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Momentum-space entanglement and the gravity of entanglement in AdS/CFT McDermott, Michael 2014

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Momentum-space entanglement andthe gravity of entanglement inAdS/CFTbyMichael McDermottB.Sc., Simon Fraser University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2014c© Michael McDermott 2014AbstractIn the first part of this thesis we explore the entanglement structure of rela-tivistic field theories in momentum space. We discuss a Wilsonian path in-tegral formulation and a perturbative approach. Using perturbation theorywe obtain results for specific quantum field theories. These are understoodthrough scaling and decoupling properties of field theories. Convergence ofthe perturbation theory taking loop diagrams into account is also discussed.We then discuss the entanglement structure in systems where Lorentz invari-ance is broken by a Fermi surface. The Fermi surface helps the convergenceof perturbation theory and entanglement of modes near the Fermi surfaceis shown to be amplified, even in the presence of a large momentum cutoff.In the second part of this thesis we explore the connection between en-tanglement and gravity in the context of the AdS/CFT correspondence. Weshow that there are certain thermodynamic-like relations common to all con-formal field theories, which when mapped via the AdS/CFT correspondenceto the bulk are tantamount to Einstein’s equations, to lowest order in themetric.iiPrefaceThis thesis contains some of the work published by the author of this disser-tation which has previously appeared in the peer reviewed journals PhysicalReview D and the Journal of High Energy Physics.The subject of Chapter 2 was published in Physical Review D [74] underthe title “Momentum-space entanglement and renormalization in quantumfield theory”. This work, which has been edited, was a collaboration betweenthe author of this dissertation, the author’s supervisor Mark Van Raamsdonkand Vijay Balasubramanian, a professor at the University of Pennsylvania.The subject of Chapter 3 was published in the Journal of High EnergyPhysics [24] under the title “Momentum-space entanglement for interactingfermions at finite density.” This work, which has been edited, was a col-laboration between the author of this dissertation, the author’s supervisorMark Van Raamsdonk and Leo Hsu, an undergraduate student at the time.The subject of Chapter 4 was published in the Journal of High EnergyPhysics [26] under the title “Gravitational Dynamics From Entanglement“Thermodynamics”.” This work, which has been edited, was a collaborationbetween the author of this dissertation, the author’s supervisor Mark VanRaamsdonk and Nima Lashkari, a post-doctoral researcher.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Quantum entanglement . . . . . . . . . . . . . . . . . 21.1.2 Measures of entanglement . . . . . . . . . . . . . . . 31.1.3 Entanglement in field theory . . . . . . . . . . . . . . 121.2 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Holography . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Why AdS? . . . . . . . . . . . . . . . . . . . . . . . . 201.2.3 Matching degrees of freedom . . . . . . . . . . . . . . 211.2.4 Partition functions . . . . . . . . . . . . . . . . . . . 221.2.5 Scalar field on AdS . . . . . . . . . . . . . . . . . . . 241.2.6 Geometric entropy in AdS/CFT . . . . . . . . . . . . 27ivTable of Contents1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . 352 Momentum-space entanglement and renormalization in quan-tum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2 The low-energy density matrix . . . . . . . . . . . . . . . . . 402.3 Measures of entanglement . . . . . . . . . . . . . . . . . . . . 442.3.1 Entanglement observables in perturbation theory . . 462.3.2 Entanglement observables in quantum field theory . . 492.4 Scalar field theory: entanglement between scales . . . . . . . 522.4.1 The φ3 theory in 1+1 dimensions . . . . . . . . . . . 542.4.2 The φ3 theory in higher dimensions . . . . . . . . . . 572.4.3 φ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.4 General remarks . . . . . . . . . . . . . . . . . . . . . 602.5 The extent of entanglement between scales . . . . . . . . . . 632.5.1 An aggregate measure of the range of entanglement . 632.5.2 Single mode entanglement . . . . . . . . . . . . . . . 662.5.3 Mutual information between individual modes . . . . 682.5.4 Convergence and validity of leading order expressions. 692.6 Entanglement in the Thirring model . . . . . . . . . . . . . . 712.6.1 The correspondence . . . . . . . . . . . . . . . . . . . 722.6.2 The ground state . . . . . . . . . . . . . . . . . . . . 722.6.3 Thirring ground state . . . . . . . . . . . . . . . . . . 742.6.4 Checks . . . . . . . . . . . . . . . . . . . . . . . . . . 752.6.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 782.6.6 Direct calculation . . . . . . . . . . . . . . . . . . . . 812.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Momentum-space entanglement for interacting fermions atfinite density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Momentum-space entanglement in perturbation theory . . . 94vTable of Contents3.3 Entanglement entropy for lattice fermions with nearest neigh-bor interactions . . . . . . . . . . . . . . . . . . . . . . . . . 973.4 Entanglement entropy for continuum non-relativistic fermions1013.4.1 Mutual information between modes . . . . . . . . . . 1033.5 Entanglement entropy for relativistic fermions . . . . . . . . 1053.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134 Gravitational Dynamics From Entanglement “Thermody-namics” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.2 Entropy-energy relation . . . . . . . . . . . . . . . . . . . . . 1164.3 Gravitational implications of dS = dE in holographic theories 1184.3.1 Gravitational calculation of dS . . . . . . . . . . . . . 1184.3.2 Gravitational calculation of dE . . . . . . . . . . . . 1204.4 Derivation of linearized Einstein’s equations from dE = dS . 1214.4.1 Proof that δS = δE for solutions of Einstein’s equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.4.2 Proof that δS = δE implies the linearized Einstein’sequations . . . . . . . . . . . . . . . . . . . . . . . . . 1244.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133AppendicesA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.1 Momentum-space entanglement and correlators . . . . . . . . 142A.2 Entanglement entropy in a fermionic system . . . . . . . . . 143B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.1 Two qubit system example . . . . . . . . . . . . . . . . . . . 145viTable of ContentsC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.1 Alternative derivation of linearized Einstein’s equations fromδE = δS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.1.1 Expansion of δE = δS in powers of R . . . . . . . . . 150C.1.2 Checking that solutions of Einstein’s equations satisfyδS = δE . . . . . . . . . . . . . . . . . . . . . . . . . 151viiList of Tables2.1 Spatial dimensions where momentum space mutual informa-tion and entanglement entropy converge. The results applyfor any bounded regions A and B in momentum space. . . . . 71viiiList of Figures1.1 Mutual information . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Entanglement entropy in a two spin system. . . . . . . . . . . 91.3 Density matrix path integral boundary conditions . . . . . . . 141.4 Reduced density matrix path integral boundary conditions . . 141.5 Diagram depicting Bekenstein-Hawking entropy as maximal . 181.6 Diagram illustrating some requirements for a holographic theory 191.7 Diagram illustrating the holographic dual of geometric entropy 271.8 Diagram illustrating area law divergence . . . . . . . . . . . . 301.9 Diagram used in strong subadditivity proof . . . . . . . . . . 321.10 Ryu-Takayanagi surface for global AdS . . . . . . . . . . . . . 331.11 UV/IR connection in AdS/CFT . . . . . . . . . . . . . . . . . 341.12 Figure of possible momentum space Ryu-Takayanagi relation 351.13 Regions of spacetime separating due to loss of entanglement . 362.1 Leading contributions to S(µ) for φ3 theory in 1+1 dimen-sions. Full result for S(µ) is proportional to λ2(log(1/λ2)+1)times bottom function plus λ2 times top function. . . . . . . 572.2 (A) Integration regions for φ3 theory in 2+1 dimensions. (B)The function F (x) appearing in the entanglement entropy forφ4 theory in 1 + 1 dimensions. . . . . . . . . . . . . . . . . . . 582.3 Ratio of first and second terms in (2.53) vs µ = (µ2 − µ1)/mfor (A) µ1 = 1 and (B) µ1 = 4. This is a measure of therange of entanglement in φ4 theory in 1 + 1 dimensions. Wehave taken the mass to be m = 1. . . . . . . . . . . . . . . . . 66ixList of Figures2.4 Single-mode entanglement entropy vs magnitude of mode mo-mentum for φ3 field theory in 1+1 (bottom), 2+1 (middle),and 3+1 (top) dimensions. The entropies are normalized bytheir values at p = 0. . . . . . . . . . . . . . . . . . . . . . . . 683.1 Leading perturbative contribution to single mode entangle-ment entropy S(k) as a function of mode momentum for lat-tice fermions at half filling. This diverges logarithmically atthe Fermi points k = ±pi/2. . . . . . . . . . . . . . . . . . . 1003.2 Leading perturbative contribution to single mode entangle-ment entropy S(pFk) as a function of mode momentum (asa fraction of the Fermi momentum) for weakly interactingcontinuum non-relativistic fermions. This diverges logarith-mically at the Fermi points. . . . . . . . . . . . . . . . . . . 1023.3 Regions of (k, l) space with different behaviors for I(k, l).Results for the unshaded regions may be obtained from theresults for the shaded regions using the indicated symmetriesof I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.4 Mutual information I(pFk, pF l) between individual modes fornon-relativistic fermions with Fermi momentum pF . . . . . . 1063.5 Mutual information I(pFk, pF l) vs l for non-relativistic fermionswith k = 0 (top left), k = 0.75 (top right), k = 0.95 (bottomleft), and k = 1.25 (bottom right). The overall scale for I isarbitrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.6 Mutual information I(p, q) vs q for relativistic fermions withµ = 10m and p = 0 (top left), p = 7.5 (top right), p = 9.5(bottom left), and p = 12.5 (bottom right). Momenta aregiven in units of m, so the Fermi points are at ±10. Theoverall scale for I is arbitrary. . . . . . . . . . . . . . . . . . . 1113.7 Mutual information I(p, q) as a function of p and q for rela-tivistic fermions with µ = 0. Momenta are given in units ofm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112xGlossaryQFT Quantum field theoryCFT Conformal field theoryAdS Anti-de-SitterAdS/CFT The correspondence between gravitational theories on asymptoticallyAdS spacetime and CFT on the boundaryDOF Degrees of freedomIR InfraredUV UltravioletLHS Left hand sideRHS Right hand sideRT Ryu-TakayanagiBH Bekenstein-HawkingEPR Einstein, Podolski and RosenxiAcknowledgementsI wish to sincerely and gratefully thank my supervisors Mark Van Raams-donk and Gordon Semenoff.It is to Gordon’s energy and brilliance that I owe my original impetusto explore string theory, his joy in physics is delightfully contagious.Without Mark’s intellectual force this thesis would not have been pos-sible. His patient guidance and lucid explanations have been more thanindispensable, they have been appreciated.I would also like to express gratitude to Louis Deslauriers, Moshe Rozaliand Ariel Zhitnitsky.xiiDedicationTo my Family.xiiiChapter 1IntroductionThe concept of entanglement and the oddities it entails were famously con-sidered in a paper by Einstein, Podolsky and Rosen (EPR) [16]. Making useof an entangled state they argued that a quantum mechanical description ofreality could not be considered complete. It was thus their view that eventhough it accurately captured some aspects of physical reality, there wereothers that it was in principle incapable of describing.It wasn’t until 29 years later that Bell [4] studied what a completion ofquantum mechanics would imply. He termed a theory with completeness inthe sense demanded by EPR a local hidden variable model. Such a theory islocal so that spacelike separated events and observations are independent, ithas definite physical properties (some of which are “hidden” from quantummechanics) before measurement and furthermore these are unaffected bythe measurement. Using these assumptions he was able to show that a localhidden variable model would give rise to experimental results inconsistentwith quantum mechanics. Interestingly, the difference in predictions wasentirely due to the possibility of entanglement in the quantum description.Many experiments, e.g [2], have since been performed, excluding thepossibility of local hidden variable models. It would thus appear that en-tanglement is a physical aspect of reality. This means that systems whichare far apart really can display “spooky action at a distance” 1 whereby themeasurement of systems separated by incredibly large distances can “havean effect” on each other. Furthermore, complete knowledge of the stateof a physical system does not in fact imply knowledge of the state of itsconstituent subsystems.Bizarre as its properties may be, entanglement has become a very im-1This sentiment was