54], where one of the ways that the nonlocality manifested itselfwas by softening divergences in nonplanar diagrams.3.6 Final remarksIn this paper we computed the first perturbative correction to the entanglemententropy in two nonlocal theories, a φ 4 theory defined on the noncommutative planeand a dipole theory.The contribution to the entanglement entropy in each of these theories at firstorder in coupling comes from vacuum bubble diagrams. The planar diagrams givethe same contribution in all three theories. However, the nonplanar diagram isaffected by the modified ?-product. Never the less, these diagrams do not modifythe area law observed in the commutative theory. Thus, at this order in perturbationtheory and for the region considered at least, all these theories follow an area lawwith no sign of a volume law, as opposed to the strongly coupled case where thesignature of the volume law could be seen even for large regions.In the commutative theory it has been shown that the modification to the en-78tanglement entropy at first order in perturbation theory can be absorbed into therenormalisation of the mass [85]. It would be interesting to see if a similar inter-pretation can be made in the case of the theories considered here.Finally, a comment about the commutative limit. Since the quantities dealt within the paper are not UV finite, this is not a straightforward issue. The general pat-tern is that the nonlocality has served as an additional regulator that softens certaindivergences. Thus, if the nonlocality is removed, these divergences reappear andthe commutative limit applied to the final results is not smooth.79Part IIThe Structure of HolographicEntanglement Entropy80Chapter 4Inviolable Energy Conditions4.1 IntroductionThe AdS/CFT correspondence provides a remarkable connection between quan-tum gravitational theories and non-gravitational quantum systems [4, 5]. There arebelieved to be many examples of the correspondence; indeed, it may be that anyconsistent quantum gravity theory for asymptotically AdS spacetimes can be usedto define a CFT on the boundary spacetime. In this paper, we focus on exampleswith a classical limit described by Einstein’s equations coupled to matter. We seekto derive results that are universally true for all such theories, by translating togravitational language results that are universally true in all quantum field theories.Specifically, we will translate some basic constraints on the structure of entangle-ment in quantum systems to derive some fundamental constraints on spacetimegeometry that must hold in all consistent theories of Einstein gravity coupled tomatter.Our main tool will be the Ryu-Takayanagi formula (and its covariant gener-alization due to Hubeny, Rangamani, and Takayanagi)[6, 9].1 This relates entan-glement entropy for spatial regions A in the field theory to the areas of extremalsurfaces ∂A in the dual geometry with the same boundary as A (see Section 4.2 for1A recent proof was given in [27].81Figure 4.1: Ryu-Takayanagi formula as a map from the space G of geome-tries with boundary B to the spaceS of mappings from subsets of B toreal numbers. Mappings in region Sphys (shaded) correspond to physi-cally allowed entanglement entropies. Geometries in region Gphys mapinto Sphys while the remaining geometries are unphysical in any con-sistent theory for which the Ryu-Takayanagi formula holds (plausiblyequal to the set of gravity theories with Einstein gravity coupled to mat-ter in the classical limit).a review). Generally speaking, we can understand this as a mappingRT : G →Sfrom the set G of asymptotically AdS spacetimes with boundary geometry B to thesetS of maps S from subsets of B to real numbers.2This mapping is depicted in Figure 4.1. Physically allowed entanglement struc-tures must obey constraints, such as strong subadditivity and positivity/monotonic-ity of relative entropy, so only a subset Sphys of maps represented by S can rep-resent physically allowed entanglement structures. If a geometry M ∈ G maps to apoint outside this subset, we can conclude that such a geometry is not allowed, inany theory for which the Ryu-Takayanagi formula is valid (which we believe to be2To avoid divergent quantities, we could define the map S associated with a geometry M such thatfor subset B of the boundary of M, SM(B) is the difference between the area of the extremal surfaceB˜M and the corresponding extremal surface B˜AdS in pure AdS.82all consistent gravity theories whose classical limit is Einstein gravity coupled tomatter). Another interesting point is that the space of geometries with boundary Bis much smaller than the space of functions on subsets of B, so the image of G inS will be a measure zero subsetSG. This implies that the entanglement structuresfor quantum field theory states with gravity duals are extremely constrained.This picture suggests several interesting directions for research:• Characterize the geometries Gphys that map to physically allowed entangle-ment entropies Sphys. While some of these geometries may be ruled outby additional constraints not related to entanglement, we can say that anygeometry not in Gphys cannot represent a physical spacetime.• Characterize the constraints on entanglement structure implied by the exis-tence of a holographic dual i.e. understand the subset SG. Examples in-clude the monogamy of mutual information [89], but there should be muchstronger constraints through which the entanglement entropies for most re-gions are determined in terms of the entanglement entropies for a small sub-set of regions.• Better understand the inverse mapping from Sphys to Gphys to be able toexplicitly reconstruct geometries from entanglement entropies.In this paper, we focus on the first direction, though we will have some commentson the second direction in Section 4.6. Many recent papers discuss the third direc-tion, including [90, 91, 92].Constraining geometry from entanglementThe question of which geometries give rise to allowed entanglement structures wasconsidered at the level of first order perturbations to pure AdS in [36, 37, 38] (seealso [93, 94]). Such perturbations correspond to small perturbations of the CFTvacuum state. For these first order CFT perturbations, the entanglement entropy forball-shaped regions is determined in terms of the expectation value of the stress-energy tensor3 via the “first law of entanglement,” which we review in Section 4.23The stress tensor is determined in terms of the entanglement entropy for infinitesimal ball shapedregions, so we can think of the entanglement first law as a constraint determining the entanglement83below. As shown in [36, 37] the gravitational version of this constraint is exactlythe linearized Einstein equation. For a discussion of constraints at the second-orderin the metric perturbation, see [95, 96].In this paper, we begin to unravel the implications of entanglement constraintson geometries away from this perturbative limit. One might ask whether it is pos-sible to obtain the full non-linear Einstein equations in this way. However, at theclassical level, the entanglement quantities tell us only about the dual geometry,so the entanglement constraints will be constraints on the geometries themselves,without reference to any bulk stress-energy tensor. Further, the specific constraintswe consider (strong subadditivity of entanglement entropy, and the positivity andmonotonicity of relative entropy) take the form of inequalities, so we should ex-pect that the nonlinear constraints also take the form of geometrical inequalitiesruling out some geometries as unphysical. This is a natural outcome: since theresults must apply to all consistent theories, we cannot expect specific non-linearequations to emerge, but there should be restrictions that apply to the whole classof allowed theories.In interpreting these geometrical constraints, it is useful to translate them intoconstraints on the stress-energy tensor assuming that Einstein’s equations hold.This is a very plausible assumption. Indeed, it is possible to argue [38] indirectlyusing the linearized results that Einstein’s equations must be obtained.4 Any ge-ometry provides a solution to Einstein’s equations for some stress tensor. Thus,given a geometry that violates the entanglement constraints, we can conclude thatno consistent theory of gravity can produce the associated stress tensor. Expressedin this way, the constraints from entanglement inequalities can be thought of ascertain “energy conditions.”We will see that some of the conditions we obtain are closely related to someof the standard energy conditions used in classical general relativity. However,entropies for arbitrary ball-shaped regions from the entanglement entropies for infinitesimal balls.4In [38], it was shown that by considering quantum corrections to the Ryu-Takayanagi formula,the expectation value of the bulk stress-energy tensor comes in as a source for the linearized Einsteinequations. Assuming that the source is a generally a local operator, this is enough to see that it mustbe the stress-energy tensor. It has been argued that the linearized equations together with the stress-energy tensor as a source imply the full non-linear Einstein equations if one demands conservationof the stress-energy tensor in the full theory.84we emphasize that while these standard conditions (such as the weak and null-energy conditions) are simply plausible assumptions on the properties of matter, theconditions we derive follow from fundamental principles of quantum mechanics(assuming the Ryu-Takayanagi formula holds) and cannot be violated.Summary of resultsIn this paper, we take a few modest steps towards understanding the general con-straints on non-linear gravity due to entanglement inequalities, investigating theseconstraints in the case of highly symmetric spacetimes. Specifically, we determineconstraints on static, translationally invariant spacetimes in 2+1 dimensions, andstatic, spherically-symmetric spacetimes in general dimensions. We find the fol-lowing main results:• For spacetimes dual to the vacuum states of 1+1 dimensional Lorentz-invariantfield theories flowing between two CFT fixed points, the constraints due tostrong-subadditivity are satisfied if and only if the spacetime satisfies a setof averaged null energy conditions∫γdsTµνuµuν ≥ 0where γ is an arbitrary spatial geodesic and uµ is a null vector generating alight-sheet of γ defined such that translation by uµ produces an equal changein the spatial scale factor at all points (Section 4.3).• For static translation-invariant spacetimes dual to excited states of 1+1 di-mensional CFTs, we show that the monotonicity of relative entropy impliesthat the minimum scale factor reached by an RT surface for spatial intervalis always less than the scale factor reached by the corresponding RT sur-face in the geometry for the thermal state with the same stress-energy tensor(Section 4.4).• For these spacetimes, we find that asymptotically, the positivity of relativeentropy is exactly equivalent to the statement that observers near the bound-ary moving at arbitrary velocities in the field theory direction cannot ob-serve negative energy. That is, we get a subset of the weak energy condition85Tµνuµuν ≥ 0 where uµ is an arbitrary timelike vector with no component inthe radial direction.• For static spherically symmetric asymptotically AdS spacetimes, the pos-itivity of relative entropy implies that the area of a surface bisecting thespacetime symmetrically is bounded by the mass of the spacetime. For four-dimensional gravity, the specific result is (Section 4.5)∆A≤ 2piGNM`AdS .We offer a few concluding remarks in Section 4.6.Previous connections between energy conditions and entanglement inequalitiesappeared in [97, 98, 73, 99] who noted that the null energy condition is sufficientto prove certain entanglement inequalities holographically. The use of relative en-tropy in holography was pioneered in [100] and applied to derive gravitationalconstraints at the perturbative level in [95, 96].Note: While this manuscript was in preparation, the paper [41] appeared, whichoverlaps with the results in Section Background4.2.1 Entanglement inequalitiesIn this section, we review various entanglement inequalities that should place con-straints on possible dual spacetimes via the holographic entanglement entropy for-mula.5Strong subadditivityTo begin, we recall that the entanglement entropy S(A) for a subsystem A of aquantum system is defined as S(A) = − tr(ρA log(ρA)), where ρA is the reduceddensity matrix for the subsystem.5See, for example [14], for a more complete discussion of entanglement inequalities.86xx12ξξ12ABCFigure 4.2: Spacelike intervals for strong subadditivity.The strong subadditivity of entanglement entropy states that for any three dis-joint subsystems A, B, and C,S(A∪B)+S(B∪C)≥ S(B)+S(A∪B∪C). (4.1)Considering only spatial regions of a constant-time slice in a time-invariant statecorresponding to a static dual geometry, this constraint places no constraints onthe dual geometry, as shown in [74]. However, in the time-dependent cases, orfor regions of a time-slice that do not respect the symmetry, this inequality givesnon-trivial constraints, as we will see below.For our analysis below, we will be interested in applying the constraints ofstrong subadditivity in the case of 1+1 dimensional field theories. Entanglemententropy is the same for any spacelike regions with the same domain of dependence,so for any connected spacelike region A, entanglement entropy is a function of thetwo endpoints of the region. We write S(x1,x2) to denote the entanglement entropyof the interval [x1,x2] (or any spacelike region with the same domain of depen-dence). We focus on the case where A, B, and C in (4.1) are adjacent spacelikeintervals, as shown in Figure 4.2.We note first that the full set of strong subadditivity constraints for adjacentintervals follow from the constraints in the case where the intervals A and C areinfinitesimal. For suppose the strong-subadditivity