expressed by Einstein in a correspondence with Max Born.11.1. Entanglementportant topic in physics in the last 20 years and has found many usefulapplications. A few key areas where entanglement plays a fundamental roleare quantum cryptography [5], quantum teleportation [6], and in a verygeneral sense quantum information theory. It has been used as a tool forunderstanding disordered systems with long ranged interactions [15], super-conductivity [42], and properties of frustrated system ground states [13]. Italso serves as an order parameter in understanding quantum phase transi-tions, see e.g. [43] and references therein.The main motivation for the study of entanglement in this thesis, how-ever, comes in the context of AdS/CFT. This will be discussed at the endof the chapter after a brief review of some salient features of entanglementand AdS/CFT. Throughout the thesis we use units where c = ~ = k = 1.1.1 Entanglement1.1.1 Quantum entanglementQuantum mechanics as a theory is incredibly successful, however, it appliesto scales far outside the realm of human experience. As a result, it hasmany counter-intuitive features such as quantization, superposition, tun-nelling and summing over paths. Not least of these features is quantumentanglement, which has no classical counterpart. To gain an understand-ing of this consider a physical system which may be divided into multiple(k) subsystems.As a concrete classical example this could be k isolated boxes of identicalgas, each with number of particles nk. Then we can specify the classical stateof the system by giving the position and momenta of all the particles in eachbox. So denoting the list of coordinates in box j as {~qi, ~pi}j , the total stateof the system would be [{~qi1 , ~pi1}1, {~qi2 , ~pi2}2, ..., {~qik , ~pik}k].The quantum mechanical description of k subsystems will, however, bequite different. Suppose that the j-th subsystem has a complete basis ofstates |s(j)i〉. Then because of the superposition principle an arbitrary state21.1. Entanglement|φ〉 of the system may be written as:|φ〉 =∑i1,...,ikci1,...,ik |s(1)i1〉 ⊗ |s(2)i2〉 ⊗ ...⊗ |s(k)ik〉. (1.1)The first difference to note between the classical and quantum case isthat, for the classical case the total state is just a product of the states of thesubsystem. The quantum system will generally not be writable as a productof states of the subsystem |φ〉 = |φ1〉⊗ |φ2〉⊗ ...⊗|φk〉. We are unsure aboutthe state of the subsystems, the state is entangled.The second thing to note, which is closely related to the first, is thatin the classical system knowledge of the state of the larger physical systemimplies knowledge of the state of the subsystems. In an entangled quantumstate, however, we run into the strange situation where complete knowledgeof the large physical system (i.e. specifying its state as in (1.1)) does notimply knowledge of the states of the subsystems.1.1.2 Measures of entanglementShannon entropyThe Shannon entropy is a classical entropy for a system with random vari-ables X. Thus event Xi occurs with probability pi, and the Shannon entropyis defined as:H(X) = −∑ipi log pi. (1.2)It measures the amount of information gained about X, on average, when anevent occurs. Equivalently, and this view of entropy will generally be moreuseful to us, the Shannon entropy quantifies the uncertainty in X we havebefore an event occurs. 2 The statement that the Shannon entropy mea-sures the average amount of information gained when we learn the valueof a random variable X can be heuristically “derived” from the following2A nice illustration to have in mind is a dice thrown under a box. The entropy measuresour lack of knowledge about what the dice will read when we lift the box.31.1. Entanglementarguments. First one demands continuity: we want a measure of gainedinformation G(p) to depend continuously on the probabilities of the randomvariable. Second one wants additivity of information: the amount of in-formation gained from knowledge of two independent events simultaneously(with probability p1p2) should be the same as the sum of the informationgained from individual events separately:G(p1p2) = G(p1) +G(p2). (1.3)From this it follows immediately that G(p) = G(1) +G(p) so that G(1) = 0,which is saying that the amount of information gained from an event certainto happen is 0. Next, taking p2 = c/p1 (where 0 ≤ c ≤ p1 is introduced tokeep p2 ≤ 1) we have G(p) = −G(1/p). Making use of continuity we alsohaveG′(p)dp =G(p+ dp)−G(p) (1.4)=G(1 +dpp) (1.5)=G(1) +G′(1)pdp (1.6)=G′(1)pdp (1.7)Integrating both sides with G(1) = 0 gives G(p) = G′(1) log(p). Note thatit makes sense to gain more information from an unlikely event. To get theaverage gain we simply weight each gain by the probability of it occuring:〈G〉 = G′(1)∑ipi log(pi). (1.8)Thus we see that the Shannon entropy is proportional to the average gainin information when we learn a random variable.41.1. EntanglementS(X) S(Y)I(X,Y)Figure 1.1: Diagram illustrating the mutual information.Classical mutual informationSince mutual information deals with two sets of random variables X andY we define the joint probability p(x, y) (probability of obtaining both xand y). Then the joint Shannon entropy (information gained on average byobtaining a joint event) is simply:H(X,Y ) = −∑p(x, y) log p(x, y), (1.9)the sum being over all possible events.The mutual information between two random variables X and Y canthen be defined as:I(X,Y ) = H(X) +H(Y )−H(X,Y ). (1.10)Any information which is gained in the independent regions is canceled out(counted twice in H(X) together with H(Y ) and twice in H(X,Y )) andany information gained in the mutual region is counted once in total (twicein H(X) together with H(Y ) and once in H(X,Y )). Thus the mutualinformation is a measure of how much information is in common between Xand Y , see Figure 1.1.To illustrate mutual information consider an example involving two 3-sided dice.1. The two dice are thrown independently. Then the probability distri-51.1. Entanglementbutions are independent p(x, y) = p(x)p(y) which means H(X,Y ) =H(X) + H(Y ). Thus in this case the mutual information I1(X,Y )vanishes.2. The two dice are attached in such a way that they always come up withthe same value. In other words they are the “same”. The joint optionsare (1, 1), (2, 2) and (3, 3). In terms of probabilities, this is the sameas the individual options (1), (2) and (3), and so H(X,Y ) = H(X) =H(Y ). We then have I2(X,Y ) = H(X) = H(Y ) = H(X,Y ) = log(3).Thus the mutual information in the case where the two random vari-ables are perfectly correlated (and so are basically the same) is justthe total information.3. The two dice are arranged in such a way that whenever a 1 is rolledon one die, the other also rolls a 1. There are 5 options for the jointdistribution: (1, 1), (2, 2), (2, 3), (3, 2), (3, 3). Then I3(X,Y ) = H(X)+H(Y ) − H(X,Y ) = log(3) + log(3) − log(5) = log(9/5). Thus in theintermediary case where the two dice aren’t independent but aren’tcompletely correlated either, the mutual information is between thatof the two limiting cases: I1 < I3 < I2.Von Neumann entropyIn classical mechanics systems which are in a statistical mixture of classicalstates x are described by a probability distribution p(x). Expectation valuesof functions of these classical variables are simply obtained as:〈F (x)〉 =∑xp(x)F (x). (1.11)The situation is different in quantum mechanics. Say a physical systemis in a mixture of quantum states |φi〉 which form a complete basis, eachwith probability pi. Then if Oi is the expectation value of some physicalobservable O in the state i, then its expectation value in the statisticallymixed state is:61.1. Entanglement〈O〉 =∑ipiOi (1.12)=∑ipi〈φi|O|φi〉 (1.13)=∑ipi〈φi|O|∑n|n〉〈n|φi〉 (1.14)=∑n〈n|∑ipi|φi〉〈φi|O|n〉 (1.15)= tr (ρO) (1.16)where we have defined ρ =∑pi|φi〉〈φi|. Thus a quantum mixed state isdescribed by a density matrix ρ, which encodes the classical probability piof a system being in a quantum mechanical state |φi〉.We can associate an entropy to the mixed state, the Von Neumann en-tropy:S(ρ) = − tr ρ log ρ = −∑pi log pi. (1.17)Notice that in the eigenbasis, this is the Shannon entropy for the eigenvalues.Thus in the eigenbasis of the density matrix, the Von Neumann entropyquantifies how uncertain we are about which state the system is in.Entanglement entropyConsider a physical system which can be decomposed into two subsystems Aand its complement A¯, so that its Hilbert space is a product H = HA⊗HA¯.In such a case we can form a reduced density matrix ρRA for degrees offreedom in the subsystem A by tracing the full density matrix ρAA¯ overdegrees of freedom in the subsystem A¯:ρRA = trA¯ ρAA¯. (1.18)Note that if ρAA¯ satisfies all the properties of a density matrix, then so willρRA. Physically, ρRA is the density matrix for an observer who only has access71.1. Entanglementto degrees of freedom in A. 3We are now ready to define the entanglement entropy. Assuming the fullsystem starts in a pure state |ψ〉, so that ρAA¯ = |ψ〉 〈ψ|, then the entangle-ment entropy is the Von Neumann entropy for the reduced density matrixρA = trA¯ |ψ〉 〈ψ|.As an example consider a system of two sites with spin 1/2. Prepare thesystem in the one parameter family of states:|ψ〉 =√u |↑↑〉+√1− u |↓↓〉 , (1.19)with u ∈ [0, 1]. Notice that at u = 0 and u = 1 the system is in a productstate and so is not entangled. At u = 1/2 each subsystem is equally likelyto be spin up or down and the system is maximally entangled.Let us calculate the entanglement entropy. The reduced density matrix(for either spin) is:ρs1/2 =u 00 1− u (1.20)Thus the entanglement entropy isS(ρs1/2) = −u log u− (1− u) log(1− u) (1.21)This is plotted in Figure 1.2. The entanglement entropy vanishes for u = 0and u = 1 and attains its maximum at u = 1/2, which is what we expectedfrom considering the entanglement in the original state.Entanglement between mixed statesIn order to discuss mutual information let us first discuss the concept ofentanglement between mixed states. Two subsystems A and B in mixed3An example of such a scenario is an accelerated Minkowski observer. Due to theacceleration there is an entire half of the space from which the observer may never receivelight signals, and as such the state is properly described by a reduced density matrix. Thishas an observable consequence in the Unruh effect, where the observer detects radiationat a temperature proportional to his/her acceleration [40] .81.1. EntanglementFigure 1.2: Entanglement entropy in a two spin system.91.1. Entanglementstates ρA and ρB are said to be entangled with each other if we cannot writethe joint state ρAB as [23]:ρAB =∑ipiρiA ⊗ ρiB, (1.22)where pi are probabilities, and ρiA and ρiB are some density matrices fordegrees of freedom in A and B respectively.This can be understood as follows [44]. Consider a system with Hilbertspace H = HA⊗HB. An uncorrelated state is one where each subsystem isin a state independent of the other, i.e.ρ = ρA ⊗ ρB (1.23)where ρ is the full state. An uncorrelated state has the property that ex-pectation values of joint observables O = OA ⊗OB factorize:〈O〉 = tr(ρO) (1.24)= tr(ρAOA ⊗ ρBOB) (1.25)=〈OA〉A 〈OB〉B (1.26)We can create a situation where the states of the two subsystems are cor-related. Simply have a random generator output numbers i = 1, ..., n withprobability pi. When i is generated, prepare the system in a state ρiA ⊗ ρiB.In this situation the expectation for O should be:〈O〉 =∑ipi tr(ρiAOA) tr(ρiBOB) (1.27)= tr(∑ipiρiA ⊗ ρiB OA ⊗OB) (1.28)= tr(ρcO). (1.29)101.1. EntanglementThus we conclude the system is in a correlated state:ρc =∑ipiρiA ⊗ ρiB. (1.30)The important point to note is that the device used is simply a classicalnumber generator (it could have been rolling a weighted n-side die) andso these correlations are classical. Now we see why density matrices whichcannot be written as ρc are entangled. Density matrices which can be writtenas ρc are termed separable.Quantum mutual informationWe are now ready to define mutual information. Take a system admitting aHilbert space H = HA ⊗HB ⊗HC . Then the mutual information betweendegrees of freedom in A and B is defined analogously to the classical case,in terms of the entanglement entropy as:I(A,B) = S(ρA) + S(ρB)− S(ρAB). (1.31)Now, for any regions R1 and R2 for which entanglement entropy S can bedefined, it satisfies certain inequalities, one of the more useful being strongsubadditivity which states that:SR1 + SR2 ≥ SR1∪R2 + SR1∩R2 (1.32)from which directly follows subadditivity of the entropySR1 + SR2 ≥ SR1∪R2 . (1.33)This last inequality implies that I ≥ 0. It is also true that the mu-tual information saturates the bound I = 0 iff ρAB = ρA ⊗ ρB 4, i.e. the4This follows as a simple application of Klein’s inequality, which states that for twodensity matrices ρ and d we havetr ρ log ρ− tr ρ log d ≥ 0, (1.34)with the bound saturated iff ρ = d. To prove subadditivity set ρ = ρAB and d = ρA⊗ ρB .111.1. Entanglementmutual information vanishes only when there is no classical correlation orentanglement in the state ρAB. Thus the mutual information measures allthe correlations between A and B, both classical and quantum mechanical.This has been shown rigorously in [19].1.1.3 Entanglement in field theoryNormalized replica methodConsider an unnormalized density matrix ρ, the normalized density matrixbeing ρˆ = ρ/ tr ρ. Then−(ddn− 1) ln tr ρn∣∣∣∣n=1=−1tr