constraint is true for regions A,B, and C with the proper length of A and C less than Lmax. Then we can show thatthe constraint holds for intervals with A and C less than 2Lmax, and so forth. For87example, if A, B, C1 and C2 are adjacent intervals with C1 and C2 having properlength less than Lmax, we haveS(A∪B)+S(B∪C1) ≥ S(A∪B∪C1)+S(B)S(A∪B∪C1)+S(B∪C1∪C2) ≥ S(A∪B∪C1∪C2)+S(B∪C1)Adding these, we findS(A∪B)+S(B∪{C1∪C2})≥ S(A∪B∪{C1∪C2})+S(B) .In this way, we can combine two strong subadditivity constraints for which therightmost interval has length smaller than Lmax to obtain a constraint where therightmost interval is any interval with length less than 2Lmax.6Now, consider the strong subadditivity constraint where B is the interval [x1,x2]while A and C are the intervals [x1+ εξ1,x1] and [x2,x2+δξ2], as shown in Figure4.2. In this case, the constraint (4.1) givesS([x1+ εξ1,x1]∪ [x1,x2])+S([x1,x2]∪ [x2+δξ2])≥ S([x1+ εξ1,x1]∪ [x1,x2]∪ [x2+δξ2])+S([x1,x2])=⇒ S(x1+ εξ1,x2)+S(x1,x2+δξ2)−S(x1+ εξ1,x2+δξ2)−S(x1,x2)≥ 0Expanding to first order in both δ and ε , this givesξα1 ξβ2 ∂1α∂2βS(x1,x2)≤ 0 .Since this constraint is linear in the spacelike vectors ξ1 and ξ2, it is sufficient torequire that the constraint be satisfied in the lightlike limit of ξ1 and ξ2, i.e. whenξ1 and ξ2 lie along the dotted lines in Figure 4.2. Thus, a minimal set of strongsubadditivity constraints that imply all constraints for connected regions is∂ 1+∂2+S(x1,x2)≤ 0 ∂ 1+∂ 2−S(x1,x2)≤ 0 ∂ 1−∂ 2+S(x1,x2)≤ 0 ∂ 1−∂ 2−S(x1,x2)≤ 0 .6Essentially the same argument works in general dimensions to show that the full set of strongsubadditivity constraints are implied by considering the constraint (4.1) where B is an arbitrary regionand where A and C are taken to be infinitesimal.88In the special case of states invariant under spacetime translations, the entan-glement entropy for an interval can only depend on the difference between the end-points so S(x1,x2) = S(x2− x1). In this case, the basic constraints may be writtenas7∂+∂+S(x)≤ 0 ∂−∂−S(x)≤ 0 ∂+∂−S(x)≤ 0 ∂−∂+S(x)≤ 0; . (4.2)Only the latter two constraints here are saturated for the vacuum state, so we expectthese will provide more useful constraints.Finally, in the case of a Lorentz-invariant state, the entanglement entropy candepend only on the proper length of the interval, so is described by a single functionS(R). In this case, the constraints reduce tod2SdR2± 1RdSdR≤ 0 , (4.3)where the first two constraints in (4.2) give the − sign and the latter two give the+ sign. In particular, the constraint with the + sign (which is saturated for vacuumstates) is equivalent toc′(R)≤ 0 c(R)≡ R dSdR.This was shown by Casini and Huerta [101] in their proof of the c-theorem usingstrong-subadditivity.Positivity and Monotonicity of Relative EntropyA very general class of constraints on the entanglement structure of a quantumsystem are related to relative entropy. This gives a measure of distinguishability ofa density matrix ρ to a reference state σ , defined asS(ρ||σ) = tr(ρ logρ)− tr(ρ logσ) .7Similar constraints were noted in [42], which appeared while the current version of this paperwas in preparation.89Relative entropy is always positive, increasing from zero for identical states ρ andσ to infinity for orthogonal states. Furthermore, for reduced density matrices ρAand σA obtained by a partial trace operation from ρ and σ , we haveS(ρA‖σA)≤ S(ρ‖σ). (4.4)This decrease in ρ under restriction to a subsystem is known as the monotonicityof relative entropy, or the data processing inequality [14].It is useful to define the modular Hamiltonian associated with the referencestate as Hσ = − log(σ), in analogy with thermodynamics. Using this, and thedefinition S(ρ) = − tr(ρ log(ρ)) for entanglement entropy, we can rewrite the ex-pression for relative entropy asS(ρ||σ) = tr(ρ logρ)− tr(σ logσ)+ tr(σ logσ)− tr(ρ logσ)= 〈− logσ〉ρ −〈− logσ〉σ −S(ρ)+S(σ)= ∆〈Hσ 〉−∆S. (4.5)For nearby states, ρ−σ = εX with ε  1 and X an arbitrary traceless Hermi-tian operator, one can expand relative entropy in powers of ε . To the first order in εrelative entropy vanishes. This is typically referred to as the first law of entangle-ment since it implies δ 〈Hσ 〉 = δS. The expression at second order in ε is knownas Fisher information, and is discussed in detail in Section 4.3.The rewriting in (4.5) becomes useful in cases where we can compute the mod-ular Hamiltonian Hσ . Generally this is possible when the reference state is thermalwith respect to some Hamiltonian. For example, the density matrix for a half-space in the vacuum state of a Lorentz-invariant field theory on Minkowski spaceis thermal with respect to the Rindler Hamiltonian (boost generator), so we haveHmod = c∫ddxxT00. The cases we consider below can all be obtained by conformaltransformations from this example [102, 100].For a ball shaped region in the vacuum state of a CFT on Rd,1, we have [100]HB = 2pi∫|x|<RddxR2−|x|22RT CFT00 . (4.6)90For a ball-shaped region in the vacuum state of a CFT on a sphere, we haveHB = 2pi∫Bddxcos(θ)− cos(θ0)sin(θ0)T00 . (4.7)In the special case of 1+1 dimensional CFTs the modular Hamiltonian can alsobe calculated for thermal states. For a spatial interval [−R,R] in an unboostedthermal state with temperature T = β−1, the modular Hamiltonian isHB =2βsinh(2piRβ) ∫ R−Rdxsinh(pi(R− x)β)sinh(pi(R+ x)β)T00(x) , (4.8)We can also obtain the expression for the modular Hamiltonian of an interval in aboosted thermal state. This is derived in Appendix B.Optimal relative entropy constraints for a family of reference statesIn various situations, we may have a family of reference states σα depending onparameters αi (e.g. temperature), and we would like to find the strongest rela-tive entropy constraint coming from this family. We will assume that the modularHamiltonians for these reference states take the form of an integral over linearcombination of local operators with α-dependent coefficients,Hα =∫ddx fn(x,α)On(x) . (4.9)According to the entanglement first law, under first order variation of the referencestate σα , the entanglement entropy of this state changes asδSα =∫ddx fn(x,α)δ 〈On(x)〉α .Here the right side corresponds to the variation in the expectation value of themodular Hamiltonian for the reference state under a variation of the state (whilekeeping the modular Hamiltonian fixed). Using this result and the definition (4.5),we haveδS(ρ||σα) = δ{〈Hβ 〉ρ −〈Hβ 〉σβ −S(ρ)+S(σβ )}91=∫ddxδ fn(x,α)[〈On(x)〉ρ −〈On(x)〉σα ] . (4.10)Thus, the relative entropy will be extremized with respect to parameters αi if wecan choose a reference state such that∫ddx∂ fn(x,α)∂αi[〈On(x)〉ρ −〈On(x)〉σα ]= 0 . (4.11)In the special case where the initial state and reference states are translation invari-ant, this becomes∂ In(α)∂αi[〈On〉ρ −〈On〉σα ]= 0 , (4.12)whereIn(α) =∫ddx fn(x,α) .so we see that an extremum will be obtained if we can choose a reference state withthe same expectation value as our state for each of the operators,〈On〉ρ = 〈On〉σα (4.13)The same state will also be provide an extremum for the monotonicity constraint,since if R parameterizes a region whose size increases with R,ddαiddRS(ρ||σα) = ∂2In(α,R)∂R∂αi[〈On〉ρ −〈On〉σα ]= 0 .Thus the reference state σα∗ whose operator expectation values match the state ρwill also give the minimum dS(ρ||σα)/dR (and thus the strongest monotonicityconstraint), assuming that the extremum is a minimum.8The matching of operator expectation values and the form (4.9) of the Hamil-tonian implies that ∆〈Hα∗〉 = 0, so in this case, the constraint from positivity and8In practice, we should also check whether other extrema exist, and check the boundary of theparameter space. However, since the relative entropy provides a measure of how close our state is tothe reference state, it is plausible that the relative entropy is minimized by matching the expectationvalues of operators. For the cases below, we have explicitly checked that this is the case using theexplicit form of the modular Hamiltonian.92monotonicity of relative entropy are simply that9S(ρR)−S(σα∗R )≤ 0ddR(S(ρA)−S(σα∗A ))≤ 0. (4.14)4.2.2 Holographic formulae for entanglement entropyIn this paper, we consider general theories of gravity dual to holographic QFTssuch that the leading order (in the 1/N expansion) entanglement entropy for spatialregions of the field theory is computed by the Ryu-Takayanagi formula [6], or itscovariant generalization [9]. This states that the entanglement entropy of a regionA is given byS(A) =Area(A˜)4GN,where A˜ is the extremal surface in the dual geometry with ∂ A˜= ∂A (i.e. such that Aand ∂A have the same boundary). The surface A˜ is also required to be homologousto A, and in cases where multiple extremal surfaces exist, it is the extremal surfacewith least area.The Ryu-Takayanagi formula receives quantum corrections from the entangle-ment entropy of bulk quantum fields, but we consider only the classical limit in thispaper. We note also that for theories of gravity with higher powers of curvature orhigher derivatives, the entropy is computed using a more complicated functionalthan area. However, we restrict attention in this paper to theories for which thegravitational sector is Einstein gravity.4.2.3 Energy conditionsTo end this section, we briefly review a few of the standard energy conditionsdiscussed in the gravitational literature. These are statements about the stress-energy tensor that are taken to be plausibly true, but which are generally not derivedfrom any underlying quantum theory.10 The weak energy condition states that theenergy density in any frame of reference must be non-negative. Specifically, if uµ9A similar simplification of relative entropy was noted in [103] when considering the problem offinding entropy-maximizing states consistent with local data.10See [104] for a recent argument for the null-energy condition based on perturbative string theory.93is a timelike vector, thenTµνuµuν ≥ 0 .The null energy condition takes the same form, but with u is taken to be a nullvector. This is implied by the weak energy condition.Various authors have also considered averaged energy conditions, in which theconditions are only required to hold when averaged over some geodesic or spatialregion. This is the type of contraint that we will find below.4.3 Constraints on spacetimes dual to Lorentz-invariant1+1D field theoriesIn this section, we consider Lorentz-invariant holographic two-dimensional fieldtheories that flow from some CFT in the UV to another CFT in the IR. For suchtheories, the vacuum state is dual to a spacetime of the form11ds2 =F2(r)r2dr2+ r2(−dt2+dx2) , (4.15)where F(r) approaches constants both at r= 0 and at r=∞ (giving AdS geometriescorresponding to the IR and UV fixed points).12 We would like to understand theconstraints on the function F(r) that arise from entanglement inequalities in theCFT. Specifically, we consider the constraints arising from strong subadditivity.For any spacelike interval, Lorentz-invariance implies that the entanglemententropy depends only on the proper length of the interval, so entanglement entropyfor connected regions is captured by a single function S(R). As we reviewed inSection 4.2, Casini and Huerta have shown [101] starting from strong subadditivitythat the function c(R) = dS/d(ln(R)) = RdS/dR obeys c′(R) ≤ 0. The functionc(R) therefore decreases monotonically for increasing R, which leads immediatelyto the Zamolodchikov c-theorem, since c(R) reduces to the UV and IR centralcharge for small and large R respectively.The holographic version of the statement c′(R) ≤ 0 was obtained previously11In special cases, there may be additional compact directions in the dual spacetime. In thesecases, we consider the KK-modes of the metric and other fields as part of the matter sector.12This choice of coordinates assumes that the spatial scale factor is monotonic in the radial direc-tion. At the end of this section, we comment on the case where this doesn’t hold.94in [97], but we review the calculation here since we will be generalizing this inthe next section. Using the Ryu-Takayanagi formula, the entanglement entropy foran interval of length R in the geometry (4.15) is obtained by the minimum of theactionS =∫dλ√F2(r)r2(drdλ)2+ r2(dxdλ)2(4.16)with boundary conditions (r(λi),x(λi)) = (rmax,0) and (r(λ f ),x(λ f )) = (rmax,R),where rmax is a regulator that we will take to infinity. In Appendix C, we derivea general formula for the variation of the entanglement entropy under a variationin the endpoints of the interval for translation-invariant geometries. For the caseof variations in the size of spatial interval, the result (derived previously in [97]) isthat dSdR equals the minimum spatial scale factor reached by the RT surface. Thus,for our choice of coordinates,dSdR= r0 c(R) = r0R . (4.17)To find an explicit relation between r0 and R (and check that r0 has a well-defined limit as we remove the regulator), we note that the equation for curves x(r)extremizing the action (4.16) isddr r2 dxdr√F(r)2r2 + r2(dxdr)2= 0 .In terms of the r0, the value of r where dr/dx vanishes, we have(dxdr)2=F2(r)r4(r2r20−1) . (4.18)Thus, we obtainR = 2∫ ∞r0drF(r)r21√r2r20−1= 2∫ ∞1dxF(r0x)r0x2√x2−1 . (4.19)95We can now translate the strong-subadditivity condition c′(R) ≤ 0 to a conve-nient bulk expression. Starting from the relation (4.17), we have thatddRc(R) =dr0dRddr0(Rr0) =d2SdR2∫ ∞1dxF ′(rx)x√x2−1 (4.20)Strong subadditivity implies that13dr0dR=d2SdR2≤ 0 , (4.21)so we have finally that ddR c(R)≤ 0 is equivalent to the condition on F(r) that∫ ∞r0drF ′(r)r√r2r20−1≥ 0 (4.22)for every r0. This result was derived originally in [97].4.3.1 An averaged null energy conditionWe will now show that the condition (4.22) can be interpreted as a particular aver-aged null energy condition in this geometry.We start by considering the light sheet emanating from the curve B, pointingin the forward direction in time with light rays going towards the boundary. Wecan define a null vector field on B directly along this lightsheet by the conditionsthat u · u = 0, u · ∂λ xB = 0 and uµ∂µr = 1. Here, the scale factor r can be definedas r =√ξ ·ξ , where ξ is the Killing vector corresponding to spatial translationsalong the field theory direction. In our coordinates, we have(ut ,ur,ux) =F(r)rr0,−1,±F(r)√r2− r20r0r2 .13To see this, apply the strong subadditivity constraint (4.1) to the case where B is an interval oflength R and A and C are intervals of length δR to the left and right. Then strong subadditivityimplies that 2S(R+ δR)− S(R)− S(R+ 2δR) ≥ 0 which gives S′′(R) ≤ 0 in the limit δR → 0.Holographically, this implies that Ryu-Takayanagi surfaces for larger intervals must penetrate deeperinto the bulk.96Physically, this null vector field is normalized so that translation by the vector fieldproduces the same (additive) change in the scale factor everywhere.Defining Tµν to be the stress tensor giving rise to the geometry (4.15) via Ein-stein’s equations, we find thatTµνuµuν ∝F ′(r)rF(r),where we have used that the Einstein tensor in our geometry isGrr =1r2Gtt =−Gxx = r3F(r)3F ′(r)− r2F(r)2.From (4.15) and (4.18) the distance element along an RT curve B with minimalradial coordinate r0 is given byds =drF(r)r0√r2r20−1It follows that the condition (4.22) is equivalent to the condition that for every RTcurve B ∫BTµνuµuνds≥ 0. (4.23)Thus, the positivity of Casini and Huerta’s entanglement c-function is equivalentin holographic theories (at the classical level) to this averaged null-energy con-dition.14 This is clearly implied by the null energy condition, but is a weakercondition, since it is possible for Tµνuµuν ≤ 0 to be negative locally while all theintegrals are positive.We can give an alternative statement of the energy condition in terms of aglobally defined null vector field uˆ, defined by replacing the condition u ·∂λ xB = 0with u ·ξ = 0, where ξ is the spatial Killing vector. In our coordinates, (uˆt , uˆr, uˆx) =(F(r)/r2,1,0). Physically, this null vector field is defined so that it points only inthe radial and time directions, and so that translation by the vector field produces14This is not equivalent to what is usually called the averaged null energy condition, which in-volves an average over null geodesics.97the same (additive) change in the scale factor everywhere. In terms of this nullvector, the energy condition is also expressed as (4.23). In this case, the condition(4.23) may be expressed by saying that the “Radon transform”15 of Tµν uˆµ uˆν iseverywhere non-negative.4.3.2 Non-monotonic scale factorsThe coordinate choice (4.15) assumed the scale factor to be monotonic in the radialcoordinate. In this section, we briefly consider the case where it is not. Here, wecan choose coordinatesds2 = dr2+a(r)2(−dt2+dx2) . (4.24)Asymptotically, a(r) must be increasing, but suppose that a′(r) < 0 in some in-terval with upper bound rc, such that a′(rc) = 0. Note that any such geometryviolates the null energy condition d2/dr2(ln(a)) ≤ 0 which forbids local minimaof a. However, we would like to understand whether such a geometry can stillsatisfy the constraints coming from strong subadditivity.It is straightforward to check that a′(rc) = 0 implies that r = rc is an extremalsurface, so as r0 approaches rc, there will be a family of extremal surfaces end-ing on boundary intervals whose length diverges. These extremal surfaces are re-stricted to the region r ≥ rc, so their regulated length will scale with the intervalsize R in the limit of large R. This is inconsistent with our assumption that the IRphysics is some conformal fixed point, so it must be that beyond some R∗, these ex-tremal surfaces are no longer minimal. Let a1 = limR→R−∗ a(r0(R)) be the minimalvalue of a attained by this branch of extremal surfaces.In the present coordinates, the equations for an extremal surface penetrating tosome minimum radial value r0 are(drdx)2= a2(r)(a2(r)a2(r0)−1).Thus, we see that only when a(r0) = minr≥r0 a(r) can an extremal surface reach15Here we mean the map from a function on a space to a function on the space of geodesic curvesobtained by integrating the original function over the curve.98the boundary. Otherwise, the previous equation would imply some negative valuefor( drdx)2at locations where a(r) < a(r0). Thus, the branch of extremal surfaceswhich become minimal for R > R∗ have r0 greater than the value where a(r) againdecreases past a(rc). Let a2 be the maximal value of a for this R > R∗ branch ofsolutions. We see that a2 < a1.Using the result (4.17) in the previous section, we havedSdR= a(r0) c(R) = Ra(r0)so we see that non-monotonic scale factors, the entanglement c-function is discon-tinuous, jumping from R∗a1 to R∗a2 at R = R∗. This was emphasized previously in[97].Despite the discontinuous behavior of the RT-surfaces, the constraint frommonotonicity of the c-function can still be expressed as (4.23), as we can showby repeating the calculations from the previous section in the coordinates (4.24).In this case, the constraint applies only to the extremal surfaces with minimal area.4.4 Constraints on spacetimes dual to states of 1+1DCFTsIn this section, we place restrictions on translation and time-translation invariantspacetimes dual to states of 1+1 dimensional holographic CFTs on Minkowskispace.4.4.1 Constraints from positivity and monotonicity of relativeentropyWe start by considering constraints arising from the positivity and monotonicity ofrelative entropy for spacelike intervals.For our CFT state Ψ, we can choose to work in a frame of reference wherethe stress tensor is diagonal. We consider the density matrices ρI for a spacelikeinterval I from (0,0) to (Rx,Rt). We will compare these to the density matricesσβ ,vT calculated from a reference state, which we take to be a boosted thermal statewith temperature β and boost parameter v. For these states the relative entropy99S(ρT ||σβ ,vT ) must be positive and increase with the size of the interval,δ+I S(ρI||σβ ,vI )≥ 0 (4.25)where δ+I represents a deformation (Rx,Rt)→ (Rx+δx,Rt +δ t) that increases theproper length of the interval. Note that positivity follows from this monotonicitycondition since the relative entropy is zero for a vanishing interval.According to the result (4.14) and the discussion in that section, the optimalrelative entropy constraints will be obtained by choosing the reference state pa-rameters (β ,v) such that the stress tensor of the boosted thermal state matches thestress-tensor of our state. This requires v = 0 and β = β ∗ such that the energydensity of the thermal state matches that of our state. From (4.14) the optimalmonotonicity constraint reduces simply toδ+I{S(ρI)−S(σβ∗I )}≤ 0 (4.26)A general expression for the variation of the holographic entanglement entropyunder a variation in the interval is given in Appendix C. The result is:δ+I S = δx[Ax0γ0]−δ t[At0γ0β0] (4.27)where A0x and A0t are the spatial and temporal scale factors at the deepest pointr0 on the extremal surface, defined for a general diagonal choice of the metric byA0x =√gxx(r0) and A0t =√−gtt(r0), and γ0 = (1−β 20 )− 12 with β0 = (Atdt)/(Axdx)measuring the “tilt” of the geodesic at the point r0.Using this result, the monotonicity constraint may be expressed asδx{[Ax0γ0]I− [Ax0γ0]β∗I}−δ t{[At0γ0β0]I− [At0γ0β0]β∗I}≤ 0 (4.28)where ∆ refers to difference between our state and the reference thermal state withthe same stress-tensor expectation values. Here we require δx> 0 and |δ t| ≤ δx, sothe strongest constraint will either be for δ t = δx or δ t =−δx. Thus, an equivalentstatement is∆[γ0(Ax0±β0At0)]I ≤ 0 , (4.29)100where ∆ refers to the result for our state minus the result for the thermal state.Spatial constraintIt is interesting to write the our constraint more explicitly for the special case of aspatial interval. We choose coordinates for which the metric takes the formds2 =F2(r)r2dr2+ r2dx2− r2G2(r)dt2 , (4.30)so that the radial coordinate measures the spatial scale factor. In this case, thegeodesics lie on constant time slices, so β0 = 0, γ0 = 1, and the constraint (4.28)givesr0(R)≤ rβ∗0 (R) , (4.31)Thus, the monotonicity of relative entropy constraint for spatial intervals is equiva-lent to the statement that the minimum scale factor reached by an extremal surfacein the geometry associated with |Ψ〉 is never less than the value in the thermal stategeometry with the same 〈T00〉.Since r is a decreasing function of R according to (4.21), the condition (4.31)is equivalent toR(r0)≤ Rβ (r0) , (4.32)Using the coordinates (4.30) and the result (4.19), we can express this as∫ ∞1dx1x2√x2−1(F(r0x)−Fβ (r0x))≤ 0 . (4.33)As we show in the next section, this constraint agrees asymptotically with the con-dition of positive energy T00 ≥ 0.More generally, we can show that the condition (4.33) is implied by but doesnot imply the constraint of positive energy. To see this, we note that F(∞) =Fβ (∞) = 1 and that for large r, F(r)−Fβ (r) = ar−n +O(r−(n+1)) with n ≥ 3. Inour coordinates, the positive energy constraint gives rF ′(r)−F(r)+F3(r)≥ 0 withequality for Fβ (r) describing the thermal state. Thus,(F−Fβ )′ ≥1r(Fβ −F)(F2β +FβF +F2−1) .101To leading order in large r this is a(n−2)≤ 0, so that F(r)−Fβ (r) must initiallydecrease below zero as we move in from r = ∞. Then since Fβ (r) ≥ 1, (F(r)−Fβ (r))′ ≥ 0 and F(r)−Fβ (r) must continue to decrease as r decreases, ensuringthat (4.33) holds.Asymptotic ConstraintsIt is interesting to work out the implications of the relative entropy constraint(4.29) on the asymptotic geometry of the spacetime. For this purpose, we chooseFefferman-Graham coordinatesds2 =1z2(dz2+ f (z)dx2−g(z)dt2) . (4.34)To apply the constraint (4.29) we need an expression relating the parameters β0,Ax0, and At0 to the parameters (Rx,Rt) describing the boundary interval. Startingfrom the area functionalArea(B˜) =∫ dzz√1−g(z)(dtdz)2+ f (z)(dxdz)2, (4.35)we find that the surface is extremal ifddz f (z)dxdzz√1−g(z)( dtdz)2+ f (z)(dxdz)2 = 0ddz g(z) dtdzz√1−g(z)( dtdz)2+ f (z)(dxdz)2 = 0 . (4.36)Let z0 be the maximum value of z reached by the surface, and define as aboveβ0 =√g(z0)f (z0)dtdx(z = z0) ,102such that |β0|< 1 for a spacelike path. In terms of these parameters, we get(dxdz)2=z2 f0z20 f 21[1− z2 f0z20 f]−β 20[1− z2g0z20g](dtdz)2= β 20z2g0z20g21[1− z2 f0z20 f]−β 20[1− z2g0z20g] (4.37)where we have defined f0 = f (z0) and g0 = g(z0). Using these, we obtainRx =∫ z00dzz√f0z0 f1√[1− z2 f0z20 f]−β 20[1− z2g0z20g]Rt =∫ z00dzzβ0√g0z0g1√[1− z2 f0z20 f]−β 20[1− z2g0z20g] (4.38)To understand the asymptotic constraints, we can write f and g asymptotically as16f (z) = 1+ z2 f2+ z3 f3+ z4 f4+ . . . g(z) = 1− z2 f2+ z3g3+ z4g4+ . . . .(4.39)where we have used tracelessness of the CFT stress tensor to conclude that[g]z2 +[ f ]z2 ∝ 〈−Ttt +Txx〉= 0 .Defining the proper length L =√R2x−R2t and v = Rt/Rx, we can use (4.38) toexpress L and v as power series in z0 with β0-dependent coefficients. Invertingthese, we can express z0 and β0 as power series in L with v-dependent coefficients.Finally, we can write the expressionδIS = γ0(Ax0±β0At0) =1√1−β 20(√f (z0)z0+β0√g(z0)z0)16Note that in purely gravitational solutions, f3 and g3 vanish, but more generally, these could besourced by another bulk field corresponding to an operator with sufficiently low dimension.103appearing in (4.29) as a power series in L with v-dependent coefficients. Here wehave chosen the plus sign in (4.29) without loss of generality, since the constraintis invariant under a swap of the sign and v− > −v. The monotonicity constraintimplies a negative difference between this expression for general f and g and theexpression with the thermal state valuesfβ ∗ = 1+ f2z2+14f 22 z4 gβ ∗ = 1− f2z2+14f 22 z4 .Since we are working in the limit of small L, the negativity implies that the leadingorder nonzero terms in the power series must have a negative coefficient.In the case where f3 and g3 are nonzero, the leading order term is at order L2,and negativity of the coefficient gives:v(3v−2)g3+(2v−3) f3 ≥ 0This is required to be true for all |v| < 1 (corresponding to the tilt of the interval),and we find that the combination of these conditions is equivalent tof3 ≤ g3 f3 ≤ 3√5−72g3 ≈−0.1459g3 (4.40)In the case where f3 and g3 vanish, the constraint becomes the positivity of the L3term, which givesv(2v−1)(g4− 14 f22 )+(v−2)( f4−14f 22 )≥ 0Again, this is required to be true for all |v|< 1, and the combination of constraintsgivesf4 ≤ g4 ( f4− 14 f22 )≤ (4√3−7)(g4− 14 f22 )≈−.07178(g4−14f 22 )(4.41)Comparison with standard energy conditionsWe can compare our results to the standard weak and null energy conditions Tµνuµuν ≥0 for various timelike or null vectors u. The non-vanishing components of the stress10442222gf33gf (f  /4,f  /4)4Figure 4.3: Relative entropy constraints on coefficients in the Fefferman-Graham expansion of the metric (striped region). Constraints on theright apply only if f3 = g3 = 0. Dark blue shaded region are the con-straints from the null-energy condition. Full shaded region correspondsto constraints from positivity of relative entropy, equivalent to con-straints from the weak energy condition for timelike vectors with nocomponent in the radial direction.tensor areTzz = − 12zg′g− 12zf ′f+14f ′fg′gTtt =g4z(2f ′f+ z(f ′f)2−2z f′′f)Txx = − f4z(2g′g+ z(g′g)2−2zg′′g)(4.42)Assuming that f3 and g3 are nonzero, the weak energy condition applied to timelikevectors with no radial component (i.e. the non-negativity of energy for observersmoving in the field theory directions) givesf3 ≤ g3 f3 ≤ 0 , (4.43)while including uµ in the radial direction strengthens the conditions tof3 ≤ g3 f3 ≤−12g3 . (4.44)105When f3 = g3 = 0, the weak energy condition applied to timelike vectors with noradial component givesf4 ≤ g4 f4− 14 f22 ≤ 0 , (4.45)while the full weak/null energy condition givesf4 ≤ g4 f4− 14 f22 ≤−13(g4− 14g22) . (4.46)The conditions (4.40) and (4.41) coming from monotonicity of relative entropyare intermediate between the weak/null energy condition considering only u in thefield theory directions and the conditions for general u. An interesting point is thatthe weaker conditions (4.43) and (4.45) are exactly equivalent to the conditionsobtained