ρntr(ln ρ ρn)∣∣∣∣n=1+ ln tr ρ (1.39)=− tr(ρˆ ln ρˆ) (1.40)We thus haveS(ρˆ) = −(ddn− 1) ln tr ρn∣∣∣∣n=1. (1.41)This is a very useful relation since it allows us to determine the entropyof ρˆ from tr ρn (since we are differentiating with respect to n, we will needthis as a function of n). This is known as the replica method for computingthe entropy of a density matrix. Notice the other nice feature which is thatwe need not worry about the normalization of the density matrix, sincethe quantity in (1.41) always computes the entropy for the correspondingnormalized state.Then the Klein inequality givesS(ρAB) ≤− tr ρAB log ρA ⊗ ρB (1.35)=− tr ρAB log ρA − tr ρAB log ρB (1.36)=− tr ρA log ρA − tr ρB log ρB (1.37)=S(ρA) + S(ρB). (1.38)Thus S(ρA) + S(ρB) ≥ S(ρAB). The equality condition is ρ = d which yields ρAB =ρA ⊗ ρB .121.1. EntanglementLet us now see how the ground state entanglement entropy betweenspatial regions can be calculated using this method. Locality guaranteesthat the Hilbert space factorizes for spatial regions. Consider for simplicitya scalar field theory in one spatial dimension with some action S(ϕ). Thesystem starts in the ground state:ρ = |Ω〉 〈Ω| . (1.42)In position basis this isρ(ϕa, ϕb) = 〈ϕa|Ω〉〈Ω |ϕb〉 . (1.43)We can represent the transition amplitude as a path integral (after Wickrotation to Euclidean space)〈ϕa|Ω〉 =∫ ϕ(t=0,x)=ϕa(x)t=−∞Dϕ e−S(ϕ) (1.44)The total density matrix is thus a path integral over fields ϕ takingvalues in R2, and with different boundary conditions at t = 0 depend-ing on wether we approach from below (ϕ(t = 0−, x) = ϕa(x)) or above(ϕ(t = 0+, x) = ϕb(x)). This is depicted in Figure 1.3.Now divide space into two regions A and A¯ with xA consisting of allthe points x ∈ A and xA¯ consisting of all the points x ∈ A¯. We obtain thereduced density matrix for the region xA by tracing out the fields in A¯. Thisamounts to setting the fields equal on (t = 0, xA¯) and integrating over themin the path integral, see Figure 1.4. This path integral can be written interms of functional delta functions as: 5ρ(ϕa(xA), ϕb(xA)) =∫Dϕδ(ϕ(0−, xA)− ϕa(xA))δ(ϕ(0−, xA)− ϕa(xA))e−S .(1.45)5We need not keep track of the normalization by the vacuum partition function in thiscase, since our expression for entropy is independent of the normalization.131.1. EntanglementtFigure 1.3: Boundary conditions for the path integral representation of theground state density matrix.tFigure 1.4: Boundary conditions for the path integral representation of thereduced density matrix.141.1. EntanglementNow in order to compute tr ρnA we need:tr ρnA =∫ n∏iDϕAi ρ(ϕA1 , ϕA2 )ρ(ϕA2 , ϕA3 )...ρ(ϕAn , ϕA1 ). (1.46)Each ρ(ϕAi , ϕAi+1) is a path integral on R2, and the total path integral (1.46)relates them by requiring that the upper (t = 0+) boundary condition forthe j integral be the same as the lower (t = 0−) on the j + 1 integral.We can thus instead think of the full path integral (1.46) as being overa surface Σn, made up of n sheets, connected by these boundary conditions:tr ρnA =∫ΣDϕe−S(ϕ). (1.47)One important point to note is that if one were to stand at the regionseparating A from A¯ then one would need to travel in a circle n times tocome back to the original spot. There is thus a deficit angle of 2pi − 2pin atsuch locations.Area lawEntanglement entropy in local quantum field theory, between a connectedregion A and its complement A¯ is a divergent quantity, owing to the factthat in any region, no matter how small, a field theory has infinitely manydegrees of freedom. If we cut off the theory in the UV by placing it ona lattice with size a then one finds [8, 33] that the entanglement entropydiverges proportionally to the area ∂A bounding the region A:S(ρA) = k ·∂Aad+O(1ad−1)(1.48)where d is the spatial dimension of the theory 6. This result can be under-stood heuristically as arising because the entanglement between two spatial6Note that the mutual information is a nice quantity in this regard since it will ingeneral not suffer from this divergence. Recall that I(A,B) = S(A) + S(B) − S(A ∪ B),thus as long as the regions A and B don’t share any area in common and divergenceproportional to the area cancels out.151.1. Entanglementregions is concentrated near the boundary region dividing them.There is, however, a violation of this area law in 1+1 dimensional con-formal field theory. There one finds a logarithmic dependence on the size ofthe region A. If the region size is L then in a theory with central charge cone has [9, 22]:S =c3logL/a. (1.49)A nice scaling argument [38] which captures these features of the entan-glement entropy exists. The idea is to add up all the contributions to theentanglement entropy from each distance scale x. Thus we start with a mi-croscropic Hamiltonian at some cutoff scale Ha, then we imagine changingthe scale of interest to x, under which the Hamiltonian renormalizes to Hx.The theory can be considered local for scales larger than the renormalizionscale (i.e. for a lattice cutoff the theory is local at the lattice scale). Becauseof locality the theory will be entangled accross the boundary, but becausewe are considering scales larger than x this sets a minimum scale for degreesof freedom. Thus the boundary may only contribute as ∂Ad−1/xd−1 to theentanglement. Couple this with the fact that in renormalization group the-ory the canonical measure that always appears is dµ/µ or dx/x one getsthat each scale contributes to the entropy as:dS =∂Ad−1xd−1dxx. (1.50)The total contribution from all scales is then:S =∫ xIRxUV∂Ad−1xddx. (1.51)Taking xUV = a and xIR large this reproduces the area divergence for d ≥ 2.It also reproduces the conformal result for d = 1 since in that case the onlyavailable length scale (the system is infinite in extent) for xIR is L.161.2. AdS/CFT1.2 AdS/CFTMuch of what is discussed in this chapter is based on excellent reviews suchas [1, 14, 29–31].1.2.1 HolographyAlthough the specifics of the so-called AdS/CFT duality can be quite in-volved, and the literature on the subject is already quite substantial andever growing, the basic idea is very simple. It states that a quantum theoryof gravity (on asymptotically Anti-de-Sitter spacetime) is exactly the sameas a regular quantum field theory (with conformal symmetry) on its bound-ary. It relates a d+ 1 dimensional gravitational theory to at d dimensionalfield theory. As such it is an example of a more general idea called the holo-graphic principle [35, 39]: that our physical universe is a “hologram” whichmay in fact be described by a theory in one less dimension.The canonical example of holography, AdS/CFT, is provided by stringtheory. The logic of holography, however, does not rely on string theory. 7The first motivation for holography comes from considering the entropyof a black hole. Black holes were known to obey analogous laws to the fourlaws of thermodynamics, but it wasn’t until Hawking [20] proved that blackholes radiate a thermal spectrum that they were shown to be more than justan analogy; blackholes really were thermodynamic objects. Along with thiscame the famous result for the black hole entropy, the Bekenstein-Hawkingentropy:SBH =A4GN, (1.52)where A is the area of the black hole. Because this is a true entropy itsuggests that the degrees of freedom for a black holes scale with its areainstead of its volume.Furthermore the Bekenstein-Hawking entropy is actually the maximum7It should be noted that in practice all known examples of holographic dualities knownend up having interesting connections to string theory.171.2. AdS/CFTform black holeFigure 1.5: Diagram depicting Bekenstein-Hawking entropy as maximal.entropy for a region ΣA delimited by a sphere of area A. To see this, assumeto the contrary that there is something else inside the spherical region ΣAbounded by area A and which has entropy S > SBH (Figure 1.5). Thestarting configuration then cannot be a larger black hole since that wouldn’tfit inside ΣA, it can only be a smaller black hole or something with lessenergy. Thus since the starting configuration must have less energy thanthe black hole of size ΣA, we can always just throw in energy until wecreate the black hole which fits. This process, however, can only increasethe entropy, but since we end up with S = SBH this is a contradiction.Then SBH = A/4GN gives the maximum entropy for a region, and so,somewhat astoundingly, we conclude that the number of degrees of freedomwe need to describe a gravitating region ΣA scales with its area rather thanits volume.A second argument for holography can be given as follows [28]. Holog-raphy relates fields in a bulk region to fields on the boundary of this region.This on its own is perhaps not surprising since in order to describe what ishappening at point A Alice can send out light signals, which will eventually181.2. AdS/CFTABABFigure 1.6: Diagram depicting some requirements for a holographic theory.reach the boundary at point B 8 where Bob can be sitting to collect the sig-nals. Bob can then account for the interactions with the signal on the wayto him to reconstruct Alice’s message. In this way we can write boundaryquantities in terms of bulk ones (this is depicted in the left side of Figure1.6).Holography, however, demands more than this. It demands that quanti-ties in the bulk can be written in terms of quantities on the boundary at thesame time. This is because the boundary theory has its own HamiltonianH∂ which can be used to write fields at later times in terms of earlier ones(this is depicted in the right side of Figure 1.6). Typical field theories willcertainly not have this stronger property 9.Theories of quantum gravity, however, may have this property. This isbecause the gravitational Hamiltonian HQG is known to reduce to a bound-ary term on shell. If this remains true in the full quantum gravity then thestronger step required by holography is easily accomplished, we can simply8This is of course except in the case where Alice is inside an eternal black hole. Sincewe are assuming a unitary theory with no loss of information, even if Alice started in ablack hole, as long as it evaporates the information would escape.9Classical theories of gravity won’t either since we can specify initial conditions on atime slice, so that quantities on the boundary are independent of those in the bulk.191.2. AdS/CFTtake the observable at time tB where Bob was and map it back to an earliertime t:O(tB) = eiHQG(tB−t)O(t)e−iHQG(tB−t). (1.53)In this way one can relate boundary observables at any time to those attime t.1.2.2 Why AdS?In this section we wish to further motivate the correspondence by askingwhy it is anti-de-Sitter space in particular which is related to a conformalfield theory on the boundary. Given that the d + 1 dimensional CFT is inMinkowski space, it has the Poincare´ invariant metric:ds2CFT = −dt2 + d~x2. (1.54)The most general higher dimensional metric (z being the extra dimension)which respects the Poincare´ symmetry isds2AdS = f(z)(−dt2 + d~x2)+ g(z)dz2, (1.55)which after a coordinate redefinition can be written:ds2AdS = c(z)(−dt2 + d~x2 + dz2). (1.56)Demanding that this be invariant under conformal transformations (t, ~x, z)→λ(t, ~x, z) (since z has dimensions of length it should transform along withboundary coordinates) requires c(z) ∝ 1/z2. Inserting a length scale L tomake c(z) dimensionless gives:ds2AdS =L2z2(−dt2 + d~x2 + dz2), (1.57)which is the metric for pure AdS spacetime in d+ 2 dimensions. 1010This metric actually only covers the part of AdS spacetime visible to certain accel-erated coordinates, which is termed the Poincare´ patch. This is similar to the Rindler201.2. AdS/CFTAnother check that it is AdS which should correspond to the CFT isthat the symmetries of the theories match. In this case the isometries(transformations which leaves the metric invariant) of pure AdSd+2 forma SO(2, d+ 1) symmetry, which is indeed the same as the conformal groupin one temporal and d spatial dimensions. 11Also note that from (1.57) we can immediately obtain the so-calledUV/IR correspondence in AdS/CFT which states that high energy exci-tations in CFT are related to length scales far away from the interior ofAdS (near the boundary). From (1.57) we have1EAdS∼1z1ECFT. (1.58)ECFT is scaling as 1/z so that approaching the boundary z → 0 correspondsto higher energies in the boundary theory (and vice versa).1.2.3 Matching degrees of freedomIn order for the AdS/CFT duality to be correct there must be a matching ofdegrees of freedom between the two theories. Field theories boast infinitelymany degrees of freedom, however, and so one must impose an IR cutoff bylimiting the spatial extent of the system to R (i.e. we put the field theoryin a box of size R) and a UV cutoff by imposing a lattice cutoff a.On the d + 1 dimensional field theory side the number of degrees offreedom NCFT is simply the number of lattice sites Nl times the degrees offreedom per site c. 12 ThusNCFT = c ·Nl = c ·Rdad. (1.59)We can calculate the degrees of freedom in AdS by using the Bekenstein-coordinates in flat spacetime.11The conformal group in 3 + 1 dimensions consists of 15 symmetries. The Poincare´group: 4 translations, 6 boosts and rotations, then the leftover conformal part has 1dilation and 4 special conformal transformations.12The nomenclature comes from the fact that in a conformal field theory the centralcharge c measures the number of degrees of freedom per site.211.2. AdS/CFTHawking entropy:NAdS ≈ SBH =ABound4GN, (1.60)where ABound is the area of the boundary of AdS. Using the metric (1.57),with the UV cutoff at z = a (since z is related to energy scales in the fieldtheory) we have:ABound =∫z=addx√g =∫ddxLdad= RdLdad, (1.61)where the boundary coordinates were cutoff by the size of the box, R. Thusup to overall constants, the matching NCFT = NAdS requiresc =LdGN=Ld`dP, (1.62)where `P is the planck length.A particularly important limit for us will be when the classical gravitytheory is a good approximation. This is the regime where general relativityis valid, i.e. when the length scale of the spacetime is much larger than thePlanck length. This means that Ld`dP 1. Thus in the CFT, according to(1.62) this means that c  1. So classical gravity will be a good approxi-mation when the number of degrees of freedom per lattice site in the CFTis very large. 