by positivity of relative entropy (without demanding monotonicity).4.4.2 Constraints from strong subadditivityWe now consider the constraints arising from the strong subadditivity of entangle-ment entropy. For a state invariant under spacetime translations, the entanglemententropy for any spacelike interval will be a single function S(Rx,Rt) where (Rx,Rt)represents the difference between the two endpoints. According to the discussionin Section 4.2, the requirements of strong subadditivity in this case are impliedby the minimal set of strong subadditivity constraints (4.2). In these formulae,we have defined R± = Rx±Rt . To obtain explicit expressions for these, we canevaluate the first derivatives using the result (4.27). We have∂±S = γ0(Ax0∓β0At0) (4.47)where At , Ax, β0, and γ0 are defined in the previous subsection. From here, we canwrite the constraints (4.2) explicitly by taking one more derivative. For example,we have∂+∂−S =∂∂R+[γ0(Ax0+β0At0)]=∂ r0∂R+∂∂ r0[γ0(Ax0+β0At0)]+∂β0∂R+∂∂β0[γ0(Ax0+β0At0)]106=1∆{−∂R−∂β0∂∂ r0[γ0(Ax0+β0At0)]+∂R−∂ r0∂∂β0[γ0(Ax0+β0At0)]}where∆= det ∂R−∂ r0 ∂R−∂β0∂R+∂ r0∂R+∂β0 .The strong subadditivity constraint is then that ∂+∂−S ≤ 0. Here, the determinant∆ is positive for geometries in some neighborhood of pure AdS (and possibly moregenerally); in this case, the constraint simplifies to the statement that the expressionin curly brackets is non-positive.We can write an explicit expressions for R− and R+ using the steps leading to(4.38). We findR± =∫γdsγ0{Ax0(Ax)2±β0 At0(At)2}(4.48)where the integral is along the extremal surface, with length elementds =dr√grrγ0√[1− (Ax(r0))2(Ax(r))2]−β 20[1− (At(r0))2(At(r))2] .From this, the constraint ∂+∂−S≤ 0 for each spacelike interval I can be expressedas an integral over the extremal curve γ ending on I. It is natural to expect that theresult can be expressed in a covariant form similar to (4.23), but we leave this forfuture work.Asymptotic constraintsUsing the tools from Section 4.4.1, it is straightforward to work out the con-straints on the asymptotic geometry implied by the strong subadditivity constraint∂+∂−S≤ 0. Note that the conditions ∂+∂+S≤ 0 and ∂−∂−S≤ 0 are always satisfiedasymptotically.We work again in the Fefferman-Graham expansion (4.34) with metric func-tions expanded as (4.39). We can write the expression (4.47) as a power seriesin the proper length L of the interval, with coefficients depending on the ratio107β = Rt/Rx and the coefficients appearing in (4.39). Acting with∂+ =∂L∂R+∂L+∂v∂R+∂v=12√1− v1+ v{∂L+(1− v2)1L∂v}gives a power series for ∂+∂−S, and the strong subadditivity constraint implies thatthe leading non-zero coefficient must be negative.In the case where f3 and g3 are nonzero, the leading order term is at order L,and negativity of the coefficient gives:(2−7v2)g3 ≤−(7−2v2) f3This is required to be true for all |v| < 1 (corresponding to the tilt of the interval),and we find that the combination of these conditions is equivalent tof3 ≤ g3 f3 ≤−27g3 (4.49)In the case where f3 and g3 vanish, the constraint becomes the negativity of the L2term, which gives(1−7v2)(g4− 14 f22 )≤−(7− v2)( f4−14f 22 )≥ 0 (4.50)Again, this is required to be true for all |v|< 1, and the combination of constraintsgivesf4 ≤ g4 ( f4− 14 f22 )≤−17(g4− 14 f22 ) (4.51)These constraints take a similar form to the constraints (4.40) and (4.41) frommonotonicity of relative entropy, but are slightly stronger. However, they are stillweaker than the constraints (4.44) and (4.46) arising from the null energy condi-tion.1084.5 Constraints on spherically-symmetric asymptoticallyAdS spacetimesIn this section, we point out a simple constraint on the geometries of static, spheri-cally symmetric asymptotically AdSd+2 spacetimes. This would apply for exampleto spherically symmetric “stars” made of any allowable type of matter in a theoryof gravity whose classical limit is Einstein gravity coupled to matter.For these spacetimes, the dual state is an excited state of the dual CFT on asphere with a homogeneous stress tensor. If the mass of the spacetime (relative toempty AdS) is M, the field theory energy is M` (taking the sphere radius equal toone for the CFT), so we can say that the energy density expectation value for thisstate relative to the vacuum state is∆〈T00〉= M`Ωd , (4.52)where Ωd is the volume of a d-sphere.Now, consider a ball-shaped region Bθ of angular radius θ0 on the sphere. Forthis region, the relative entropy for our state with respect to the vacuum state isSBθ (ρ||0) = ∆〈Hmod〉−∆S= 2pi∫BdΩdcos(θ)− cos(θ0)sin(θ0)∆〈T00〉−∆Swhere we have used the expression (4.7) for the modular Hamiltonian.Since the stress tensor (4.52) is constant on the sphere, we can perform theintegral explicitly to obtainSBθ (ρ||0) = −∆S+2piM`Ωd−1ΩdId(θ0)whereId(θ0)=∫ θ00dθ sin(θ)d−1cos(θ)− cos(θ0)sin(θ0)=(sinθ0)d−1d[1− 2F1(12,d2;d2+1;sin2 θ0)cosθ0].Then, using the Ryu-Takayanagi formula, the positivity of relative entropy gives109the constraint∆Area(θ0)≤ 8√piGNM`Id(θ0)Γ(d2 +12)Γ(d2) .where ∆Area is the area of the bulk extremal surface with boundary δBθ .For the special case of a hemisphere (θ0 = pi/2), we have that∆Area(pi/2)≤ 8√piGNM`Γ(d2 +12)dΓ(d2) .which reduces for 3+1 dimensional gravity to∆A≤ 2piGNM`AdS .Typically, the minimal area extremal surface bounded by an equator on the spherewill be the surface bisecting the spacetime symmetrically, so this constraint boundsthe change in area for this bisecting surface by the mass contained in the space-time.17 Roughly, the constraint places a bound on how much a certain amount oftotal energy in the spacetime can curve the spacetime.4.6 DiscussionIn this paper, we have explored constraints from entanglement inequalities onhighly symmetric spacetimes. It will be interesting to see how these results gener-alize to less symmetric cases. In our analysis, we have used only the classical termin the Ryu-Takayanagi formula, so our constraints apply to gravitational theoriesin the classical limit. It would be interesting to understand how the constraints arecorrected when the contribution of bulk quantum fields are taken into account. Thisshould be possible using the quantum-corrected holographic entanglement entropyformula proposed by [28].17In some cases, however, there may exist more than one extremal surface bounded by an equator,and in this case, the minimal area surface may not be the symmetrical one.1104.6.1 Constraints on entanglement structure from geometryBefore concluding, we offer a few remarks on the orthogonal research directionof understanding which entanglement structures are consistent with the existenceof a geometrical dual spacetime. In the language of Figure 4.1, we would like toprecisely characterize the image of G in S (or in (Sphys). Here, we make a fewqualitative observations that hopefully illuminate how severe these constraints are.Consider a general asymptotically AdSd+2 spacetime. In a Fefferman-Grahamdescription of the metric,ds2 =1z2[dz2+Γµν(z,x)dxµdxν]the information about the geometry is contained in the functions Γµν(z,x) of (d+1) variables.A set of entanglement entropies that includes a similar amount of informationas one of these functions is the set {S(R,x)} for ball-shaped regions with any ra-dius R centered at any point x. At least close to the boundary (where the geometryis similar to AdS), we expect that there is a one-to-one correspondence betweenpairs (R,x) and bulk points (z,xbulk), obtained by choosing the point on the RT sur-face with the largest value of z. For pure AdS, we have simply (z,xbulk) = (R,x).Thus, given the entanglement entropies for ball-shaped regions in one spatial slice,it is plausible that we can reconstruct some combination of the metric functionsΓµν(z,x). The other combinations are related by Lorentz-transformations, so it isfurther plausible that we can reconstruct the remaining functions (in some neigh-borhood of the boundary) by considering entanglement entropies for ball-shapedregions in other Lorentz frames.Assuming this reconstruction is possible, we now have enough information(the full geometry in a neighborhood of the boundary) to calculate entanglemententropies for regions of any other shape. Thus, it is plausible that for a quan-tum state with gravity dual, the entanglement entropies for regions of arbitraryshape (assuming they are not too large) are completely determined from the en-tanglement entropies for ball-shaped regions (in the various frames of reference).Furthermore, they are determined in a very specific way, via construction of a dualgeometry and calculation of extremal surface areas. A natural question is then to111understand which field theory Hamiltonians can give rise to low-energy states withthis entanglement structure, and/or why the known examples of holographic CFTshave this property.112Chapter 5Entanglement Entropy ofHolographic States in Terms ofOne-point Functions5.1 IntroductionIn holographic conformal field theories, states with a simple classical gravity dualinterpretation have a remarkable structure of entanglement: according to the holo-graphic entanglement entropy formula [6, 7, 9], their entanglement entropies forarbitrary regions (at leading order in large N) are completely encoded in the ex-tremal surface areas of an asymptotically AdS spacetime. In general, the space ofpossible entanglement entropies (functions on a space of subsets of the AdS bound-ary) is far larger than the space of possible asymptotically AdS metrics (functionsof a few spacetime coordinates), so this property of geometrically-encodable en-tanglement entropy should be present in only a tiny fraction of all quantum fieldtheory states [35]. It is an interesting question to understand better which CFTstates have this property1, and which properties of a CFT will guarantee that fami-1Even in holographic CFTs, it is clear that not all states will have this property. For example, if|Ψ1〉 and |Ψ2〉 are two such states, corresponding to different spacetimes MΨ1 and MΨ2 , the super-position |Ψ1〉+ |Ψ2〉 is not expected to correspond to any single classical spacetime but rather to asuperposition of MΨ1 and MΨ2 . Thus, the set of “holographic states” is not a subspace, but some113lies of low-energy states with geometric entanglement exist.For a hint towards characterizing these holographic states, consider the gravityperspective. A spacetime MΨ dual to a holographic state |Ψ〉 is a solution to thebulk equations of motion. Such a solution can be characterized by a set of initialdata on a bulk Cauchy surface (and appropriate boundary conditions at the AdSboundary). The solution away from the Cauchy surface is determined by evolvingthis initial data forwards (or backwards) in time using the bulk equations. Alter-natively, we can think of the bulk solution as being determined by evolution in theholographic radial direction, with “initial data” specified at the timelike boundaryof AdS. In this case, the existence and uniqueness of a solution is more subtle, butthe asymptotic behavior of the fields determines the metric at least in a perturba-tive sense (e.g. perturbatively in deviations from pure AdS, or order-by-order inthe Fefferman-Graham expansion). It is plausible that in many cases, this bound-ary data is enough to determine a solution nonperturbatively to some finite distanceinto the bulk, or even for the whole spacetime. Thus, for geometries dual to holo-graphic states, we can say that the bulk spacetime (at least in a perturbative sense)is encoded in the boundary behavior of the various fields.According to the AdS/CFT dictionary, this boundary behavior is determined bythe one-point functions of low-dimension local operators associated with the lightbulk fields. On the other hand, the bulk spacetime itself allows us to calculatedentanglement entropies (and many other non-local quantities). Thus, the assump-tion that a state is holographic allows us (via gravity calculations) to determine theentanglement entropies and other non-local properties of the state (again, at leastperturbatively) from the local data provided by the one-point functions:|Ψ〉 → 〈Oα(xµ)〉 → φα asymptotics→ φα(xµ ,z)→ entanglement entropies S(A)(5.1)where φ here indicates all light fields including the metric.2The recipe (5.1) could be applied to any state, but for states that are not holo-graphic, the results will be inconsistent with the actual CFT answers. Thus, wegeneral subset.2Here, the region A should be small enough so that the bulk extremal surface associated with Ashould be contained in the part of the spacetime determined through the equations of motion by theboundary values; we do not need this restriction if we are working perturbatively.114have a stringent test for whether a CFT state has a dual description well-describedby a classical spacetime: carry out the procedure in (5.1) and compare the resultswith a direct CFT calculation of the entanglement entropies; if there is a mismatchfor any region, the state is not holographic.3In this paper, our goal is to present some more explicit results for the gravityprediction SgravA (〈Oα〉) in cases where the gravitational equations are Einstein grav-ity with matter and the region is taken to be a ball-shaped region B. We will workperturbatively around the vacuum state to obtain an expression as a power seriesin the one-point functions of CFT operators. At first-order, the result depends onlyon the CFT stress tensor expectation value [102]:SB(|Ψ〉) = SvacB +2pi∫Bdd−1xR2− r22R〈T00〉+O(〈Oα〉2) . (5.2)This well-known expression is universal for all CFTs since it follows from the firstlaw of entanglement δ (1)SB = δ 〈HB〉, whereHB ≡− logρvacB = 2pi∫Bdd−1xR2− r22RT00 (5.3)is the vacuum modular Hamiltonian for a ball-shaped region. Thus, to first-order,the gravity procedure (5.1) always gives the correct CFT result for ball-shapedregions, regardless of whether the state is holographic.General second-order result for ball entanglement entropyOur focus will be on the second-order answer; in this case, it is less clear whetherthe gravity results from (5.1) should hold for any CFT