131.2.4 Partition functionsOne can package all the information in a quantum theory into its partitionfunction, and so it shouldn’t be surprising that a statement of the AdS/CFTduality is [45]:ZAdS = ZCFT . (1.63)13Note that for an SU(N) gauge theory NCFT = c ∼ N2, and thus classical gravity isa good description at large N .221.2. AdS/CFTMore precisely, to every field ϕi 14 in the gravity side there corresponds anoperator Oi in the CFT which couples to the field ϕi at the boundary, as itwould to a source. In other words the presence of a field ϕi on the gravityside with boundary conditions ϕi(x, z = 0) = ϕi0(x) gives rise to a sourceterm∫ϕi0Oi for the CFT. ThusZAdS |ϕi→ϕi0 = 〈e∫ϕi0Oi〉CFT ≡ ZCFT [ϕi0]. (1.64)We will be particularly interested in the limit where classical gravityapplies so that the gravity partition function is dominated by a saddle point,and thus well approximated by its classical solution:ZAdS |ϕi→ϕi0 =∑all fields∣∣∣∣∣ϕi→ϕi0eSG ≈ eSclassicalG∣∣∣ϕi→ϕi0. (1.65)So with (1.64) we now have a way to compute quantum correlation functionsin terms of a classical gravity action as〈Ok(x1)...Oj(xn)〉 =δδϕk0(x1)...δδϕj0(xn)logZCFT [ϕi0] (1.66)=δδϕk0(x1)...δδϕj0(xn)SclassicalG∣∣∣ϕi→ϕi0. (1.67)One should be careful when performing these manipulations since gener-ically the gravitational action and the fields at the boundary are divergent.To cure these divergences one typically adds a counter term action and ex-tracts the finite part of the boundary behaviour of the fields. An exampleof this is shown for a scalar field in the next section.14We represent any index or set of indices under which the field ϕ transforms (suchas spinor or vector) as i, with inner product denoted by contraction between raised andlowered indices.231.2. AdS/CFT1.2.5 Scalar field on AdSIn this section we wish to calculate the expectation value of an operator Oin the CFT vacuum, dual to a scalar field ϕ in pure AdSd+2 gravity (we willuse the Poincare´ coordinates (1.57)).The action for a free massive field in Euclidean AdSd+2 is:S = −12∫dzdd+1x√g(∂µϕ∂µϕ+m2ϕ2)(1.68)Since we are assuming ϕ to be a classical solution we may integrate by partsand use the equations of motion to get a boundary term:Sclassical =12∫dd+1x√g (ϕgzz∂zϕ)z= , (1.69)where we have cutoff the boundary at z = . In order to evaluate this one caneasily find the behaviour of the field ϕ near the boundary by considering theequations of motion obtained from 1.68 at small z. After a Fourier transformϕ(z, x) =∫dd+1k(2pi)d+1eikxϕk(z) (1.70)one finds that near the boundary z = 0 the field ϕ behaves as:ϕk(z) ∼ C1(k)zd+1−∆ + C2(k)z∆, (1.71)where ∆ = (d+ 1)/2 +√(d+ 1)2/4 +m2L2.Substituting into (1.69) gives at z = Sclassical =Ld2∫dd+1k(2pi)d+1[−(2∆−(d+1))(d+ 1−∆)C1(−k)C1(k) + (d+ 1)C1(−k)C2(k)],(1.72)where the C2(−k)C2(k) term is of order z2∆−(d+1) and so vanishes as z → 0for scalar fields which satisfy the Breitenlohner-Freedman (BF) bound. 1515Since ∆ = (d+ 1)/2 +√(d+ 1)2/4 +m2L2, we have2∆− (d+ 1) = 2√(d+ 1)2/4 +m2L2. (1.73)241.2. AdS/CFTNow we notice that the expression for Sclassical diverges as  → 0. Toremedy this we should add a counterterm action Sct. Like any good actionit should be local in the field ϕ. Note that to cancel the divergence theaction must be a boundary action, and it must be quadratic in the field inorder to be able to cancel the term of the form C1C1. A natural guess forthe counter-term action is thenSct ∼∫dd+1x√γ ϕ2, (1.75)γ being the induced metric for the z =  boundary. This action indeeddiverges as −(2∆−(d+1))C1(−k)C1(k), and so we just need to find the rightprefactor to cancel the divergence in Sclassical. This isSct = −d+ 1−∆2L∫dd+1x√γ ϕ2. (1.76)In this way we finally arrive at the finite piece of the action Srenorm =Sclassical + Sct, evaluated on the classical solution:Srenorm =Ld(2∆− (d+ 1))2∫dd+1k(2pi)d+1C1(−k)C2(k). (1.77)So now to get the expection value, from (1.67) we need:〈O〉 =δδϕ0Sclassical. (1.78)We already removed the divergences in Sclassical by using Srenorm, but thereis still the issue of the field ϕ diverging at the boundary.From (1.71) we see that ϕ diverges as zd+1−∆ near the boundary, and sowe can remove this divergence (and the divergence in the action) by insteadThis quantity is thus positive (or 0) wheneverm2 ≥ −(d+ 12L)2, (1.74)which is known as the BF (Breitenlohner-Freedman) bound on the mass.251.2. AdS/CFTusing the expression:〈O(x)〉 =δδ(z−(d+1−∆)ϕ0(x))Srenorm. (1.79)Then using (1.71), along with a Fourier transform we have〈O(k)〉 =δδC1(−k)Srenorm. (1.80)Combining this with (1.77) we can finally find the expectation value of theoperator in the CFT to be 16〈O(k)〉 = Ld(2∆− (d+ 1))C2(k). (1.84)Thus, for a scalar field in pure AdS, the expectation value of the dualoperator in the CFT is related to the subleading behaviour of the field atthe boundary.If one similarly calculates the 2-point function, one finds〈O(x1)O(x2)〉 ∼1|x1 − x2|2∆. (1.85)This is precisely the result for correlation function between operators ofdimension ∆ in a CFT.16Actually when one solves the equations of motion for the full solution ϕk(z) =C1(k)f1(k, z) + C2(k)f2(k, z), one finds that demanding that it stays finite as z → ∞fixes the ratio C1(k)/C2(k) = R(k). Thus when taking the functional derivative (1.80)one hasδδC1(−k)∫dqC1(−q)C2(q) =C2(k) +∫dqC1(q)δδC1(−k)C2(−q) (1.81)=C2(k) + C1(k)R(k) (1.82)=2C2(k) (1.83)261.2. AdS/CFTFigure 1.7: Diagram illustrating the holographic dual of geometric entropy.1.2.6 Geometric entropy in AdS/CFTRyu-Takayanagi formulaSuppose we divide up a d + 1 dimensional CFT into two spatial regionsA and A¯, with boundary between the two A∂ . We can then calculate theentanglement entropy SA between degrees of freedom in A and A¯. In thecontext of AdS/CFT a natural question to ask is “how do we compute SAin the dual gravity theory?”. Since we are working in the classical gravitylimit we may expect something geometric, which also satisfies a least action(or saddle point) condition. The correct dual quantity is given by the areaA˜ of the minimal surface whose boundary is given by A∂ 17, see Figure 1.7.The precise result, known as the Ruy-Takayanagi formula [32] is:SA =A˜4GN. (1.86)17We will only be concerned with static metrics and entanglement entropies, thus wecan think everything taking place at a given time slice.271.2. AdS/CFTThe numerical factors are the same as in the Bekenstein-Hawking entropy,as is required for consistency in the thermal state.Sketch of proofThe formula (1.86) was proved in [27]. An outline of the proof can be givenas follows. We know from the discussion in section 1.1.3 that we shouldcompute the CFT partition function on a space Σn composed of n sheetsof R1,d glued together in the region A, and with deficit angles 2pi(1− n) atthe dividing surface A∂ . To compute this partition function the holographicduality states we need to compute the partition function of the correspondinggravity solution. Assuming that the dual gravity solution turns out to beone with a surface σA ending on A∂ which preserves the deficit angle (suchthat σA also carries a deficit angle of 2pi(1 − n)). This can be shown toinduce a curvature singularity [17] on σA, R ∼ 4pi(1 − n)δ(σA). Since thegravitational action integrates over the curvature R, this induces a termproportional to the area of the surface σA, which will be the only termwhich contributes to the entanglement entropy. 18 At this point we see thatsince the area term appears in the action it should be minimal at the saddlepoint, so that Area(σA) = A˜, the minimal area extending into the bulk withboundary A∂ . So we finally find18The gravitational action with cosmological constant isS(n)grav = −116piGN∫dd+2x√g (R+ Λ) , (1.87)so R having a delta function singularity on some surface produces a term proportional tothe area of that surface. Alsotr ρˆn =ZnZn1, (1.88)where Zn is the partition function on the n-sheeted space Σn. Thus any term in the totalpartition function Zn which is just n copies of single copy partition function cancels out.This is why only the piece which comes from the deficit angle is important.281.2. AdS/CFTSA = −ddnln tr ρˆn∣∣∣∣n=1(1.89)= −ddnlnZnZn1∣∣∣∣n=1(1.90)= −ddnlnZAdS(n)ZnAdS∣∣∣∣n=1(1.91)= −ddn4pi(1− n)Area(σA)16piGN∣∣∣∣n=1(1.92)=A˜4GN, (1.93)which is the Ryu-Takayanagi formula.Consistency checksArea law The first consistency check comes from our knowledge that ina CFTd+1 the geometric entropy obeys an area law divergence (see section1.1.3):SA ∼ k ·A∂ad−1(1.94)for some small distance cutoff a.On the gravity side a small distance cutoff a corresponds to a cutoff onthe spacetime at z = a, see Figure 1.8. Set up d − 1 coordinates σi on theboundary, such that the surface A∂ is parameterized as x(σ), with inducedmetric on the surface d then∫A∂dd−1σ√d = A∂ , (1.95)where we are using A∂ to describe both the surface and its area. In the bulk(with pure AdS metric (1.57)), however, on a slice of constant z the area ismodified by the induced metric h as well:291.2. AdS/CFTFigure 1.8: Diagram illustrating area law divergence.Abulk(z) =∫A∂dd−1σ√d√h, (1.96)=Ldzd∫A∂dd−1σ√d, (1.97)=LdzdA∂ . (1.98)Then near the boundary the contribution the the area A˜ can be obtained byintegrating over the physical area Abulk(z) close to the cutoff. The divergentpiece of this comes from the cutoff surface itself so thatA˜ ∼∫ adzAbulk(z) ∼ Ld A∂ad−1, (1.99)and so we haveSA =A˜4GN∼LdGNA∂ad−1. (1.100)Thus the Ryu-Takayanagi formula reproduces the known area law (1.94) in301.3. MotivationCFT.Entropic inequalities It is also possible to check that the Ryu-Takayanagiformula reproduces the strong subadditivity relation [21]:SA + SB ≥ SA∪B + SA∩B. (1.101)Denote the area of a holographic surface with boundary A as ΣA, if it isminimal we’ll call it σA. Then from Figure 1.9 it is clear thatσA + σB =ΣA∪B + ΣA∩B, (1.102)≥σA∪B + σA∩B. (1.103)In going from the first to second line we have gone from surfaces whichare not necessarily minimal to ones which are. Since the Ryu-Takayanagiformula in this context simply states thatSA =σA4GN, (1.104)the area inequalities (1.103) imply strong subadditivity of the entanglemententropy (1.101).1.3 MotivationAs we have seen, AdS/CFT (see Section 1.2) relates a gravitational theoryon asymptotically Anti-de-Sitter (AdS) space to a conformal field theory(CFT) on its boundary. Since our main interest lies in understanding theinterplay between geometry and entanglement, we will assume that we arein a regime in the gravitational theory where the general relativistic space-time picture is valid.19 In this regime there is a very nice geometric way ofquantifying the entanglement between two complementary spatial regions -A and A¯ - in the CFT. This is captured by the Ryu-Takayanagi (RT) formula19This is analogous to a regime in Quantum Electrodynamics where Maxwell’s equationsapply.311.3. MotivationFigure 1.9: Diagram used in strong subadditivity proof.(see Section 1.2.6). A measure of how entangled degrees of freedom in Aand A¯ are in a given state is computed by the entanglement entropy SA (seeSection 1.1). The RT formula states that SA is the area of a minimal surfacein the gravity theory (having as its boundary the region A∂ separating Aand A¯), see Figure 1.10 :SA =A˜4GN. (1.105)In this way a purely quantum mechanical phenomenon in one theoryis related to an entirely classical quantity in another. Roughly speaking,in Figure 1.10, we can think of A˜ as separating that region MA in thebulk corresponding to A on the boundary, from the region MA¯ in the bulkcorresponding to A¯ on the boundary. 