or whether they represent aconstraint from holography. To obtain explicit formulae at this order, we begin bywritingSB(|Ψ〉) = SvacB +∆〈HB〉−S(ρB||ρvacB ) (5.4)which follows immediately from the definition of relative entropy S(ρB||ρvacB ) re-viewed in Section 5.2 below. We then make use of a recent result in [105]: to3Another interesting possibility is that the one-point functions could give boundary data that isnot consistent with any solution of the classical bulk equations; this possibility exists since the “initialdata” for the radial evolution problem obeys certain constraints.115second-order in perturbations from the vacuum state, the relative entropy for aball-shaped region in a holographic state4 is equal to the “canonical energy” as-sociated with a corresponding wedge of the bulk spacetime. We provide a briefreview of this in Section 5.2 below. On shell, the latter quantity can be expressedas a quadratic form on the space of first-order perturbations to pure AdS spacetime,so we haveS(ρB||ρvacB ) = ∆〈HB〉−∆SB =12E (δφα ,δφα)+O(δφ 3) . (5.5)Rearranging this, we have a second-order version of (5.2):SB(|Ψ〉) = SvacB +δ (1)SB+δ (2)SB+O(δφ 3)= SvacB +∆〈HB〉−12E (δφα ,δφα)+O(δφ 3) (5.6)= SvacB +2pi∫Bdd−1xR2− r22R〈T00〉− 12E (δφα ,δφα)+O(δφ3) .As we review in Section 5.2 below, the last term can be written more explicitly asE (δφα ,δφα) =∫Σω(δg,£ξδg)−∫Σξ aT (2)ab εb , (5.7)where Σ is a bulk spatial region between B and the bulk extremal surface B˜ withthe same boundary, ω is the “presymplectic form” whose integral defines the sym-plectic form on gravitational phase space, T (2)ab is the matter stress tensor at second-order in the bulk matter fields, and ξ is a bulk Killing vector which vanishes onB˜. The first-order bulk perturbations δφα (including the metric perturbation) maybe expressed in terms of the boundary one-point functions via bulk-to-boundarypropagatorsδφα(x,z) =∫DBKα(x,z;x′)〈Oα(x′)〉 , (5.8)where DB is the domain of dependence of the ball B. Given the one-point functionswithin DB, we can use (5.8) to determine the linearized bulk perturbation in Σ andevaluate (5.7).The expression (5.6), (5.7), and (5.8) together provide a formal result for the4This second-order relative entropy is known as quantum Fisher information.116ball entanglement entropy of a holographic state, expanded to second-order in theboundary one-point functions.Explicit results for 1+1 dimensional CFTsIn order to check the general formula and provide more explicit results, we focusin Section 5.3 on the case of 1+1 dimensional CFTs, carrying out an explicit cal-culation of the gravitational contributions to (5.7) starting from a general boundarystress tensor. We find the resultδ (2)SgravB =−12∫B′dx+1∫B′dx+2 K2(x+1 ,x+2 )〈T++(x+1 )〉〈T++(x+2 )〉+{+↔−}(5.9)where the integrals can be taken over any spatial surface B′ with boundary ∂B, andthe kernel is given byK2(x1,x2) =6pi2cR2 (R− x1)2(R+ x2)2 x1 ≥ x2(R+ x1)2(R− x2)2 x1 < x2 , (5.10)where c is the central charge. In this special case, the conservation equations de-termine the stress tensor expectation values throughout the region DB from theexpectation values on B′, so as in the first-order result (5.2), our final expressioninvolves integrals only over B′. This will not be the case for the terms involvingmatter fields, or in higher dimensions. As a consistency check, we show that theexpression (5.10) is always negative, as required by its interpretation as the second-order contribution to relative entropy.We can also check the formula (5.10) via a direct CFT calculation by consider-ing states that are obtained from the CFT vacuum by a local conformal transforma-tion. In two dimensions, states with an arbitrary traceless conserved stress-tensorcan be obtained, and the entanglement entropy for these states can also be calcu-lated explicitly. We carry out this calculation in Section 5.4, and show that theresult (5.10) is exactly reproduced.In Section 5.3.2, we consider the matter terms in (5.7) providing some explicitresults for the quadratic contributions of scalar operator expectation values. Here,117as in the generic case, the result takes the formδ (2)SmatterB =−12∫DB∫DBGαβ (x,x′)〈Oα(x)〉〈Oβ (x′)〉 (5.11)with integrals over the entire domain of dependence region.Auxiliary de Sitter Space InterpretationRecently, in [43] it has been pointed out that the first-order result δ (1)S(xµ ,R) forthe entanglement entropy of a ball with radius R and center xµ can be obtained asthe solution to the equation of motion for a free scalar field on an auxiliary de Sitterspace ds2 = L2dSR2 (−dR2+dxµdxµ) with the CFT energy density 〈T00(xµ)〉 acting asa source term at R = 0. In Section 5.5, we show that in the 1+1 dimensional case,the stress tensor term (5.10) for the entanglement entropy at second-order can alsoresults from solving a scalar field equation on the auxiliary de Sitter space if weadd a simple cubic interaction term. In an upcoming paper [106], it is shownthat this agreement extends to all orders for a suitable choice of the scalar fieldpotential. The resulting nonlinear wave equation also reproduces the second-orderentanglement entropy near a thermal state in the auxiliary kinematic space recentlydescribed in [107].Including the contributions from matter fields or moving to higher dimensions,the expression for entanglement entropy involves one-point functions on the entirecausal diamond DB, so reproducing these results via some local differential equa-tion will require a more complicated auxiliary space that takes into account thetime directions in the CFT. This direction is pursued further in [108, 106].DiscussionWhile the explicit two-dimensional stress tensor contribution (5.10) can be ob-tained by a direct CFT calculation for a special class of states, we emphasize thatin general the holographic predictions from (5.1) are expected to hold only forholographic states in CFTs with gravity duals. It would be interesting to under-stand better whether all of the second order contributions we considered here areuniversal for all CFTs or whether they represent genuine constraints/predictions118from holography.5 In the latter case, and for the results at higher order in pertur-bation theory, it is an interesting question to understand better which CFT statesand/or which CFT properties are required to reproduce the results through directCFT calculations. This should help us understand better which theories and whichstates in these theories are holographic.5.2 BackgroundOur holographic calculation of entanglement entropy to second-order in the bound-ary one-point functions makes use of the direct connection between CFT quantumFisher information and canonical energy on the gravity side, pointed out recentlyin [105]. We begin with a brief review of these results.5.2.1 Relative entropy and quantum Fisher informationOur focus will be on ball-shaped subsystems B of the CFTd , for which the thevacuum density matrix is known explicitly through (5.3). More generally, we canwrite it asρvacB = e−HB , HB =∫B′ζ µB Tµνεν , (5.12)where Tµν is the CFT stress tensor operator and ε is defined asεν =1(d−1)!ενν1···νd−1dxν1 ∧·· ·∧dxνd−1 , (5.13)so that nµεµ is the volume form on the surface perpendicular to a unit vector nµ ,and ζB is a conformal Killing vector defined in the domain of dependence regionDB, with ζB = 0 on ∂B. For the ball B with radius R and center xµ0 in the t = t0slice, we haveζB =−2piR (t− t0)(xi− xi0)∂i+piR[R2− (t− t0)2− (~x−~x0)2]∂t . (5.14)5There is evidence in [109, 110, 111] that at least some of the contributions at this order can bereproduced by CFT calculations in general dimensions, since they arise from CFT two and three-point functions, though the results there most directly apply to the case where the perturbation is tothe theory rather than the state.119By the conservation of the current ζ µB Tµν associated with this conformal Killingvector, the integral in (5.12) can be taken over any spatial surface B′ in DB with thesame boundary as B.For excited states, the density matrix ρB will generally be different than ρvacB .One measure of this difference is the relative entropyS(ρB||ρvacB ) = tr(ρB logρB)− tr(ρB logρvacB )= ∆〈HB〉−∆SB , (5.15)where HB is the vacuum modular Hamiltonian given in (5.12), SB =− tr(ρB logρB)is the entanglement entropy for the region B and ∆ indicates the difference with thevacuum state.For a one-parameter family of states near the vacuum, we can expand ρB asρB(λ ) = ρvacB +λ δρ1+λ2δρ2+O(λ 3) . (5.16)The first-order contribution to relative entropy vanishes (this is the first law of en-tanglement δ (1)SB = δ 〈HB〉) so the leading contribution to relative entropy appearsat second-order in λ . This quadratic in δρ1 with no contribution from δρ2,S(ρB(λ )||ρvacB ) = λ 2 〈δρ1,δρ1〉ρvacB +O(λ 3) , (5.17)where〈δρ,δρ〉σ ≡ 12 tr(δρddλlog(σ +λδρ)∣∣∣λ=0). (5.18)This quadratic form, which is positive by virtue of the positivity of relative entropy,defines a positive-(semi)definite metric on the space of perturbations to a generaldensity matrix σ . This is known as the quantum Fisher information metric.Rearranging (5.15) and making use of (5.17), we haveSB = SvacB +∫B′ζ µB 〈Tµν〉εν −λ 2〈δρ1,δρ1〉ρvacB +O(λ 3) . (5.19)This general expression is valid for any CFT, but the O(λ 2) term generally hasno simple expression in terms of local operator expectation values. However, for120holographic states we can convert this term into an expression quadratic in the CFTone-point functions by using the connection between quantum Fisher informationand canonical energy.5.2.2 Canonical energyConsider now a holographic state, which by definition is associated with some dualasymptotically AdS spacetime M. Near the boundary, we can describe M using ametric in Fefferman-Graham coordinates asds2 =`2AdSz2(dz2+dxµdxµ + zd Γµν(x,z)dxµdxν)(5.20)where Γµν(z,x) has a finite limit as z→ 0 and Γ= 0 for pure AdS.The relative entropy S(ρB||ρvacB ) can be computed at leading order in large Nby making use of the holographic entanglement entropy formula, which relatesthe entanglement entropy for a region A to the area of the minimal-area extremalsurface A˜ in M with boundary ∂A,SA ≡ Area(A˜)4GN . (5.21)This yields immediately that ∆SA = (Area(A˜)M−Area(A˜)AdS)/(4GN). The result(5.21) also allows us to relate the ∆〈HB〉 term in relative entropy to a gravitationalquantity, since it implies that the expectation value of the CFT stress tensor isrelated to the asymptotic behaviour of the metric through [37]〈Tµν〉=d`d−1AdS16piGNΓµν(x,z = 0) . (5.22)Thus, for holographic states, we can writeS(ρB||ρvacB ) =d`d−1AdS16piGN∫Bζ µB Γµν(x,0)εν − Area(A˜)M−Area(A˜)AdS4GN. (5.23)For a one-parameter family of holographic states |Ψ(λ )〉 near the CFT vacuum,the dual spacetimes M(λ ) can be described via a metric and matter fields φα =121(g,φmatter) with some perturbative expansiong = gAdS+λδg1+λ 2δg2+O(λ 3) ,φmatter = λδφmatter1 +λ2δφmatter2 +O(λ3) . (5.24)By the result (5.19) from the previous section, the second-order contribution toentanglement entropy is equal to the leading order contribution to relative entropy.This is related to a gravitational quantity via (5.23). The main result in [105] isthat this second-order quantity can be expressed directly as a bulk integral over thespatial region Σ between B and B˜ where the integrand is a quadratic form on thelinearized bulk perturbations δg1 and δφmatter1 .zxtBDBΣ B˜Figure 5.1: The Rindler wedge RB associated to the ball-shaped region B onthe boundary. The blue lines indicate the flow of ζB, and the red linesξB. The surface Σ lies between B and the extremal surface B˜.To describe the general result, consider the region Σ between B and B˜ in pureAdS spacetime, and define RB as the domain of dependence of this region, as shownin Figure 5.1. Alternatively, RB is the intersection of the causal past and the causalfuture of DB; it can be thought of as a Rindler wedge of AdS associated with B. OnRB, there exists a Killing vector which vanishes at B˜ and approaches the conformal122Killing vector ζB at the boundary. In Fefferman-Graham coordinates, this isξB =−2piR (t− t0)[z∂z+(xi− xi0)∂i]+piR[R2− z2− (t− t0)2− (~x−~x0)2]∂t (5.25)The vector ξB is timelike hence defines a notion of time evolution within the regionRB; the “Rindler time” associated with this Rindler wedge.The “canonical energy”, dual to relative entropy at second-order, can be under-stood as the perturbative energy associated with this time, as explained in [112].This is quadratic in the perturbative bulk fields including the graviton, and givenexplicitly byE (δg1,δφ1) = WΣ(δφ1,£ξBδφ1)=∫Σω f ull(δφ1,£ξBδφ1)=∫Σω(δg1,£ξBδg1)+∫Σωmatter(δφ1,£ξBδφ1)=∫Σω(δg1,£ξBδg1)−∫Σξ aBT(2)ab εb . (5.26)In the first line, WΣ is the symplectic form associated with the phase space of grav-itational solutions on Σ, and £ξBδφ1 is the Lie derivative with respect to ξ on δφ1,the first-order perturbation in metric and matter fields. The symplectic form isequal to the integral over Σ of a “presymplectic” form ω f ull which splits into agravitational part and a matter part as in the third line. The matter part can bewritten explicitly in terms of T (2)ab , the matter stress tensor at quadratic order in thefields, while the gravitational part ω is given explicitly byω(γ1,γ2) =116piGNεaPabcde f (γ2bc∇dγ1e f − γ1bc∇dγ2e f ) (5.27)Pabcde f = gaeg f bgcd− 12gadgbeg f c− 12gabgcdge f − 12gbcgaeg f d +12gbcgadge f .In deriving (5.26) it has been assumed that the metric perturbation has been ex-pressed in a gauge for which the coordinate location of the extremal surface B˜ doesnot change (so that ξB continues to vanish there), and the vector ξB continues to123satisfy the Killing equation at B˜. Thus, we require thatξB|B˜(λ ) = 0, (5.28)£ξBg(λ )|B˜(λ ) = 0. (5.29)As shown in [112], it is always possible to satisfy these conditions; we will see anexplicit example below.5.3 Second-order contribution to entanglement entropyUsing the result (5.7), we can now write down a general expression for the ballentanglement entropy of a general holographic state up to second-order in pertur-bations to the vacuum state, in terms of the CFT one-point functions. Accordingto (5.19) and (5.26), the second-order term in the entanglement entropy for a ballB can be expressed as an integral over the bulk spatial region Σ between B and thecorresponding extremal surface B˜, where the integrand is quadratic in first-orderbulk perturbations.These linearized perturbations are determined by the boundary behavior of thefields via the linearized bulk equations. In general, to determine the linearizedperturbations in the region Σ (or more generally in the Rindler wedge RB), we onlyneed to know the boundary behavior in the domain of dependence region DB, asdiscussed in detail in [113]. The relevant boundary behaviour of each bulk field iscaptured by the one-point function of the corresponding operator. We can expressthe results as(δφ1)α(x,z)|Σ =∫DBddx′Kα(x,z;x′)〈Oα(x′)〉CFT (5.30)where Kα(x,z;x′) is the relevant bulk-to-boundary propagator. As discussed in[114, 113, 70], Kα should generally be understood as a distribution to be integratedagainst consistent CFT one-point functions, rather than a function. Since the ex-pression (5.30) is linear in the CFT expectation values, the result (5.7) is quadraticin these one-point functions and represents our desired second-order result.To summarize, for a holographic state, the second-order contribution to entan-glement entropy in the expansion (5.19) is the leading order contribution to the124relative entropy S(ρB||ρvacB ). This is dual to canonical energy, given explicitly by:δ (2)SB =−〈δρ1,δρ1〉ρvacB =−12E (δφ1,δφ1)=−12∫Σω(δg1,£ξBδg1)+12∫Σξ aBT(2)ab εb .(5.31)This is quadratic in the linearized perturbations δφα (including the metric pertur-bation, and these can be expressed in terms of the CFT one-point functions on DBas (5.30).5.3.1 Example: CFT2 stress tensor contributionIn this section, as a sample application of the general formula, we provide an ex-plicit calculation of the quadratic stress tensor contribution to the entanglemententropy for holographic states in two-dimensional conformal field theories. Thisarises from the first term in (5.7).For a general CFT state, the stress tensor is traceless and conserved,〈T µµ〉= 〈∂µT µν〉= 0 . (5.32)In two dimensions, these constraints can be expressed most simply using light-conecoordinates x± = x± t, where we have〈T+−〉= ∂+〈T−−〉= ∂−〈T++〉= 0 . (5.33)Thus, a general CFT stress tensor can be described by the two functions, 〈T++(x+)〉and 〈T−−(x−)〉.Assuming that the state is holographic, there will be some dual geometry of theform (5.20). According to (5.22), the stress tensor expectation values determine theasymptotic form of the metric asΓ++(x,0) = 8piGN`AdS〈T++(x+)〉 Γ−−(x,0) = 8pi GN`AdS〈T−−(x−)〉 (5.34)Now, suppose that our state represents a small perturbation to the CFT vacuum,so that the stress tensor expectation values and the asymptotic metric perturbations125are governed by a small parameter λ :Γ++(x,0)≡ λh+(x+) Γ−−(x,0)≡ λh−(x−) . (5.35)Then the metric perturbation throughout the spacetime is determined by this asymp-totic behavior by the Einstein equations linearized about AdS. Here, we need onlythe components in the field theory directions, which give1z3∂z(z3∂zΓµν)+∂ρ∂ ρΓµν = 0 . (5.36)The solution in our Fefferman-Graham coordinates with boundary behaviour (5.35)isΓ(1)++(x,z) = λh+(x+) Γ(1)−−(x,z) = λh−(x−) (5.37)with the linearized perturbation Γ(1)µν independent of z.Satisfying the gauge conditionsWe would now like to evaluate the metric contribution to (5.7)δ (2)SgravB =−12∫Σωgrav(δg1,£ξBδg1) . (5.38)This formula assumes the gauge conditions (5.28) which differ from the Fefferman-Graham gauge conditions we have been using so far. Thus, we must find a gaugetransformation to bring our metric perturbation to the appropriate form. In general,we can writeγab = hab+(£V g)ab = hab+∇aVb+∇bVa . (5.39)where γ is the desired metric perturbation satisfying the gauge condition, and h isthe perturbation in Fefferman-Graham coordinates (equivalent to Γ for d = 2).The procedure for finding an appropriate V and evaluating (5.38) is described indetail in [105], but we review the main points here. Defining coordinates (XA,X i)so that the extremal surface lies at some fixed value of XA with X i describing coor-dinates along the surface, the gauge condition (5.28) (equivalent to requiring that126the coordinate location of the extremal surface remains fixed) gives(∇i∇iVA+[∇i,∇A]V i+∇ihiA−12∇Ahii)|B˜ = 0 (5.40)while the condition (5.29) that ξB continues to satisfy the Killing equation at B˜gives(hiA+∇iVA+∇AVi) |B˜ = 0 , (5.41)(hAD−12δADhCC +∇AVD+∇DV A−δDD∇CVCC)∣∣∣∣B˜= 0 . (5.42)To solve these, we first expand our general metric perturbation in a Fourierbasis.hµν(t,x,z) = λ∫ [δ+µ δ+ν hˆ+(k)eikx+ +δ−µ δ−ν hˆ−(k)eikx−]dk , (5.43)with a gauge choice hza(t,x,z) = 0.For each of the basis elements, we use the equations (5.40), (5.41) and (5.42)to determine V and its first derivatives at the surface V . For these calculations, itis useful to define polar coordinates (z,x) = (r cosθ ,r sinθ). Since the gauge con-ditions are linear in V , the conditions on V for a general perturbation are obtainedfrom these by taking linear combinations as in (5.43),Va(t,x,z) = λ∫ [Vˆ+a (k)eikx+ +Vˆ−a (k)eikx−]dk . (5.44)127After requiring Va remain finite at θ =±pi2 , we findVˆ−t (k; t,r,θ) =e−iktk3r2 cos2 θ(−icos(kr)+ sinθ sin(kr)− i (k2r2 cos2 θ −1)eikr sinθ2)Vˆ−r (k; t,r,θ) =e−iktk3r2 cos2 θ(sin(kr)− isinθ cos(kr)− (k2r2 cos2 θ sinθ + ikr cos2 θ +2isinθ)eikr sinθ2)∂tVˆ−θ (k; t,r,θ) =e−ikt2k2 r cosθ((2+ k2r2 cos2 θ −2 ikr sinθ)eikr sinθ − 2sin(kr)k3r2)∂rVˆ−θ (k; t,r,θ) =e−iktk3r2 cosθ(2icos(kr)+[2kr sinθ + r3k3 sinθ cos2 θ + i(r2k2 cos2 θ − kr2+2)]eikr sinθ)(5.45)where the V± solutions are related through Vˆ+r (k; t,r,θ)= Vˆ−r (k;−t,r,θ) and Vˆ−t (k; t,r,θ)=−Vˆ+t (k;−t,r,θ). The results here give the behavior of V and its derivatives only atthe surface B˜ (r = R in polar coordinates). Elsewhere, V can be chosen arbitrarily,but we will see that our calculation only requires the behavior at B˜.Evaluating the canonical energyGiven the appropriate V , we can evaluate (5.38) usingω(g,γ,£ξ γ) = ω(h+£V g,£ξB(h+£V g)) (5.46)= ω(g,h,£ξh)+ω(g,h+£V g,£[ξ ,V ]g)−ω(g,£ξh,£V g)where[ξ ,V ]a = ξ b∂bV a−V b∂bξ a (5.47)and we have used that £ξg = 0. We can simplify this expression using the gravita-tional identityω(g,γ,£ξg) = dχ(γ,X) (5.48)128whereχ(γ,X) =116piGNεab{γac∇cXb− 12γcc∇aXb+∇bγacXc−∇cγacXb+∇aγccXb}.(5.49)Thus, we haveω(g,γ,£ξ γ) = ω(g,h,£ξh)+dρ (5.50)whereρ = χ(h+£V g, [ξ ,V ])−χ(£ξh,V ) . (5.51)Finally, choosing V so that it vanishes at B, we can rewrite (5.38) asE =∫Σω(g,h,£ξh)+∫B˜ρ(h,V ) . (5.52)In this final expression, we only need V and its derivatives at the surface B˜. Thus,we can now calculate the result explicitly for a general perturbation. In the Fourierbasis, the final result in terms of the boundary stress tensor isE =∫dk1∫dk2 Kˆ2(k1,k2)〈T++(k1)〉〈T++(k2)〉+{+↔−} , (5.53)where the kernel isKˆ2(k1,k2) =256pi2 R4 GN`AdSK3(K−κ)3(K+κ)3[(K5−2(κ2+4)K3+κ4K)cosK−(5K4−6K2κ2+κ4)sinK+8K3 cosκ] ,(5.54)with K ≡ R(k1 + k2),κ ≡ R(k1− k2). We note in particular that the result splitsinto a left-moving part and a right-moving part with no cross term.Transforming back to position spaceE =∫B′dx+1∫B′dx+2 K2(x+1 ,x+2 )〈T++(x+1 )〉〈T++(x+2 )〉+{+↔−} , (5.55)where the kernel K2 is symmetric under exchange of x±1 and x±2 , and has support129only on x±i ∈ [−R,R]. Focusing only on the domain of support, we haveK2(x1,x2) =4pi2GNR2`AdS(R− x1)2(R+ x2)2 x1 ≥ x2(R+ x1)2(R− x2)2 x1 < x2 . (5.56)Using the relation c= 3`AdS/(2GN) between the CFT central charge and the gravityparameters, we recover the result (5.10) from the introduction.Like the leading order result in (5.19), the integrals can be taken over any sur-face B′ with boundary ∂B. The fact that we only need the stress tensor on a Cauchysurface for DB is special to the stress tensor in two dimensions, since the conserva-tion relations allow us to find the stress tensor expectation value everywhere in DBfrom its value on a Cauchy surface. For other operators, or in higher dimensions,the result will involve integrals over the full domain of dependence. We will see anexplicit example in the next subsection.5.3.2 Example: scalar operator contributionWe now consider an explicit example making use of the bulk matter field term in(5.7) in order to calculate the terms in the entanglement entropy formula quadraticin the scalar operator expectation values. The discussion for other matter fieldswould be entirely parallel. This example is more representative, since the formulawill involve scalar field expectation values in the entire domain of dependence DB,i.e. a boundary spacetime region rather than just a spatial slice. The results hereare similar to the recent work in [109, 110, 111], but we present them here to showthat they follow directly from the canonical energy formula.We suppose that the CFT has a scalar operator of dimension ∆with expectationvalue 〈O(x)〉. According to the usual AdS/CFT dictionary, this corresponds to abulk scalar field with mass m2 = ∆(∆−d) and asymptotic behaviorφ(x,z)→ γz∆〈O(x)〉 , (5.57)where γ is a constant depending on the normalization of the operator O . Theleading effects of the bulk scalar field on the entanglement entropy (5.7) come130from the matter term in the canonical energyδ (2)SmatterB =12∫Σξ aBT(2)ab εb . (5.58)Using the explicit form of ξB from (5.25) and ε from (5.13), this gives (for aball centered at the origin)δ (2)SmatterB =−`d−1AdS2∫ R0dzzd−1∫x2<R2−z2dd−1xpiR(R2− z2− x2)T (2)00 (x,z) . (5.59)This expression is valid for a general bulk matter field. For a scalar field, we haveT (2)ab = ∂aφ1∂bφ1−12gab(gcd∂cφ1∂dφ1+m2φ 21 ) , (5.60)where gab is the background AdS metric and φ1 represents the solution of the lin-earized scalar field equation on AdS,1zd−1∂z{zd−1∂zφ}+∂µ∂ µφ − m2z2φ = 0 , (5.61)with boundary behavior as in (5.57). This solution is given most simply in Fourierspace, where we haveφ1(k,z) =2νΓ(ν+1)(2pi)d∫k20>~k2ddkeikµxµ(k20−~k2)ν/2 z d2 Jν(√k20−~k2z) γ〈O(k)〉 ,(5.62)where ν = ∆−d/2, but we can formally write a position-space expression using abulk-to-boundary propagator K(x,z;x′) as [115, 116]φ1(x,z) = γ∫dx′K(x,z;x′)〈O(x′)〉 . (5.63)The integral here is over the boundary spacetime, however it has been argued (see,for example [113, 114]) that to reconstruct the bulk field throughout the Rindlerwedge RB (and specifically on Σ), we need only the boundary values on the do-main of dependence region. We recall some explicit formulae for this “Rindlerbulk reconstruction” in Appendix D. Combining these results, we have a general131expression for the scalar field contribution to entanglement entropy at second-orderin the scalar one-point functions,δ (2)SscalarB = −`d−1AdS2∫ R0dzzd−1∫x2<R2−z2dd−1xpiR(R2− z2− x2) (5.64){(∂0φ1)2+(∂iφ1)2+(∂zφ1)2+m2z2φ 21}where φ1 is given in (5.62) or (5.63) .As a simple example, consider the case where the scalar field expectation valueis constant. In this case it is simple to solve (5.61) everywhere to find thatφ1(x,z) = γ〈O〉z∆ . (5.65)Inserting this into the general formula (5.64), and performing the integrals, weobtainδ (2)SscalarB =−pi`d−1AdS4γ2〈O〉2R2∆Ωd−2∆Γ(d2 − 12)Γ(∆− d2 +1)Γ(∆+ 32). (5.66)This reproduces previous results in the literature [100, 110].5.4 Stress tensor contribution: direct calculation forCFT2In Section 5.3.1, we used the equivalence between quantum Fisher information andcanonical energy to obtain an explicit expression for the second-order stress tensorcontribution to the entanglement entropy for holographic states in two-dimensionalCFTs. This is applicable for general holographic states, whether or not other matterfields are present in the dual spacetime (in which case there are additional terms inthe expression for entanglement entropy). In special cases where there are no mat-ter fields, the spacetime is locally AdS and we can understand the dual CFT state asbeing related to the vacuum state by a local conformal transformation. We show inthis section that in this special case, we can reproduce the holographic result (5.56)through a direct CFT calculation, providing a strong consistency check. We notethat the result does not rely on taking the large N limit or on any special properties132of the CFT, so the formula holds universally for this simple class of states.Our approach will be to develop an iterative procedure to express the entan-glement entropy as an expansion in the stress tensor expectation value for thisspecial class of states. We evaluate the entanglement entropy for these states froma correlation function of twist operators obtained by transforming the result for thevacuum state.6 Similarly, the stress tensor expectation values follow directly fromthe form of the conformal transformation. Inverting the relationship between therequired conformal transformation and the stress tensor expectation value allows usto express the entanglement entropy as a perturbative expansion in the expectationvalue of the stress tensor.5.4.1 Conformal transformations of the vacuum stateIn two-dimensional CFT, under a conformal transformation w = f (z), the stresstensor transforms asT ′(w) =(dwdz)−2(T (z)+c12{ f (z);z}). (5.67)Here c is the central charge of the CFT and the inhomogeneous part is the Schwarzianderivative{ f (z);z} ≡ f′′′(z)f ′(z)− 3 f′′(z)22 f ′(z)2. (5.68)For an infinitesimal transformation f (z) = z+ λ ε(z), the Schwarzian derivativecan be expanded as{z+λε(z);z}= λ ε ′′′(z)−λ 2(ε ′′′(z)ε ′(z)+32ε ′′(z)2)+λ 3(ε ′(z)2ε ′′′(z)+3ε ′(z)ε ′′(z)2)+· · ·(5.69)The CFT vacuum is invariant under the SL(2,C) subgroup of global conformaltransformations. However, for transformations which are not part of this subgroup,the vacuum state transforms into excited states. The action of the full confor-mal group includes the full Virasoro algebra which involves arbitrary products and6A similar approach was recently used to derive the modular Hamiltonian of these excited statesin [117].133derivatives of the stress tensorId∼ 1,T,∂mT,T 2,T∂ nT, · · · . (5.70)These states capture the gravitational sector of the gravity dual. Other excitedstates can be obtained by the action of other primary operators and their descen-dants. However we restrict ourselves to the class states that are related to ‘puregravity’ excitations, which are the states obtained by conformal transformation ofthe vacuum state.We denote the excited state as | f 〉=U f |0〉 where U f is the action of a confor-mal transformation on the vacuum |0〉. The expectation value of the stress tensorfor the state perturbed state | f 〉 is〈 f |T (z)| f 〉= 〈0|U†f T (z)U f |0〉= 〈0|T ′(w)|0〉=(d fdz)−2 c12{ f (z);z} , (5.71)where we used that 〈0|T (z)|0〉= 0. The anti-holomorphic component of the stresstensor T¯ (z¯) is similarly related to the anti-holomophic part of the conformal trans-formation f¯ .To leading order in a conformal transformation near the identity, this equationrelates the conformal transformation to 〈T (z)〉 by a third-order ordinary differen-tial equation. The three integration constants correspond to the invariance of 〈T (z)〉under the global conformal transformations. Thus we have an invertible relation-ship between the conformal transformations modulo their global part and 〈T (z)〉,at least near the identity.5.4.2 Entanglement entropy of excited statesIn a two-dimensional CFT, the entanglement entropy can be explicitly computedusing the replica method [118, 83]. The computation can be reduced to a corre-lation function of twist operators Φ±, which are conformal primaries with weight(hn, h¯n) = c24(n−1/n,n−1/n).The Re´nyi entropy isexp((1−n)S(n))= 〈Φ+(z1)Φ−(z2)〉= (z2− z1)−2hn . (5.72)134The entanglement entropy is obtained by taking the n→ 1 limit of S(n).Svac = limn→1S(n) = limn→1(1−n)−1 log(z2− z1)−2hn = c12 log(z2− z1)2δ 2. (5.73)For the excited states obtained by conformal transformations z→ w = f (z) theRe´nyi entropy isexp((1−n)S(n)ex)= 〈 f |Φ+(z1)Φ−(z2)| f 〉 (5.74)=(d fdz)−hnz1(d fdz)−hnz2(d f¯dz¯)−h¯nz¯1(d f¯dz¯)−h¯nz¯2〈0|Φ+(z1)Φ−(z2)|0〉 .