20Now in AdS/CFT there is also a relation between the length scales on theboundary and “radial” distance in the gravity, the so-called UV/IR relation(see Section 1.2.2). It states that the physics of small length (large) scales onthe boundary CFT is understood by studying regions near (far away from)the boundary in the gravity theory. This can actually be seen in the RT20This correspondence was made more precise in [12].321.3. MotivationFigure 1.10: Diagram displaying the Ryu-Takayanagi surface A˜. The cylin-der on the left is global AdS spacetime (with time running vertically). Onthe right is a constant time slice where we consider the Ryu-Takayanagisurface for SA, separating spacetime into regions MA and MA¯.formula (see Figure 1.11). If A is a small region, then the minimal surfaceA˜ stays relatively near the boundary. As we increase the size of A, A˜ dipsdeeper and deeper into the bulk and so probes longer and longer distancescales in the gravity side.A natural question then is: what region in the bulk (if any) correspondsto degrees of freedom on the boundary CFT below or above a certain lengthscale? Divide up the boundary into degrees of freedom below and abovea length scale ` as `< (UV) and `> (IR) and denote the correspondingbulk regions as M`< and M`> . Now we can think of MA and MA¯ ascorresponding to A and A¯ respectively, and MA and MA¯ are obtained byconsidering the entanglement between regions A and A¯. Thus we argue thatthe regions M`< and M`> should be related to the entanglement in theCFT between UV and IR degrees of freedom.Fortuitously field theories provide a natural basis of states to study onemeasure of the UV/IR entanglement; the Fock basis. This basis of states canbe written as a product of occupation number npi of modes with momentum331.3. MotivationFigure 1.11: Diagram displaying a manifestation of the UV/IR connectionwith RT surfaces.pi below a certain scale µ (IR) |pi| < µ, and occupation nPi of modeswith momentum Pi above that scale (UV) |Pi| > µ. This gives us a wayto calculate the entanglement entropy Sµ between UV and IR degrees offreedom. Studying the question in enough detail we may hope to find amomentum space RT-like geometric picture in the gravity theory, depictedin Figure 1.12.The RT result also leads us to consider the “connectedness” of spacetimeregions MA and MA¯ in terms of the entanglement between A and A¯. Thiswas explored in [41]. Consider subregions a ⊂ A and a¯ ⊂ A¯. For a generalstate correlation functions between operators at spacelike points xa and xa¯are exponentially decaying; 〈OaOa¯〉 ∼ e−s(xa,xa¯)/ξ where s(xa, xa¯) is theproper length of the minimal geodesic between xa and xa¯, and ξ is somelength scale. A measure of the entanglement between degrees of freedom ina and a¯ is given by the mutual information I(a, a¯) (it also includes classicalcorrelations, see Section 1.1.2). This measure obeys a certain inequality [55]:I(a, a¯) ≥12(〈OaOa¯〉|Oa||Oa¯|)2(1.106)for any operators O referring to regions a and a¯ with vanishing expectationvalue. Thus if the entanglement as measured by I(a, a¯) vanishes, so must341.4. Outline of thesisFigure 1.12: Figure of possible momentum space Ryu-Takayanagi relation.the correlations between operators O on these regions, and this implies thatthe proper length between these regions diverges.We thus arrive at the following picture (see Figure 1.13) of what happensas we remove the entanglement between A and A¯: the bounding surface A˜separatingMA andMA¯ shrinks in accordance with RT, and the boundariesA and A¯ get further and further away from each other as measure by theproper distance or geodesics connecting them. So we get a situation whereMA andMA¯ are pinching off from one another. From this it is apparent thatentanglement in the CFT is serving to hold the dual space-time together.A natural question to ask is: precisely how do changes in the entanglement(perhaps subject to certain conditions) affect the dual geometry ?1.4 Outline of thesisThe structure of this thesis is as follows. In Chapter 2 we introduce and ex-plore entanglement in momentum space. We begin by giving a path integraldescription of the reduced density matrix for low momentum degrees of free-dom. We then develop a perturbative approach for computing entanglement351.4. Outline of thesisFigure 1.13: Diagram illustrating the separation of spacetime due to lossof entanglement in the CFT.361.4. Outline of thesisentropy and mutual information and apply these to example field theories.Entanglement properties are understood in terms of scaling arguments andrelated to properties of decoupling. Conditions under which our measuresare well defined in perturbation theory are also discussed.We change gears from Lorentz invariant theories to exploring systemswith Fermi surfaces in Chapter 3. We study the entanglement between sin-gle modes in a lattice theory and its continuum version. Surprisingly thepresence of a Fermi surface allows to compute the entanglement entropy indimensions where the Lorentz invariant version would have diverged. In-teresting behaviour near the Fermi surface is shown not to depend on highenergy details.In Chapter 4 we discuss how gravity “emerges” from certain thermodynamic-like relations on the boundary in AdS/CFT. We show how one arrives at therelation dS = dE in the CFT and explore the implications of each side ofthis equation on the AdS gravity side. The equation of the two quantities onthe AdS side can be shown (under suitable conditions) to imply Einstein’sequations.37Chapter 2Momentum-spaceentanglement andrenormalization in quantumfield theory2.1 IntroductionA quintessential property that distinguishes quantum mechanics from classi-cal mechanics is the entanglement of otherwise distinct degrees of freedom.When certain degrees of freedom are entangled with the rest of a quan-tum system, it is not possible to describe them by a pure state. Rather,the most complete description of a subsystem A at a particular time is viathe reduced density matrix obtained by tracing over the degrees of free-dom in the complement, ρA = trA¯(|Ψ〉〈Ψ|), where |Ψ〉 is the state of theentire system. The entropy constructed from the reduced density matrix,S(ρA) = − tr(ρA log(ρA)), quantifies the amount of entanglement betweenA and its complement. The entanglement entropies corresponding to re-duced density matrices for diverse subsets of degrees of freedom provide arich characterization of the quantum state for systems with many degreesof freedom.21In physical systems, we typically only have access to a subset of the de-grees of freedom, namely the low-energy or long-wavelength modes that are21For a basic review of density matrices, entanglement entropy, and related concepts,see for example [68].382.1. Introductiondirectly accessible to experiments. In an interacting theory, it will generallybe true that these degrees of freedom are entangled with the inaccessiblehigh-energy degrees of freedom. Thus, the long-wavelength modes will bedescribed by a density matrix. A more familiar description of low-energydegrees of freedom is due to Wilson [47] – one carries out the complete pathintegral over the inaccessible degrees of freedom, arriving at an effective ac-tion capturing the dynamics of the remaining system. Here we will indexthe degrees of freedom by their spatial momenta and consider integratingout modes of high spatial frequency. We show that the resulting Wilsonianprescription is compatible with the description in terms of a density matrix:given a Wilsonian effective action (defined more precisely below) we cancanonically associate the corresponding density matrix via equation (2.5)below.For continuous physical systems described by interacting quantum fieldtheories, understanding the variation with scale of the Wilsonian effectiveaction SW (µ) provides key insights into the nature of the quantum fieldtheories, revealing a striking insensitivity of the low-energy physics to thedetails of the ultraviolet description. Correspondingly, it is natural to con-sider the variation with scale of the density matrix ρ(µ) for the degrees offreedom with momentum |~p| < µ and the associated entanglement entropyS(µ). To make our considerations concrete, we derive a formula for thislow-energy entanglement entropy in perturbative quantum field theory, andapply it to scalar field theories with φn potentials in various dimensions. Thescale-dependence of the entropy S(µ) in such theories can be understood interms of the variation of the coupling and number of degrees of freedom withscale.To study entanglement between scales in greater detail, we can considerthe entanglement entropy associated with any subset of the allowed mo-menta, or the mutual information between any two subsets of the allowedmomenta, for example between individual modes with momenta ~p and ~q.These measures characterize the extent of entanglement between specificscales in field theory, and we compute the rate at which this entanglementdeclines as the scales separate. This fall-off may give an alternative charac-392.2. The low-energy density matrixterization of the property of decoupling in quantum field theories. In theoriesthat do not enjoy the property of decoupling, e.g. noncommutative gaugetheories [48] and theories of gravity, the entanglement between degrees offreedom at different scales may play an especially important role.Entanglement entropy in quantum field theory has been considered pre-viously, but almost all previous work has focused on entanglement betweendegrees of freedom associated with spatial regions (e.g. [69, 70] ). The no-tion of a density matrix for low-momentum modes or entanglement betweendifferent momentum modes has appeared earlier in the context of cosmologyand condensed matter physics (e.g. [51, 53, 71]), but there is little overlapwith the present work.2.2 The low-energy density matrixA quantum system with many degrees of freedom has a Hilbert space of theform H = H1⊗H2⊗· · · . Given a subset of these degrees of freedom (A, withcomplement A¯), we can write H = HA⊗HA¯ where HA is the tensor productof Hilbert spaces for the degrees of freedom in A. If ρ is the density matrix forthe full system (which may be in a pure state), a reduced density matrix forAis defined by tracing over A¯: ρA = trA¯(ρ) (or, given components in a specificbasis, ρAmn =∑N ρmN,nN ). Expectation values of operators that act only onA can be computed as tr (ρ (OA ⊗ 1 )) = trA(ρAOA). If A is entangled withits complement, ρA will have a finite entropy: S = − trA(ρA log ρA) > 0.In this construction, the Hilbert space can be be decomposed into atensor product in any convenient way. For example, the Hilbert space fortwo identical oscillators could be decomposed either as a product of theHilbert spaces for the individual oscillators, or as a product of the Hilbertspaces of even and odd normal modes. A reduced density matrix could becomputed in either case – good choices of decomposition are dictated by thestructure of the interactions and restrictions on which degrees of freedomare accessible to measurements.In quantum field theories, locality makes it natural to associate inde-pendent degrees of freedom with disjoint spatial domains. Hence, given a402.2. The low-energy density matrixspatial region A (and complement A¯), one can decompose the Hilbert spaceas H = HA ⊗ HA¯ and trace over A¯ to derive the reduced density matrixof A. But since the Hamiltonians of free field theories are diagonalized bymodes of fixed momentum, it is in many ways more natural to use the Fockspace decomposition, H = ⊗~pH~p, where H~p is the Hilbert space of modesof momentum ~p.22 While this decomposition is motivated by consideringthe case of free field theory, it applies equally well once we turn on interac-tions, and is indeed the standard setting for computations in perturbativequantum field theory.23In free field theory, the vacuum state is a tensor product of the Fock spacevacuum states for each independent field mode – there is no entanglementbetween the field modes at different momenta. But in an interacting theory,the full vacuum state will be a superposition of Fock basis states – hence themodes of different momenta will generally be entangled. In this case, thereduced density matrix corresponding to a subset of the degrees of freedom(A) will necessarily have a finite entropy, indicating that A is effectivelyin a mixed state if the rest of system is traced over. Now, one is mostoften interested in the physics of the “infrared” degrees of freedom thatare accessible to experiment, i.e., the degrees of freedom with momentabelow some scale µ. The present discussion shows that tracing over theultraviolet, i.e. degrees of freedom with momenta above µ, should lead toan infrared effective description in terms of a low-momentum density matrixcorresponding to a mixed state with finite entropy.22Here, it is clearest to define the field theory as a limit of a theory on finite volumeso that the tensor product is over a discrete set of allowed momenta. For a general fieldtheory, the factors would be labeled by field type and spin/polarization in addition tomomentum.23There is a formal sense in which turning on interactions takes one out of the originalHilbert space. However, by placing a cutoff at some energy scale much higher than anyscale of interest, the Hilbert space structure of the interacting theory will be the same asthat of the free theory, and a density matrix for low-momentum modes can be preciselydefined. Furthermore, as we will see later, various observables related to the spectrum ofthe density matrix have a well-defined limit as we take the cutoff to infinity.412.2. The low-energy density matrixRelation between low-energy density matrix and low-energyeffective actionThe standard way of studying the low-energy theory is through the Wilso-nian effective action. How is this quantity related to the low-energy densitymatrix? To begin, consider a bare action SΛ defined with a cutoff Λ suchthat |p| ≤ Λ. Associated to this, we have a Hamiltonian HΛ, which will havesome ground state |Ψ0Λ〉 and corresponding density matrix ρ0Λ = |Ψ0Λ〉〈Ψ0Λ|.This density matrix can be written as a Euclidean path integral by takingthe T → 0 limit of the finite temperature density matrix〈φˆy|ρTΛ|φy〉 =1Z〈φˆy|e−βHΛ |φy〉 =1Z∫ φ(τ=β/2)=φˆyφ(τ=−β/2)=φyDφ(τ) e−SEΛ , (2.1)where {φy} is a basis of field configurations indexed by y, β = 1/T is theinverse temperature, SEΛ is the Euclidean action, and Z is the partition sumthat normalizes the path integral.Given a subset of degrees of freedom A (with complement A¯) and thetensor product structure of the Hilbert space, we can split the parametery which indexes the basis states as y = (a, a¯), and a pick a basis withφy = φa × φa¯. To define a reduced density matrix for A by tracing over A¯we write〈φˆa|ρTA|φa〉 =∫Dφ′a¯ 〈φˆaφ′a¯|ρTΛ|φaφ′a¯〉 =1Z∫ φA(β/2)=φˆaφA(−β/2)=φaDφA(τ)DφA¯(τ)e−SE .(2.2)In the last expression, the fields φA¯ are periodic in the range [−β/2, β/2],which is implied after substituting (2.1) into the trace in the middle expres-sion.Now define a conventional thermal effective action for the the subsystemA:e−STW (φA) =∫−β/2≤τ≤β/2DφA¯(τ) e−SE(φA,φA¯) . (2.3)422.2. The low-energy density matrixIn terms of this, the density matrix for A is〈φˆa|ρTA|φa〉 =1Z∫ φA(β/2)=φˆaφA(−β/2)=φaDφA(τ) e−STW (φA) =1Z∫ φA(τ=0+)=φˆaφA(τ=0−)=φaDφA(τ) e−STW (φA) .(2.4)In the last expression we translated time to put the discontinuity in theintegral at τ = 0± and the fields are taken to be periodic at τ = ±β/2. Thereduced density matrix for A in the ground state |Ψ0Λ〉 for the entire systemis extracted by taking β →∞.We now specialize to the case where A is the subset of degrees of free-dom with spatial momenta |p| < µ for any scale µ which is lower than theultraviolet cutoff Λ. The reduced density matrix ρ|p|<µ obtained by tracingover the degrees of freedom with momenta in the range µ < |p| ≤ Λ is thusgiven by:〈φˆ|p|<µ|ρ|p|<µ|φ˜|p|<µ〉 =1Z∫ φ|p|<µ(τ=0+)=φˆ|p|<µφ|p|<µ(τ=0−)=φ˜|p|<µDφ|p|<µ e−SW (φ|p|<µ) .(2.5)where now (having taken β → ∞,) SW is a Wilsonian effective action ob-tained by integrating out the degrees of freedom with large spatial mo-menta:24e−SW (φ|p|<µ) =∫Dφ|p|>µ(τ) e−SE(φ|p|<µ,φ|p|>µ) . (2.6)Equation (2.5) is our final result for the low-energy density matrix. In24In relativistic quantum field theories, it is perhaps more common to define a Wilsonianeffective action by performing the Euclidean path integral over all field variables φ(pν)with |pνpν | > µ. This is convenient, since it leads to an effective action that is manifestlyLorentz invariant. However, the remaining variables in the path integral correspond to arestricted set of frequencies for each field mode φ(~p). Hence, this Wilsonian action doesnot represent the full effective action for a particular subset of degrees of freedom, butrather the action for a restricted set of configurations of a subset of degrees of freedom.In our case, while the path integral is still Euclidean, we are integrating out all modeswith spatial momenta |~p| > µ and leaving all modes with |~p| ≤ µ, regardless of frequency.The result is an effective action that describes all possible configurations of a subset ofdegrees of freedom for the theory, the field modes with |~p| ≤ µ. This type of Wilsonianaction is more commonly discussed in field theories without Lorentz invariance, such asthose describing condensed matter systems (see, for example, [54]).432.3. Measures of entanglementparticular, if O is an observable built out of the low-momentum modes atτ = 0, it follows from (2.5) thattr(Oˆρ) =1Z∫[dφ|p|<µ]Oe−SW (φ|p|<µ) , (2.7)so the standard calculation using the effective action is equivalent to a cal-culation using the density matrix. Of course, the full Wilsonian effectiveaction contains more information than the density matrix associated withthe vacuum state of the field theory. The former is a functional of time-dependent field configurations, while the latter depends only on a pair oftime-independent field configurations.The description of low-energy degrees of freedom via a density matrixmay seem unfamiliar and one may ask why we cannot simply associatea pure vacuum state to the low-energy degrees of freedom based on theeffective action. The reason is that SW will typically contain terms withhigher time derivatives, so there is no way to associate to SW a HamiltonianHµ expressed exclusively in terms of the low-momentum variables and theirconjugate momenta. Thus, there is no canonical way to associate a purestate of the low-momentum part of the Hilbert space to the full groundstate of the theory. As we have seen, the object that can be canonicallyassociated to a Wilsonian effective action for these low-momentum degreesof freedom is a density matrix.2.3 Measures of entanglementWhat observables quantify the amount of entanglement between the de-grees of freedom in different ranges of momenta? In this section we begin bydiscussing such quantities in generality and conclude by constructing per-turbative expressions for such observables in weakly coupled field theories.First, for any density matrix ρ, the von Neumann entropyS(ρ) = − tr(ρ log(ρ)) (2.8)442.3. Measures of entanglementmeasures the classical uncertainty associated with the mixed state describedby ρ. When ρ represents a microcanonical or canonical ensemble, the vonNeumann entropy gives the thermodynamic entropy. When ρ is the reduceddensity matrix describing a subsystem A of a quantum system that is in apure state, S quantifies the entanglement between A and its complement(A¯). In this case the entanglement entropy of A is equal to that of A¯, a factthat follows from a stronger result that the spectrum of eigenvalues of ρAmatches the spectrum of eigenvalues of ρA¯.When the Hilbert space for the theory can be decomposed into a tensorproduct with three or more factors, the quantum entanglement and classicalcorrelations between pairs of these subsystems are jointly quantified by themutual information. For instance, if the Hilbert space is of the form H =HA ⊗HB ⊗HC ⊗ · · · , the mutual information between A and B is definedasI(A,B) = S(A) + S(B)− S(A ∪B) . (2.9)where S(X) is the von Neumann entropy of the reduced density matrix ofsubsystem X. Mutual information is always greater than or equal to zero,with equality if and only if the density matrix for the AB subsystem is atensor product of the reduced density matrices for subsystems A and B. Inother words, mutual information is zero if and only if there is neither anyentanglement nor any classical correlation between the two subsystems.25Mutual information provides an upper bound on all correlators between thetwo regions: for any bounded operators OA and OB, acting only on thesubsystems A and B, we have [55]I(A,B) ≥(〈OAOB〉 − 〈OA〉〈OB〉)22|OA|2|OB|2. (2.10)If the Hilbert space consists of three factors H = HA ⊗ HB ⊗ HC and25The systems A and B are said to be entangled if and only if the density matrix forthe AB subsystem cannot be written as∑i piρiA ⊗ ρiB . Separable density matrices ofthis form represent states which have no quantum entanglement, but may have classicalcorrelations. The mutual information for such a state can be nonzero.452.3. Measures of entanglementthe complete system is in a pure state it follows from the definitions thatI(A ∪B,C) = I(A,C) + I(B,C) . (2.11)But if A, B, and C together comprise only a part of the system, anotherinteresting observable is the tripartite information which quantifies the ex-tent to which the mutual information between A ∪ B and C is determinedby the pairwise mutual informations I(A,B) and I(B,C):I(A,B,C) = I(A ∪B,C)− I(A,C)− I(B,C) . (2.12)In general, this quantity can be positive, negative, or zero. For a pure stateof the full system, I(A,B,C) is symmetric between the four subsystemsA,B,C, and A ∪B ∪ C.2.3.1 Entanglement observables in perturbation theoryFor weakly coupled quantum field theories, we can use perturbation theorymethods to calculate the entanglement observables described in the previoussection. To begin, it is useful to derive a set of perturbative results that applymore broadly.Consider a general quantum system whose Hilbert space may be decom-posed into a tensor product H = HA ⊗ HB, and start with a Hamiltonianof the formH = HA ⊗ 1 + 1 ⊗HB . (2.13)Denote the energy eigenstates of HA by |n〉 and the energy eigenstates of HBby |N〉, with energies En and E˜N respectively. Before adding interactions,the ground state is|0, 0〉 ≡ |0〉 ⊗ |0〉 . (2.14)Now, perturb the Hamiltonian by an interaction λHAB, where λ is a smallparameter. The perturbed ground state may be written (before normaliza-462.3. Measures of entanglementtion) as|Ω〉 = |0, 0〉+∑n6=0An|n, 0〉+∑N 6=0BN |0, N〉+∑n,N 6=0Cn,N |n,N〉 , (2.15)where A,B, and C are coefficients starting at order λ that may be com-puted in perturbation theory. To normalize, we should multiply by 1/N12 ,where N = 1 +∑|An|2 +∑|BN |2 +∑|Cn,N |2 . Now, the density matrixcorresponding to the subsystem A isρA =11 + |A|2 + |B|2 + |C|21 + |B|2 A† +BC†A+ CB† AA† + CC† , (2.16)where the elements of this matrix correspond to |0〉〈0|,|0〉〈n|,|m〉〈0|,|m〉〈n|terms respectively. By a symmetry transformation ρ → MρM−1, we cansimplify the form toρˆA =1− |C|2 00 CC†+O(λ3) (2.17)where we are using the fact that A, B, and C start at order λ. (See belowfor why this is possible.)Up to corrections of order λ3, the eigenvalues of this matrix are λ2ai and1 − λ2∑ai, where {ai} (normalized to be of order λ0) are the eigenvaluesof the matrix CC†/λ2. Thus, the entanglement entropy isSA = − tr(ρA log(ρA)) = −(1− λ2∑ai) log(1− λ2∑ai)−∑λ2ai log(λ2ai)= −λ2 log(λ2)∑ai + λ2∑ai(1− log ai) +O(λ3) .(2.18)Now, an explicit expression for the C matrix using standard perturbationtheory isCnN = λ〈n,N |HAB|0, 0〉E0 + E˜0 − En − E˜N+O(λ2) . (2.19)Using this, the leading term in the entanglement entropy for small λ is472.3. Measures of entanglementexplicitlySA = −λ2 log(λ2)∑n6=0,N 6=0|〈n,N |HAB|0, 0〉|2(E0 + E˜0 − En − E˜N )2+O(λ2) . (2.20)Interestingly, the entanglement entropy is not analytic in λ at λ = 0. Also,the leading order perturbative result (up to order λ2 terms which are notwritten explicitly) depends only on matrix elements of the interaction Hamil-tonian between the vacuum and states with both subsystems excited. Thisfollows since (2.15) can be written as|Ω〉 = (|0〉+∑n6=0An|n〉)⊗(|0〉+∑N 6=0BN |0, N〉)+∑n,N 6=0(Cn,N−AnBN )|n〉⊗|N〉 .(2.21)In this expression, the entanglement would be zero without the second term,and in this term, CnN starts at order λ while AnBN starts at order λ2.The A and B coefficients do appear in the order λ3 contributions to theentanglement entropy.Mutual informationBy a similar calculation, starting from a pure state in a theory with H =HA ⊗HB ⊗HC , the leading contribution to I(A,B) in perturbation theoryisI(A,B) = −λ2 log(λ2)2∑NA 6=0,NB 6=0,NC=0+∑NA 6=0,NB 6=0,NC 6=0|〈NA, NB, NC |Hint|0, 0〉|2(E0 − ENi)2(2.22)Similarly, when H = HA⊗HB⊗HC ⊗HD, at leading order in perturbationtheory, the tripartite information I(A,B,C) isI(A,B,C) = +λ2 log(λ2)∑Ni 6=0|〈NA, NB, NC , ND|Hint|0, 0〉|2(E0 − ENi)2+O(λ2)(2.23)482.3. Measures of entanglementWhile I(A,B,C) can in general be positive, negative or zero, we see thatthe leading perturbative result for I(A,B,C) is always less than or equal tozero. This implies that to leading order in perturbation theory,I(A ∪B,C) ≤ I(A,C) + I(B,C) . (2.24)This result is not true for general systems.26 Note also that if the matrixelement of the interaction Hamiltonian between the free vacuum and stateswith all four subsystems excited is zero27 then we will haveI(A ∪B,C) = I(A,C) + I(B,C) . (2.25)to leading order in perturbation theory. In this case, the leading order con-tribution to mutual information between any two subsystems is completelydetermined from the mutual information between any pair of minimal sub-systems.282.3.2 Entanglement observables in quantum field theoryFor all of the observables discussed above, the essential quantities we have tocompute are the reduced density matrices of the various subsystems. Giventhese quantities, we can compute the associated von Neumann entropies andmutual informations. In local quantum field theory, recent discussions of en-tanglement have focused on the density matrices associated with boundedspatial regions. These are well-defined because (by locality) there are in-dependent degrees of freedom in disjoint spatial domains, so the Hilbertspace factorizes as required. The associated spatial entanglement entropyis typically divergent, even in free field theory, because in the continuumlimit any spatial region contains an infinite number of degrees of freedom atarbitrarily short wavelengths. These divergences require regularization (e.g.26In particular, if A and B are completely uncorrelated, ρAB = ρA ⊗ ρB , the oppositeinequality, I(A∪B,C) ≥ I(A,C) + I(B,C) follows from strong subadditivity of entangle-ment entropy.27In field theory, this will be true for theories with only cubic interaction terms.28In field theory, such minimal subsystems will be the Hilbert spaces associated withmodes at a single momentum.492.3. Measures of entanglementby including an ultraviolet cutoff) and some care is needed to extract finiteregularization-independent data [70].Now, as discussed above, it is often more natural in quantum field theoryto organize degrees of freedom by momentum (or wavelength). Correspond-ing to any bounded subset of momenta in a field theory there are a finitenumber of degrees of freedom per unit spatial volume.29 As a result, the en-tanglement entropy associated with such a subset should be finite for a finitevolume system, increasing with the volume considered. For a translation-invariant state, we expect that the momentum space entanglement entropywill be an extensive quantity with a finite density S/V . We will verify thisbelow.What observables can we compute? We can define the entanglemententropy S(P ) associated to any subset P of the allowed momenta30, themutual information I(P,Q) between any two subsets of momenta, or thetripartite entanglement I(P,Q,R) for any three subsets of momenta. Wewill focus on• S(µ), the entanglement entropy between all degrees of freedom withmomenta above and below the scale µ.