(5.75)Here z1,z2 are the points f (z1) = f¯ (z¯1) = −R, f (z2) = f¯ (z¯2) = R. The entangle-ment entropy of the excited state isSex = limn→1S(n)ex =c12log∣∣∣∣ f ′(z1) f ′(z2) f¯ ′(z¯1) f¯ ′(z¯2)(z2− z1)2δ 2∣∣∣∣ . (5.76)Therefore the change in entanglement entropy respect to the vacuum state isδS≡ Sex−Svac = c12 log∣∣∣∣ f ′( f−1(R)) f ′( f−1(−R))( f−1(R)− f−1(−R))2(2R)2∣∣∣∣(5.77)+c12log∣∣∣∣ f¯ ′( f¯−1(R)) f¯ ′( f¯−1(−R))( f¯−1(R)− f¯−1(−R))2(2R)2∣∣∣∣ .By inverting (5.71), the conformal transformation required to reach the state| f 〉 can be expressed as a function of the expectation value of the stress tensor.Plugging this f into (5.77), allows us to express the entanglement entropy as afunction of the expectation value of the stress tensor alone, as we set out to do.In practice, we will invert (5.71) order by order in a small conformal trans-formation and express the entanglement entropy as an expansion in the resultingsmall stress tensor. The second-order term in this expansion will be the Fisherinformation metric.In the following, we will focus on the holomorphic term in (5.71), noting that135the anti-holomorphic part follows identically.75.4.3 Perturbative expansionConsider a conformal transformation perturbation near the identity transformationw = f (z) = z+λ f1(z)+λ 2 f2(z)+λ 3 f3(z)+ · · · , (5.78)where λ is a small expansion parameter.In this expansion,12c〈T (w)〉= λ f ′′′1 (w)+λ 2(−32f ′′1 (w)2−3 f ′1(w) f ′′′1 (w)+ f ′′′2 (w)− f1(w) f ′′′′1 (w))+O(λ 3) ,(5.79)and the entanglement entropy is12cSex = log∣∣∣∣ f ′(z1) f ′(z2)(z2− z1)2δ 2∣∣∣∣= log(2R)2δ 2+λ[R( f ′1(−R)+ f ′1(R))+ f1(−R)− f1(R)R]+λ 2(− ( f1(R)− f1(−R))24R2+− f1(−R) f ′1(−R)+ f1(R) f ′1(R)+ f2(−R)− f2(R)R− 12f ′1(−R)2−12f ′1(R)2+ f ′2(−R)+ f ′2(R)− f1(−R) f ′′1 (−R)− f1(R) f ′′1 (R))+ O(λ 3) . (5.80)Linear orderTo first-order in λ , the stress tensor is given by〈T (z)〉= λ c12f ′′′1 (z)+O(λ2) , (5.81)7Note that the potential cross-term between left and right moving contributions vanished in thegravitational computation of δ (2)S.136so that change in the expectation value of the modular Hamiltonian becomesδ 〈HB〉 = λ c24R∫ R−Rdz(R2− z2) f ′′′1 (z)=λ c24R[(R2− z2) f ′′1 (z)+2(z f ′1(z)− f1(z))]R−R=λ c12R[R( f ′1(R)+ f′1(−R))− ( f1(R)− f1(−R))]. (5.82)From (5.77) we also have that the first-order change in entanglement entropy isδ (1)S =λ c12R[R( f ′1(R)+ f′1(−R))− ( f1(R)− f1(−R))]. (5.83)Comparing with (5.82) we see that the first law of entanglement holdsδ (1)S = δ 〈HB〉 . (5.84)Second-orderThe second-order change in entanglement entropy gives the second-order relativeentropy as the modular Hamiltonian is linear in the expectation value of the stresstensor. This is the quantum Fisher metric in the state space, which is dual to thecanonical energy in gravity [105]. In this section, we obtain the expression forcanonical energy from the CFT side and find an exact match to the results of Sec-tion 5.3.1.Our procedure so far yields the entanglement entropy of a subregion in termsof a perturbative expansion in small stress tensor expectation valueδS =∫Bdz2piK1(z)〈T (z)〉− 12∫Bdz12pi∫Bdz22piK2(z1,z2)〈T (z1)〉〈T (z2)〉+ · · ·+{z↔ z¯} . (5.85)To obtain K2(z1,z2), we need to invert the relationship in (5.79) order by or-der, the lower order solutions fi−1, fi−2, · · · f1 becoming sources for the i-th ordersolution.Taking the explicit expression for 〈T (z)〉 to simplify solving the differential137equations,〈T (z)〉= λ(c1eik1z+ c2eik2z), (5.86)is sufficient to extract the Fourier transformed kernel.The first-order solution isf1(z) = F1+F2z+F3z2+12ic(c1eik1zk31+ c2eik2zk32), (5.87)where Fi are constants that corresponds to the global part of the conformal trans-formation and do not effect the final result. We take these constants to be zero forsimplicity. The second-order solution isf2(z)=− 9c2[11i16(c21e2ik1zk51+ c22e2ik2zk52)+ ic1c2k31k32ei(k1+k2)z(k41 +3k2k31 +3k22k21 +3k32k1+ k42)(k1+ k2)3].(5.88)With these solutions, we obtainK˜1(k) =2k2sin(kR)− kRcos(kR)kR, (5.89)as well asK˜2(k1,k2) =96R4c(K5−2(κ2+4)K3+κ4K)cosK− (5K4−6K2κ2+κ4)sinK+8K3 cosκK3(K−κ)3(K+κ)3 ,(5.90)with K ≡ R(k1+ k2) and κ ≡ R(k1− k2).Taking the inverse Fourier transformation of K˜1(k)K1(z) =∫dk K˜1(k)e−ikz = piR2− z2RW (R,z) (5.91)whereW (R,x)≡ (sgn(R+ x)+ sgn(R− x))2(5.92)is a window function with support x ∈ [−R,R].138The second-order position space kernel isK2(z1,z2) =6pi2cR2(R− z1)2(R+ z2)2 −R≤ z2 ≤ z1 ≤ R(R+ z1)2(R− z2)2 −R≤ z1 < z2 ≤ R . (5.93)The anti-holomorphic part is the same with z→ z¯, and the cross term vanishes.With the relationc =3`AdS2GN(5.94)this reproduces the kernel for canonical energy in (5.56).This result holds for regions defined on any spatial slice of the CFT. If wechoose the t = 0 slice, z = z¯ = x and our result becomesδS(2)EE = −12∫Bdx1∫Bdx2 K2(x1,x2) [〈T++(x1)〉〈T++(x2)〉+ 〈T−−(x1)〉〈T−−(x2)〉] .Changing variables using x1 = x− r, x2 = x+ r, the kernel is simplyK2(x,r) = K2(x,−r) = 12pi2cR2[(R−|r|)2− x2]2Θ(R−|r|− |x|) . (5.95)5.4.4 Excited states around thermal backgroundA similar analysis can be applied to perturbations around a thermal state with tem-perature T = β−1. If we denote homogeneous thermal state |β 〉, the stress tensorone-point function is〈β |T |β 〉= pi2c6β 2. (5.96)This can be obtained by a conformal transformation from the vacuum withfβ (z) =β2pilog(z) . (5.97)On top of this transformation, one could also apply an infinitesimal conformaltransformation to obtain non-homogeneous perturbation around thermal state.139A similar computation as the previous section leads to the first-order kernelKβ1 (z) =2βsinh(2piRβ )sinh(pi(R− z)β)sinh(pi(R+ z)β), (5.98)which is the modular hamiltonian of thermal state in 2d CFT.Furthermore, the second-order kernel isKβ2 (z1,z2) =24β 2c sinh2(2piRβ )sinh2(pi(R−z1)β)sinh2(pi(R+z2)β)−R≤ z2 ≤ z1 ≤ Rsinh2(pi(R+z1)β)sinh2(pi(R−z2)β)−R≤ z1 < z2 ≤ R.(5.99)Consistency check : homogeneous BTZ perturbationAs a check, consider the homogeneous perturbation example, where 〈T 〉= 〈T¯ 〉=λ8GN.8In the AdS3 this is a perturbation towards the planar BTZ black hole geometry,ds2 =1z2(dz2+(1+λ z2/2)2dx2− (1−λ z2/2)2dt2) (5.100)in Fefferman-Graham coordinates. Holographic renormalization (5.22) tells us thestress tensor expectation value of the dual CFT is〈Ttt〉= 12pi (〈T 〉+ 〈T¯ 〉) =λ8piGN. (5.101)As the black hole corresponds to the thermal state in CFT, the dual state be obtainedby the conformal transformation (5.97).First, applying (5.77) for this conformal transformation, the change in entan-glement entropy with respect to the vacuum isδS = λR26G−λ 2 R490G+λ 34R62835G+O(λ 4) , (5.102)which matches the previous known results [100, 105].8λ = 2pi2β sets the temperature.140The linear order equals δ 〈HB〉 as expected from the entanglement first law.The second-order term gives the quantum Fisher information or the canonicalenergyE =d2dλ 2(∆E−∆S)∣∣∣λ=0=R445GN. (5.103)Using the formula using the second-order kernel (5.85) and (5.93), we obtain thesame canonical energyE = 2d2dλ 2[12∫Bdz12pi∫Bdz22piK2(x1,x2)〈T 〉〈T 〉]λ=0=R445GN. (5.104)5.5 Auxiliary de Sitter space interpretationIn [43], it was pointed out that the leading order perturbative expression (5.2) forentanglement entropy, expressed as a function of the center point x and radius R ofthe ball B, is a solution to the wave equation for a free scalar field on an auxiliaryde Sitter space, with 〈T00(x)〉 acting as a source.It was conjectured that higher order contributions might be accounted for bylocal propagation in this auxiliary space with the addition of self-interactions forscalar field. In this section, we show that for two-dimensional CFTs, the second-order result (5.10) can indeed be reproduced by moving to a non-linear wave equa-tion with a simple cubic interaction to this scalar field. A slight complication is thatwe actually require two-scalar fields; one sourced by the holomorphic stress tensorT++, and the other sourced by the anti-holomorphic part T−−; the perturbation tothe entanglement entropy is then the sum of these two scalars, δS = δS++ δS−,reproducing both terms in (5.10). We will focus on δS+ since δS− follows identi-cally.To reproduce the second-order results for entanglement entropy, we consideran auxiliary de Sitter space with metricds2dS =L2dSR2(−dR2+dx2) . (5.105)141and consider a scalar field δS+ with mass m2L2dS =−2 and actionL =12∇a (δS+)∇a (δS+)+12m2 (δS+)2+4cL2dS(δS+)3 . (5.106)The equation of motion is(∇2dS−m2)δS+(R,x) =12cL2dS(δS+(R,x))2 . (5.107)As shown in [43], the first-order perturbation (5.2) obeys the linearized wave equa-tion (∇2dS−m2)δ (1)S+(R,x) = 0 . (5.108)We can immediately check that the second-order perturbation (5.10) is consistentwith the nonlinear equation by acting with the dS wave equation on the second-order kernel (5.93)(∇2dS−m2)K2(x1− x,x2− x) =− 24cL2dSK1(x1− x)K1(x2− x) . (5.109)Integration against the CFT stress tensor then gives (5.107).Alternatively, introducing the retarded9 bulk-to-bulk propagator [119]GdS(η ,x;η ′,x ′) =−η2+η ′2− (x− x ′)24ηη ′(5.110)and bulk-to-boundary propagatorKdS(η ,x;x ′) = limε→0[−4piε limη ′→εGdS(η ,x;η ′,x ′)]= piη2− (x− x′)2η.(5 111)we can show directly that the solution with boundary behaviorδS+ =4pi3〈T++〉R2+O(R3) (5.112)9These propagators are defined to be non-zero only within the future directed light-cone. This isimportant in reproducing both the support and the exact form of K2(x1,x2).142for R→ 0 isδ (1)S+(R,x0) =∫dxKdS(R,x0;x)〈T++(x)〉 (5.113)at first-order andδ (2)S+(R,x0) =12cL2dS∫dSdη ′dx′√|gdS|GdS(R,x0;η ′,x′)(∫dxKdS(η ′,x′;x)〈T++(x)〉)2,(5.114)at second-order, where the latter term comes from the diagram shown in Figure5.2.x−R x+RδS+ δS+δS+x1 x2g3(R, x)Figure 5.2: Feynman diagram which computes δ (2)S. The δS+ field propa-gates in de Sitter with a cubic interaction given by (5.106). The bold(red) line is the conformal boundary of de Sitter which is identified witha time slice of the CFT. δS+ is sourced by the CFT stress tensor on thisboundary.The integrals can be performed directly to show that these results match withthe expressions (5.2) and (5.10) respectively.A useful advantage of writing the second-order result in the form (5.114) isthat it is manifestly negative. More explicitly, we haveδ (2)S+(R,x0) = − 3cL2dS∫dηdy√|gdS| R2+η2− (x0− y)2Rη[∫BydxKdS(η ,y;x)〈T++(x)〉]2.