• S([µ1, µ2]), the entanglement entropy for a shell of momenta µ1 ≤|p| ≤ µ2.• S(~p), the entanglement entropy for a single mode with momentum p.• I({|p| < µ1}, {|p| > µ2}), the mutual information between degrees offreedom with momenta below a scale µ1 and degrees of freedom abovethe scale µ2.• I(~p, ~q), the mutual information between modes with momenta ~p and~q.29For a field theory at finite volume there will be a finite number of degrees of freedomin a bounded range of momenta. In the infinite volume limit, the set of allowed momentabecomes continuous. While there are now an infinite number of degrees of freedom withmomenta in a finite region of momentum space, the number per unit spatial volumeremains finite.30More generally, P could represent a subset of the allowed single particle states.502.3. Measures of entanglementThese quantities probe the strength and extent of entanglement in momen-tum space.In free field theory, the Hamiltonian does not couple degrees of free-dom with different momenta and thus all these measures of entanglement inmomentum space should vanish. Adding a weak interaction term that cou-ples degrees of freedom with different momenta modifies the ground stateand should introduce a small amount of entanglement between the variousfield modes. We can characterize this entanglement in perturbative quan-tum field theory by adapting the general results derived above. For thecalculation of entanglement entropy, the two subsystems correspond to twocomplementary subsets A and A¯ of the allowed momenta. The eigenstates|n,N〉 of the unperturbed Hamiltonian are elements of the Fock space basis|{ni}, {NI}〉, where ni and NI are occupation numbers for particle states inthe two subsets. The interaction Hamiltonian takes the formHI =∫ddxHI(x)for some local Hamiltonian density HI(x) that is polynomial in the fieldsand their derivatives. The matrix elements〈{ni}, {NI}|HI |0, 0〉 (2.26)may be computed by expanding the interaction Hamiltonian density in termsof creation and annihilation operators. The sum in (2.20) is now over allstates with at least one particle having momentum in the subset A and atleast one particle having momentum in the subset A¯. The nonzero matrixelements (2.26) in the sum involve states with at most k momenta, wherek is the number of fields in the interaction, and the momenta must add tozero since translation-invariance of the interaction Hamiltonian leads to amomentum-conserving delta function.The leading contribution (2.20) to the entanglement entropy can berewritten in terms of a projected two-point correlator of the interactionHamiltonian (see Appendix A.1). Below we will work directly with the512.4. Scalar field theory: entanglement between scalesexpression (2.20).2.4 Scalar field theory: entanglement betweenscalesTo develop some concrete examples of the general formalism, we will cal-culate momentum space entanglement entropy in d + 1 dimensional scalartheories with action:S =∫dd+1x(12(∂µφ)2 −12m2φ2 −λn!φn) . (2.27)For ease of formulation, we will first take the theory to be defined in a boxof size L with periodic boundary conditions, and assume a UV cutoff at ascale Λ.We will compute the entanglement entropy S(µ) of degrees of freedomwith momenta |p| < µ with the high-momentum modes. Denoting by pi andPi the allowed momenta below and above µ, the Fock space basis elementsare written as |{npi}〉 ⊗ |{nPi}〉. To use the general formula (2.20) for theleading contribution to the entanglement entropy, we need matrix elementsof the interaction Hamiltonian between the free vacuum and the states withboth low and high momenta excited. Recall that the fields can be expandedin terms of creation and annihilation operators asφ(x) =1Ld2∑p1√2ωp(ape−ip·x + a†peip·x) (2.28)where ωp =√p2 +m2. The nonzero matrix elements for the interactionHamiltonian between the Fock space vacuum and states with n particles522.4. Scalar field theory: entanglement between scalesexcited31 are〈p1 · · · pn|HI |0〉 =12n2Ld(n2−1)δp1+···+pn√ω1 · · ·ωn(2.29)From (2.20), we then haveS(µ) = −λ2 log(λ2)∑{pi}µδp1+···+pn2nLd(n−2)ω1 · · ·ωn(ω1 + · · ·+ ωn)2+O(λ2) (2.30)where the sum is over distinct sets of spatial momenta such that at leastone momentum is below the scale µ and at least one momentum is abovethe scale µ. More generally, the entanglement entropy for the field modeswith momenta in some set A is given by the same formula with the sumover distinct sets of momenta such that at least one momentum is in a setA and at least one momentum is in A¯.It is straightforward to take the limit of infinite volume. By the usualreplacements∑~p→(L2pi)d ∫ddp δ~p →(2piL)dδd(p)we find that the entanglement entropy density S(µ)/V has a well-definedlimit:S(µ)/Ld = −λ2 log(λ2)1(2pi)d(n−1)2n∫{pi}µ∏ddpiδ(p1 + · · ·+ pn)ω1 · · ·ωn(ω1 + · · ·+ ωn)2+O(λ2) .(2.31)Here, the integral is again over distinct sets of momenta such that at leastone momentum is below the scale µ and at least one momentum is abovethe scale µ.31We are only interested in matrix elements between the vacuum and states with atleast one low-momentum particle and at least one high-momentum particle. For φ3 andφ4 field theory, the only such non-zero matrix elements have 3 and 4 particles excitedrespectively. For φn theory with n > 4, matrix elements with n − 2k ≥ 3 particle statescan also contribute, but for these theories we must also include φn−2k counterterms in theaction. For now, we focus on the case of φ3 and φ4 theory.532.4. Scalar field theory: entanglement between scalesIn practice, it is often simplest to calculate the derivative dS/dµ, sincethe µ-dependence comes only in the domain of integration, and this domainfor S(µ+ dµ) is almost the same as for S(µ). In the differenceS(µ+ dµ)− S(µ)the only contributions that don’t cancel between the two terms are a positivecontribution in which one momentum is in the range [µ, µ+ dµ] and all theother momenta have magnitude larger than µ, and a negative contributionin which one momentum is in the range [µ, µ+dµ] and all the other momentahave magnitude smaller than µ.2.4.1 The φ3 theory in 1+1 dimensionsThe simplest example is the φ3 theory in 1+1 dimensions.32 From (2.31),S(µ)/V = −λ2 log(λ2)132pi2∫{pi}µ∏dpiδ(p1 + p2 + p3)ω1ω2ω3(ω1 + ω2 + ω3)2+O(λ2)≡ −λ2 log(λ2)132pi2I(µ) .LettingJ(p1, p2, p3) =1ω1ω2ω3(ω1 + ω2 + ω3)2(2.32)we find that12dIdµ=∫ ∞µdp J(µ, p,−p− µ)−∫ 0−µ/2dp J(µ, p,−p− µ) . (2.33)32We work with a massive φ3 theory, so the theory is perturbatively stable. We canassume higher order interaction terms φ2n which stabilize the theory non-perturbativelybut do not affect our leading-order perturbative calculations.542.4. Scalar field theory: entanglement between scalesEvaluating the right hand side analytically for large and small µ, we findthat33I(µ)→µm4 (pi −8√327 pi −43) µ m112µ3{2312 + ln(µ2m2)}µ m(2.34)As discussed further below, the linear behaviour for small µ is related tothe linear growth in the number of degrees of freedom below scale µ, whilethe falloff at large µ is related to fact that a φ3 interaction is relevant in1+1 dimensions so that the physics at large scales approaches that of thefree theory, for which there is no entanglement between modes at differentmomenta.Order λ2 termsAbove we computed the O(λ2 log(λ2)) term in the entanglement entropythat dominates at infinitesimal λ. At small, but finite λ, the O(λ2) term in(2.18) could compete with this. To calculate this term we must determinethe eigenvalues (and not just the trace) of the matrix CC†/λ2, whereC{pi},{Pi} = −λL12 232δ∑ pi+∑Pi(∑ωpi +∑ωPi)√∏ωpi∏ωPi(2.35)and the sets {pi} and {Pi} must have either one and two elements or twoand one elements. Thus the matrix M = CC†/λ2 has nonzero elements ofthe form Mp,q and M{p1,p2},{q1,q2}. We haveMp,q = δp,q18L∑P,Qδp+P+QωpωQωP (ωp + ωQ + ωP )2. (2.36)Thus, for each p with |p| < µ we have one eigenvalueap =18L∑|P |>µ,|Q|>µδp+P+QωpωPωQ(ωp + ωP + ωQ)2. (2.37)33To find I(µ) we evaluate the expression for dI/dµ and then integrate with respect toµ. The constant of integration is fixed by requiring that the entanglement entropy vanishas µ→ 0.552.4. Scalar field theory: entanglement between scalesThe remaining block of the matrix M has entriesM{p1,p2},{q1,q2} =δ∗p1+p2,q1+q28L1√ωp1ωp2ωq1ωq2ωp1+p2(ωp1 + ωp2 + ωp1+p2)(ωq1 + ωq2 + ωp1+p2)where δ∗ indicates that we must have |p1 + p2| > µ for a nonzero result. Tofind the remaining eigenvalues, we put M in block diagonal form, with oneblock for each P with |P | > µ, where p1 + p2 = q1 + q2 = P . For the blocklabeled by P , we can label the matrix entries by p1 and q1, withMp1,q1 =18L1√ωp1ωP−p1ωq1ωP−q1ωP (ωp1 + ωP−p1 + ωP )(ωq1 + ωP−q1 + ωP )=V (p1)V (q1)8LwhereV (p) =1√ωpωP−pωP (ωp + ωP−p + ωP ). (2.38)A matrix of this form has only one nonzero eigenvalue, equal toaP =18L∑|p|<µ,|q|<µδp+q+PωpωqωP (ωp + ωq + ωP )2. (2.39)We have one such eigenvalue for each P with |P | > µ.Having found all the eigenvalues of CC†/λ2, we can use (2.18) to writean expression for S(µ) including the order λ2 term. Recall that S(µ) =λ2(1− log(λ2))∑ai − λ2∑ai log(ai). Taking L→∞,∑ai/L =∫dp12piI(p1) (2.40)and∑ai log(ai)/L =∫dp12piI(p1) log(I(p1)) (2.41)whereI(p1) =∫∗dp2dp316piδ(p1 + p2 + p3)ω1ω2ω3(ω1 + ω2 + ω3)2. (2.42)Here, the asterisk indicates that p2 < p3, and that p2 and p3 must havemagnitude less than µ if p1 has magnitude greater than µ, while p2 and p3562.4. Scalar field theory: entanglement between scales00. 2 3 4 5Figure 2.1: Leading contributions to S(µ) for φ3 theory in 1+1 dimensions.Full result for S(µ) is proportional to λ2(log(1/λ2)+1) times bottom functionplus λ2 times top function.must have magnitude greater than µ if p1 has magnitude less than µ.We have plotted the two leading contributions (2.41) and (2.40) in figure2.1. We see that the two terms have a qualitatively similar behavior. Indetail, the term (2.41) falls off slightly more slowly for large µ, behavingas 1/µ3 log2(µ2/m2) compared with 1/µ3 log(µ2/m2) for (2.40). Thus, forfixed λ and sufficiently large µ (of order m/λ), the O(λ2) term will be largerthan the O(λ2 log(λ2)) term, although the qualitative behavior is similar.In this work, our focus is on the physics in the limit of small λ, so in theremainder of the discussion we will concentrate onO(λ2 log(λ2)) terms whichdominate as long as we stay below the parametrically large energies of order1/λ relative to the mass.2.4.2 The φ3 theory in higher dimensionsIn general dimensions, the entanglement entropy for the modes below scaleµ in the φ3 field theory is given byS(µ)/Ld = −λ2 log(λ2)18(2pi)2d∫{pi}µddp1ddp2ddp3δ(p1 + p2 + p3)ω1ω2ω3(ω1 + ω2 + ω3)2+O(λ2)572.4. Scalar field theory: entanglement between scales0ABµυmu0.070.0590.037− 6543210(A) (B)Figure 2.2: (A) Integration regions for φ3 theory in 2+1 dimensions. (B)The function F (x) appearing in the entanglement entropy for φ4 theory in1 + 1 dimensions.≡ −λ2 log(λ2)18(2pi)2dId(µ) . (2.43)It is more convenient to compute1ωd−1µd−1dIddµ= (∫B−∫A)d2pJ((µ, 0), ~p,−(µ, 0)− ~p) (2.44)where J is defined in (2.32), and A and B are the regions shown in Fig. 2.2A(symmetric between the vertical axis shown and the directions not depictedin the case d > 2). Here ωd = 2pi(d+1)/2/Γ((d + 1)/2) is the volume of theunit d-sphere.Explicitly, we have1ωd−1µd−1dIdµ= −∫ 0−µ2dpx∫ √µ2−(px+µ)20dpTωd−2pd−2T J(px, pT )+∫ µ−µ2dpx∫ ∞√µ2−p2xdpTωd−2pd−2T J(px, pT )+∫ ∞µdpx∫ ∞0dpTωd−2pd−2T J(px, pT ) (2.45)582.4. Scalar field theory: entanglement between scaleswhereJ(px, pT ) =1√µ2 +m2√p2x + p2T +m2√(µ+ px)2 + p2T +m2·1(√µ2 +m2 +√p2x + p2T +m2 +√(µ+ px)2 + p2T +m2)2.We find that in 2 + 1 dimensions, the entanglement entropy decreases withµ asI2(µ)→2pi3µ(2.46)when µ m, while in 3+1 dimensions, we haveI3(µ)→ 2pi2(ln(4)− 1)µfor µ  m. We interpret the the 3 + 1 dimensional result as saying thatin this case the µ3 growth in the number of degrees of freedom below scaleµ overwhelms the 1/µ falloff of the effective dimensionless coupling. Theseexpressions are exact (and finite) as m→ 0. For 4+1 dimensions and higher,(2.45) diverges – we will discuss this divergence below.2.4.3 φ4 theoryFinally, consider the φ4 field theory in 1+1 dimensions. From (2.31),S(µ)/V = −λ2 log(λ2)116(2pi)3∫{pi}µ∏ddpiδ(p1 + · · ·+ p4)ω1 · · ·ω4(ω1 + · · ·+ ω4)2+O(λ2)≡ −λ2 log(λ2)116(2pi)3I(µ) .Thus, we study I(µ) =∫{pi}µdp1dp2dp3dp4 δ(p1+p2+p3+p4) J(p1, p2, p3, p4),where J(p1, p2, p3, p4)−1 = ω1ω2ω3ω4(ω1 + ω2 + ω3 + ω4)2. It is again moreconvenient to evaluate12dIdµ={∫ ∞µdp1∫ p1µdp2 +∫ ∞µdp1∫ −µ− p1+µ2dp2}J(p1, p2, µ,−p1 − p2 − µ)592.4. Scalar field theory: entanglement between scales−∫ −µ3−µdp1∫ − p1+µ2p1dp2 J(p1, p2, µ,−p1 − p2 − µ) ≡1m4F (µ/m) .A numerical integration determines F , giving the final resultS(µ)/V = −λ2 log(λ2)1384pi31m3∫ µ/m0dxF (x) . (2.47)The function F (x) is plotted in Fig. 2.2B. By analyzing (analytically) thebehavior of F for large and small x, we find that the entropy S(µ) behavesas µ/m4 for small µ and asS ∼1µ3ln2(µ/m)for large µ. As for the φ3 theory, the decay at large µ is related to the factthat the φ4 theory in 1+1 dimensions is free in the