(5.115)143where√|gdS| and the squared expression are manifestly positive and the bulk-to-bulk propagator (5.110) is positive over the range of integration where (y−x0)2 ≤ (R−η)2. That this expression is negative is required by the positivity ofrelative entropy, since we showed above that −δ (2)S represents the leading orderperturbative expression for the relative entropy.Recently, it has been realized that the modular Hamiltonian in certain non-vacuum states in two dimensional CFTs can be described by propagation in a dualgeometry [107] matching the kinematic space found in [120, 121]. We find that theresults of Section 5.4.4 can be explained by the same interacting theory (5.106) onthis kinematic space. The kinematic space dual to the thermal state is [107]ds2 =4pi2L2dSβ 2 sinh2(2piRβ) (−dR2+dx2) . (5.116)The second-order perturbation to the entanglement entropy from (5.99) obeys thewave equation (5.107) with the same interactions in this kinematic space.We could imagine adding additional fields propagating in de Sitter to capturethe contributions to the entanglement entropy from scalar operators discussed inSection 5.3.2. However, unlike the contribution from the stress tensor, this con-tribution involves integration of the one-point functions over the full domain ofdependence DB. In higher-dimensions, this will also be true for the stress tensorcontribution. The R = 0 boundary of the auxiliary de Sitter space does not includethe time direction of the CFT, so any extension of these results to contributionsof other operators or higher dimensional cases will require a more sophisticatedauxiliary space. Promising work in this direction is discussed in [108, 106].144Chapter 6ConclusionThis thesis investigated entanglement entropy using holographic duality focusingon its applications to a particular class of noncommutative theories in Part I and ongeneral properties of the holographic formula for entanglement entropy in Part II.In Part I, Chapter 2 applied the holographic Ryu-Takayanagi formula for en-tanglement entropy to noncommutative theories. A violation of the area law wasfound as is to be expected in nonlocal theories. We interpreted our results as an in-dication that the vacuum states of these noncommutative theories are entangled onlength scales of the nonlocality leading to an enhancement in the effective numberof degrees of freedom involved in the entanglement entropy between two spatialregions.Chapter 3 followed up on the study of entanglement entropy in this class ofnoncommutative theories by studying them in a different perturbative regime. Theaim was to explore in what regimes similar violations to the area law could befound. It was found that noncommutative interactions did not induce long rangeentanglement in the vacuum state of these theories to leading order in perturbationstheory.Part I fits into a larger effort to apply this holographic formula to better under-stand the entanglement entropy in the full range of field theories with holographicduals. The class of noncommutative theories studied are interesting as they involvemany unusual ingredients in the context of gauge-gravity duality. These include anontrivial dilaton profile and compact dimensions and the presence of a bulk two-145form field in the gravitational dual. In addition, entanglement entropy is interestingin its own right in the larger context of the study of noncommutative field theories.It would be interesting to investigate the onset of long range entanglement in thesetheories, perturbatively at large N and through further holographic studies.Part II focused on the properties of entanglement entropy in general holo-graphic states. Chapter 4 explores the constraints imposed by the existence theholographic formula for entanglement entropies on the geometries of holographicduals. A number of constraints were identified. In particular, the strong subadditiv-ity of entanglement entropy implies that the dual geometry must obey an averagednull energy condition in three dimensional gravity.Chapter 4 describes the first step in a programme of relating the constraintson geometries imposed by the holographic entanglement entropy formula to phys-ically motivated constraints like energy conditions. It would be particularly inter-esting to extend our results to higher dimension or to explore the implication ofother field theoretic entanglement identities. The technical difficulties of dealingwith the nonlocal nature of the holographic formula provide the major barrier tosuch extension. It may be that recent approaches to reorganising the data in holo-graphic dualities using an auxiliary kinematic space along the lines of [108] mayprovide some insight into these technical difficulties.In the classical limit, the dual holographic geometry is determined by the one-point functions of the field theory. This geometry allows us to compute the en-tanglement entropies through the holographic Ryu-Takayanagi formula. Chapter5 expresses the entanglement entropies of holographic states directly in terms ofthese one-point functions. In particular the entanglement entropy in a class ofpurely gravitational asymptotically AdS3 states is expressed in an expansion in theone-point function of the stress tensor of the field theory. This is confirmed directlyin the dual conformal field theory. This result is then interpreted in terms of thepropagation of a self-interacting scalar field in an auxiliary de Sitter space.Chapter 5 describes the first steps in a program of relating entanglement en-tropy to one-point functions in holographic field theories. Finding the explicitform of this relationship could provide insights into the structure of entanglementin these field theories as well as providing constraints on the class of states whichhave holographic duals. In this way Chapter 4 can be though of as providing con-146straints on the geometries which can be dual to field theories, while Chapter 5 pro-vides constraints on the states which can have gravitational duals. 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Raju, An Infalling Observer in AdS/CFT, JHEP 10(2013) 212, [arXiv:1211.6767].157Appendix AAnalysis of the PotentialDivergences from the j > 1 TermsThis analysis follows that of [83], where it is found that the leading divergencewhen the Green’s function is evaluated at coincident points is entirely contained inthe j = 1 term.The Green’s function for the scalar field on the n-sheeted space was given in(3.25):Gn(x,x′) =∫ ∞0dkpi∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqqJk(qr)Jk(qr′)q2+ p2⊥+m2cos(k(θ −θ ′))eip⊥(x⊥−x′⊥)− 112pin2∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq∂ν [Jν(qr)Jν(qr′)]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥) (A.1)−∑j>1B2 jpin2 j(2 j)!∫ dd⊥ p⊥(2pi)d⊥∫ ∞0dqq(∂ν)2 j−1[Jν(qr)Jν(qr′)cos(ν(θ −θ ′))]ν=0q2+ p2⊥+m2eip⊥(x⊥−x′⊥).The first term is independent of n and did not enter into the calculation of theentanglement entropy. The second term was the subject of our investigation. How-ever, the third term was dropped with the claim that it could not introduce any newdivergences. This appendix will justify this claim.We start by revisiting the entanglement entropy in the commutative theory. In158this case from (3.42)S∼∫rdrG1(r,r) fn(r,r), (A.2)where only the contributions to the divergences in the final result have been kept.The Green’s function when evaluated at coincident points gives aΛ2 divergenceG1(r,r)∼∫d4 p1p2+m2∼∫dα p3d p e−α(p2+m2)− 1αΛ2∼Λ2−m2 logΛ2. (A.3)The fn term has the formfn(r,r)∼∫d2 p⊥ qdq∂ν [Jν(qr)Jν(qr)]ν=0q2+ p2⊥+m2+∑j>1∫d2 p⊥ qdq∂ 2 j−1ν [Jν(qr)Jν(qr)]ν=0q2+ p2⊥+m2.(A.4)The momentum integrals can be evaluated when the function is evaluated at coin-cident points∫d2 p⊥ qdqJν(qr)Jν(qr)q2+ p2⊥+m2=∫dβ pd p qdq Jν(qr)Jν(qr)e−β (q2+p2+m2)− 1βΛ2∼ e− 12 r2Iν(12r2) logΛ2. (A.5)This must be integrated over r∫ ∞0rdre−12 r2Iν(1− ε22r2) =1√2ε−ν+O(ε), (A.6)where a small ε has been added to regulate the integral. It is only divergent because∂ 2 j−1ν was passed though the integral sign. Once this derivative is applied, ε canbe safely taken to zero. A calculation of terms O(Λ0) would require a more careful159analysis, but this is sufficient for extracting the leading O(logΛ2) divergence. Thus∫d4x fn(x,x)∼ A⊥ logΛ2[∂ν(−ν)+∑j>1∂ 2 j−1ν (−ν)]= A⊥ logΛ2[−1+∑j>10].(A.7)This shows that all the j > 1 terms vanish when the Green’s function is eval-uated at coincident points and the divergence is entirely contained in the j = 1term.In the noncommutative and dipole theories, the Green’s functions are evaluatedat points separated by the length scale of the nonlocality rather than at coincidentpoints. However, we saw that the source of divergences in the entanglement en-tropy was regions in the integral where this separation vanishes. This analysisshows that these divergences are contained in the j = 1 term.160Appendix BModular Hamiltonian for anInterval in a Boosted ThermalState of a 1+1D CFTIn this appendix, we derive the modular Hamiltonian for a spatial interval [−R,R]in the boosted thermal state. To do this, we start by considering the domain ofdependence D1 of the interval [−r,r] for the vacuum state in Minkowski spacewith coordinates (t ′,x′). For this interval, the modular Hamiltonian is quantumoperator associated with the conformal isometry generated byH1 =pir((r2− (t ′)2− (x′)2)∂t ′−2t ′x′∂x′) .We can now apply a boostx′ = γ(x− vt) t ′ = γ(t− vx) .In this case, the region D1 maps to the domain of dependence D2 of the intervalfrom−(rt ,rx) to (rt ,rx), where r2 = r2x−r2t and v= rt/rx. In this case, the generatorH1 maps toH2 =pir2x − r2t[(rx(r2x − r2t )+2txrt − rx(t2+ x2))∂t +(rt(r2x − r2t )−2txrx+ rt(x2+ t2))∂x]161Next, we perform a transformation for which the causal development of the interval[−1,1] maps to the full Minkowski space (with coordinates (u,τ)), such that theresulting state is the thermal state on Minkowski space dual to the planar BTZgeometry with horizon at z = z0.ds2 =dz2−(1− z2z20)2dτ2+(1+ z2z20)2du2z2(B.1)The appropriate transformation (which can be obtained by finding the coordinatetransformation that maps the bulk region associated with the domain of dependenceof [−1,1] to the planar BTZ black hole) ist =sinh(2τ/z0)cosh(2u/z0)+ cosh(2τ/z0)x =sinh(2u/z0)cosh(2u/z0)+ cosh(2τ/z0). (B.2)After the map, the region D2 maps to the domain of dependence D3 of the intervalfrom −(Rt ,Ru) to (Rt ,Ru), wherert =sinh(2Rt/z0)cosh(2Ru/z0)+ cosh(2Rt/z0)rx =sinh(2Ru/z0)cosh(2Ru/z0)+ cosh(2Rt/z0). (B.3)The generator H2 maps toH3 =piz0C2u−C2t[{CuSu+CuSt sinh(2u/z0)sinh(2τ/z0)−CtSu cosh(2u/z0)cosh(2τ/z0}∂τ{−CtSt +CuSt cosh(2u/z0)cosh(2τ/z0)−CtSu sinh(2u/z0)sinh(2τ/z0}∂u](B.4)whereCu = cosh(2Ru/z0) Su = sinh(2Ru/z0) .Finally, we can perform one further Lorentz transformationu = γ(u′+ vτ ′) τ = γ(τ ′+ vu′) .162with velocity v = Rt/Rx, such that the region D3 is mapped to the domain of de-pendence of the interval [−R,R], where R2 = R2x−R2t . In terms of v,z0, and R, wefind that the generator H3 restricted to τ ′ = 0 givesH4 =piγz0C2u−C2t{−∂τ ′ (cosh(γvU)cosh(γU)(CuStv+SuCt)−sinh(γvU)sinh(γU)(SuCtv+StCu)−(StCtv+CuSu))+∂u′ (cosh(γvU)cosh(γU)(CuSt +SuCtv)−sinh(γvU)sinh(γU)(SuCt +StCuv)−(StCt +CuSuv))} (B.5)where we define U = 2u′/z0 andCt = cosh(2Rγv/z0) Cu = cosh(2Rγ/z0) St = sinh(2Rγv/z0) Su = sinh(2Rγ/z0) .The modular Hamiltonian is obtained by making the replacements ∂τ ′ → Tτ ′τ ′ and∂u′ → Tτ ′u′ and integrating over [−R,R].163Appendix CVariation in Geodesic Lengthunder Endpoint VariationIn this section, we derive a formula for the variation of the entanglement entropyof a boosted interval for some translation and time-translation invariant state in aholographic 1+1 dimensional field theory under a general variation in the endpointof the interval.1 We assume that the field theory lives on Minkowski space withcoordinates (x, t).The dual spacetime will be a 2+1 dimensional spacetime with translationalisometries in one spatial direction and one time direction, associated with Killingvectors ξ µt and ξµx . We assume that the spacetime has a conformal boundary, witha Minkowski space boundary geometry ds2 = −dt2 + dx2 such that the Killingvectors ξ µt and ξµx become ∂t = (1,0) and ∂x = (0,1) at the boundary. Considera spatial geodesic with endpoints on the boundary at points 0 and R(γ,γv), wherev < 1, γ = (1− v2)−1. We would like to determine the variation in length of thegeodesic under a variation in the proper length R of the boundary interval.The geodesic is an extremum of the actionS =∫ fidλ√gµνdxµdλdxνdλ. (C.1)1It is interesting to note that techniques similar to those in this section were used in [123] to showa relation between differential entropy and the lengths of bulk curves.164In general, the variation of an action S =∫dλL (qn, q˙n) evaluated for an on-shellconfiguration under a variation of the boundary conditions (assuming the range ofintegration remains the same) is given byδS = [pnδqn] fi ,where qn are the coordinates and pn = ∂L /∂qn are the conjugate momenta. Thisfollows immediately since the variation of the action gives a total derivative whenthe Euler-Lagrange equations are satisfied. Consider a general variation of theendpointsδxµf = δxξµx +δ tξµt .Since the conjugate momentum to xµ ispµ =∂L∂xµ=gµν dxνdλ√gµν dxµdλdxνdλ.we haveδS = δxξ µx pµ +δ tξµt pµ . (C.2)Now, for a Killing vector ξ µ , the action (C.1) is invariant under xµ → xµ + ξ µ .The corresponding conserved quantity is exactly ξ µ pµ . Thus, the right hand sideof (C.2) can be evaluated at any point on the trajectory. We choose to evaluate itat the midpoint of the geodesic, where ∂λ xµ is a linear combination of ξµt and ξµx(i.e. with no component in the radial direction). In this case,∂λ xµ = ξµtξt ·∂λ xξt ·ξt +ξµxξx ·∂λ xξx ·ξx ,so we find that our expression (C.2) becomesδS = δx[γ0Ax0]+δ t[γ0β0At0] . (C.3)where we have definedAx0 =√ξx ·ξx165At0 =√−ξt ·ξtβ0 =Ax0At0ξt ·∂λ xξx ·∂λ xγ0 =1√1−β 20,which measures the “tilt” of the geodesic at the midpoint.In the special case of a spatial interval, we will have ξt · ∂λ x = 0 everywhere,soδSδR=√ξ 2x =√gµνξµx ξ νx . (C.4)Thus, the variation of the entanglement entropy with respect to the size of a spatialinterval gives exactly the spatial scale factor.166Appendix DRindler Reconstruction forScalar Operators in CFT2In this appendix we find an expression for the matter contribution to the second-order perturbation to the entanglement entropy of a ball B using Rindler recon-struction so as to only use the one-point functions of the scalar operator in thedomain of dependence DB. We specialise to two dimensional CFTs in order toobtain a more explicit expression which can be compared to the gravitational con-tribution (5.10). Further discussions of Rindler reconstruction can be found in theliterature [115, 116, 124, 114, 113].Coordinates on the Rindler wedge RB of radius R can be given by (r,τ,φ)whichmap back into Poincare´ coordinates byz =Rr coshφ +√r2−1coshτ , (D.1)t =R√r2−1sinhτr coshφ +√r2−1coshτ , (D.2)x =Rr sinhφr coshφ +√r2−1coshτ , (D.3)where 1 < r < ∞.The scalar field dual to an operator O can be reconstructed in this Rindler167wedge using [113]φ(r,τ,φ) =∫dωdk e−iωτ−ikφ fω,k(r)Oω,k , (D.4)fω,k(r) =r−∆(1− 1r2)−iω22F1(∆2− i(ω+ k)2,∆2+i(ω+ k)2;∆;r−2), (D.5)where Oω,k is the Fourier transform of the CFT expectation value of the operatorOω,k =∫dτdφ eiωτ+ikφ 〈O(τ,φ)〉 . (D.6)This can be expressed in terms of the operator in the original coordinatesOω,k =∫DBdtdx[(R+ x+ t)ik+ω2 (R− x− t)−i k+ω2(R− x+ t)iω−k2 (R+ x− t)i k−ω2]〈O(t,x)〉 , (D.7)where the region of integration is only over the domain of dependence DB.This form of the scalar field can be combined with (5.58) to obtain an an ex-pression for δ (2)Sscalar which only depends on the expectation value of O in DB,δ (2)Sscalar =− 14∫ ∞1drdkdω1dω2 r√r2−1[fω1,k(r) fω2,−k(r)(−ω1ω2r2−1 +k2r2+∆(∆−2))+(r2−1) f ′ω1,k(r) f ′ω2,−k(r)]Oω1,kOω2,−k . (D.8)168


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