UV.The leading perturbative contribution to the entanglement entropy S(µ)of φ4 theory can be similarly evaluated in 2+1 dimensions. The integralsthere are more difficult to evaluate numerically, but are convergent. For 3+1and higher dimensions, the integral expression for the leading contributionto S(µ) in the φ4 theory has a UV divergence, which we discuss furtherbelow.2.4.4 General remarksMassless limits: We found above that in two and higher space dimen-sions, the entanglement entropy S(µ) has a finite limit as we take the massto zero. However, in 1+1 dimensions, the results for both φ3 theory and φ4theory diverge in the massless limit. These divergences suggest that S(µ) isnot an “infrared-safe” quantity for a massless scalar theory in 1+1 dimen-sions. However, the ratio S(µ)/S(µ0) has a finite limit if we hold µ and µ0fixed as we take m to zero. The result isS(µ)/S(µ0) = (µ0/µ)3 (2.48)602.4. Scalar field theory: entanglement between scalesThus, while it may not be sensible to talk about S(µ) directly for m = 0and infinite volume, the ratio for different scales appears to be well-definedeven in 1 + 1 dimensions.General understanding of large µ behavior: The results above agreewith the following heuristic derivation of the power law behavior of S(µ) forlarge µ. The behavior is influenced by two significant effects. First, thenumber of degrees of freedom per unit volume below a momentum scale µgrows like µd. All else being equal, we expect S to scale like the number ofdegrees of freedom (for example, it is extensive). However the interactionsin a general field theory depend on the scale, and this scale dependence alsocontributes to the behavior of S(µ). The dimensionless effective couplingfor a φn interaction at scale µ behaves as 1/µd+1−n(d−1)/2. Since S goes likeλ2 (plus logarithmic corrections), we can estimate that S(µ) should behaveasS ∼ µd ×(1µd+1−n(d−1)/2)2=1µd+2−n(d−1)up to possible logarithmic corrections. This is consistent with our results.At a technical level this scaling arises in the integrals for entanglement en-tropy from two sources: (a) the measure factors (i.e. the density of modes),and (b) the energy denominators in the interaction terms. These are thesame ingredients that affect the scaling of physical observables during renor-malization.Divergences: In various specific case considered above, we found thatthe leading perturbative contribution to S(µ)/V is finite in the limit wherethe IR and UV cutoffs are removed. However, for the φ3 theory in 4+1 andhigher dimensions or the φ4 theory in 3+1 and higher dimensions, the inte-gral expressions for the leading perturbative contribution to S(µ) diverge.The divergence is associated with the sum over states in (2.20), or in thesum or integral over the momenta in (2.30) or (2.31) respectively. The lead-ing divergence comes from the sum over states where one momentum hasmagnitude less than µ while the rest have magnitudes greater than µ. The612.4. Scalar field theory: entanglement between scalesdivergence is proportional to a power (or logarithm) of the UV cutoff Λ.Of course, ultraviolet divergences are commonplace in quantum field the-ory. Typically, they are associated with integrals over momenta that appearbeyond the leading order in perturbative calculations, and are dealt with byexpressing the results in terms of renormalized (physical) parameters ratherthan bare parameters. However, here the divergences appear in leading or-der perturbative results. Since the bare and renormalized parameters arethe same at leading order in perturbation theory, the divergences will not beeliminated by expressing the results in terms of renormalized parameters.34Such divergences in leading order expressions indicate a breakdown ofperturbation theory for the specific quantity that is diverging. To see this,note first that similar divergences appear even in fermionic theories, forexample fermions in 1+1 dimensions with a (ψ¯ψ)2 interaction (see AppendixA.2 for details). Furthermore, the divergence is present even at finite volume,since it is associated with the infinite number of high-momentum modeswhich are still present with an IR regulator. But for a theory of fermionicfields at finite volume, the Hilbert space associated with degrees of freedomwith momentum below a scale µ is finite-dimensional. In this case, there is anupper bound S(µ) < log(N), where N is the dimension of the Hilbert space.Now, consider the theory with a UV cutoff Λ. The leading perturbativeexpression for S(µ) will be finite for any finite Λ. But since this expressiondiverges as Λ is taken to infinity, there will be some finite Λ above whichthis leading contribution to S(µ) is larger than the bound log(N). Here, Λis still finite, so S(µ) is clearly well-defined, and the correct result for S(µ)must certainly be less than log(N), so the only possibility is that the leadingperturbative expression is not a good approximation to the correct result.Our conclusion should not be particularly surprising: regardless of howsmall the coupling parameter of a theory is, there will always be quantitiesthat cannot be computed in perturbation theory. Here, the breakdown of34We do expect the standard divergences to appear in higher order perturbative correc-tions, even for quantities whose leading order result is finite. These divergences should becured in the usual way by expressing results in terms of physical parameters, or by usingrenormalized perturbation theory with the appropriate counterterms.622.5. The extent of entanglement between scalesperturbation theory seems to be associated with computing the entangle-ment entropy between a finite set of modes with the infinite set of degrees offreedom above scale µ. We will see below that in cases where perturbationtheory breaks down for this quantity, it is still possible to perturbativelycalculate less inclusive quantities, such as the mutual information betweendegrees of freedom associated with two finite regions of momentum space.We will also encounter a case where the exact result is know but perturba-tion theory fails. In cases where no divergence appears at leading order, thefinite leading-order perturbative result should be reliable so long as subse-quent terms in the perturbative expansion (after renormalization) are smallcompared to the leading terms.aaaaaaaaaaaaaaaaaaa2.5 The extent of entanglement between scalesSo far, we have considered the entanglement between modes in a field theoryabove and below some scale µ. In this section, we ask about the entangle-ment entropy associated with a single mode of the field theory, or the mutualinformation between two individual modes. A version of the former observ-able has been considered previously in the condensed matter literature (seee.g. [71]). We will first consider the entanglement entropies of boundedregions of momentum space. These sorts of observables are useful for tworeasons: (a) they can be finite even when the entanglement entropy for thelow-energy density matrix diverges, (b) they are a much more sensitive andclear probe of the extent of entanglement since they don’t sum over theentire tower of UV modes.2.5.1 An aggregate measure of the range of entanglementThe quantity S(µ) measures entanglement between the complete set of de-grees of freedom below the scale µ and the complete set of degrees of freedomabove the scale µ. Is this entanglement largely between modes just aboveand below the scale µ, or is the entanglement “long-range” in momentum632.5. The extent of entanglement between scalesspace?One way to address this question is to consider the entanglement en-tropy for the annular region µ1 ≤ |p| ≤ µ2 in momentum space. If theentanglement is short-range, then for µ2  µ1, the entanglement entropyS([µ1, µ2]) ≡ S(µ1 ≤ |p| ≤ µ2) should be dominated by entanglement be-tween modes just above and below the scales µ1 and µ2. In addition, theseseparate contributions to the entanglement entropy should be well measuredby S(µ2) and S(µ1). Thus, for short-range entanglement we would expectS([µ1, µ2]) ≈ S(µ1) + S(µ2) µ2  µ1 . (2.49)Alternatively, consider the mutual information between the degrees of free-dom with |p| ≥ µ2 and |p| ≤ µ1:I(µ1, µ2) = S(µ1) + S(µ2)− S([µ1, µ2]) . (2.50)For short-range momentum space entanglement we expect (2.49). Hence,when µ2  µ1 we expect that I(µ1, µ2) ≈ 0. The rate of falloff of I(µ2, µ1)as µ2/µ1 increases from 1 is a characterization of the extent of entanglement.In φ4 theory the infinite volume expression for I(µ1, µ2) is (using (2.22))S([µ1, µ2])/V = −λ2 log(λ2)124∫∗∏iddpi2(2pi)d(2pi)dδ(p1 + p2 + p3 + p4)(ω1 + ω2 + ω3 + ω4)2ω1ω2ω3ω4+O(λ2)(2.51)where the asterisk indicates that we integrate over momenta such that atleast one |p| is in the range [µ1, µ2] and at least one |p| is not in this range.For simplicity, we specialize to d = 1 and take the mass m = 1. It is simplestto first evaluate the quantityd2Sdµ1dµ2. (2.52)We can see that the only contribution to this will be from regions of theintegral above where one momentum is at µ1 and another momentum is at642.5. The extent of entanglement between scalesµ2.This is equal to the integral above with p1 = ±µ2, p2 = ±µ1, and theremaining |p|s either both inside or both outside the interval [µ1, µ2]. Thedistinct choices of momenta satisfying these constraints arep1 = µ2 p2 = µ1 p3 ∈ (−∞,−2µ2 − µ1] ∪ [−2µ1 − µ2,−12(µ1 + µ2)]p1 = µ2 p2 = −µ1 p3 ∈ (−∞,−2µ2 + µ1] ∪ [−µ1, 12(µ1 − µ2)]or momenta obtained from these via pi → −pi, where in all cases, p4 isdetermined by the delta function constraint. Thus, we have1Vd2Sdµ1dµ2= −λ2 log(λ2)112116(2pi)3{ ∫ −2µ2−µ1−∞ dp J(µ2, µ1, p,−p− µ1 − µ2)+∫ − 12 (µ1+µ2)−2µ1−µ2 dp J(µ2, µ1, p,−p− µ1 − µ2)+∫ −2µ2+µ1−∞ dp J(µ2,−µ1, p,−p+ µ1 − µ2)+∫ 12 (µ1−µ2)−µ1 dp J(µ2,−µ1, p,−p+ µ1 − µ2)}whereJ(p1, p2, p3, p4) =1(ω1 + ω2 + ω3 + ω4)2ω1ω2ω3ω4.To determine S(µ1, µ2) from this expression, we can use S(µ, µ) = 0,S(0, µ) = S(µ), and ∂S∂µ2 (0, µ) =dSdµ (µ). From these, we have∂S∂µ2(µ1, µ2) =∫ µ10dµ˜1d2Sdµ1dµ2(µ˜1, µ2) +dSdµ(µ2) (2.53)andS(µ1, µ2) =∫ µ2µ1dµ˜2dSdµ2(µ1, µ˜2) =∫ µ2µ1dµ˜2∫ µ10dµ˜1d2Sdµ1dµ2(µ˜1, µ˜2)+∫ µ2µ1dµ˜2dSdµ(µ˜2) .(2.54)Here, S(µ) is the quantity that we evaluated in previous sections.To investigate whether the entropy S(µ1, µ2) is dominated by entangle-ment between degrees of freedom close to µ1 and µ2, we can vary µ2 and askwhether the variation of S(µ1, µ2) is well approximated by the variation of652.5. The extent of entanglement between scales012342 4 6 8 10 20 30 40(A) (B)Figure 2.3: Ratio of first and second terms in (2.53) vs µ = (µ2−µ1)/m for(A) µ1 = 1 and (B) µ1 = 4. This is a measure of the range of entanglementin φ4 theory in 1 + 1 dimensions. We have taken the mass to be m = 1.S(µ2) (these variations would be equal if S(µ1, µ2) = S(µ1) + S(µ2)). From(2.53), the difference between the variations is equal to the first term on theright hand side, so we ask whether this term is small compared with theother term. In Fig. 2.3, we plot the ratio of these terms as a function ofµ = (µ2 − µ1)/m. The ratio declines as 1/ lnµ increases and approaches afinite value as (µ2 − µ1)/m→ 0.The slow rate of decline is surprising given that the φ4 theory in 1+1dimensions enjoys the property of decoupling. Note however, that the quan-tity we are computing integrates over all of the UV modes. Thus it is anaggregate measure of entanglement. A more refined way to ask about therange of entanglement in momentum space is to consider the mutual infor-mation between individual modes at two different momenta p and q as wedo below. We will see that this mutual information falls off as a power lawwith |q| when |q|  |p|.2.5.2 Single mode entanglementIn this section, we calculate the entanglement entropy for a single mode withmomentum ~p. This measures the entanglement between a single mode andthe rest of the field theory. The leading result for a φn scalar field theory662.5. The extent of entanglement between scalesfollows immediately from (2.30):S(~p) = −λ2 log(λ2)∑{p2,...,pn}δp+p2+···+pn2nLd(n−2)ωpω2 · · ·ωn(ωp + · · ·+ ωn)2+O(λ2)(2.55)where the sum is over all distinct sets of (n− 1) momenta.35 In the infinitevolume limit, this givesS(~p) = −λ2 log(λ2)12n(2pi)d(n−2)∫{p2,...,pn}n∏i=2ddpiδd(p+ p2 + · · ·+ pn)ωpω2 · · ·ωn(ωp + · · ·+ ωn)2+O(λ2)≡ s1(|~p|) .By rotational invariance, the result is a function only of |~p|. All explicitfactors of the volume have canceled out without dividing by volume on theleft side.A natural interpretation of this finite quantity is that it gives the en-tanglement entropy density for degrees of freedom in an infinitesimal rangeddp around the momentum ~p. The number of modes in the box ddp is pro-portional to spatial volume, so if the entanglement entropy for one modehas no explicit volume dependence, the entanglement entropy for the set ofmodes in the box should be proportional to volume. This entropy is alsoproportional to the momentum space volume ddp (if this is infinitesimal),so the entanglement entropy associated with an infinitesimal volume ddx inposition space and volume ddp in momentum space takes the formdS(~p) =ddx ddp(2pi)ds1(|~p|) . (2.56)It is interesting that the phase space volume appears naturally here.36As an explicit example, we have plotted s1(p) for φ3 theory in two, three,and four spacetime dimensions in Fig. 2.4. In the figure, the entropies arenormalized by their value at p = 0. For 1 + 1, 2 + 1, and 3 + 1 dimensions,35For ~p = 0, we have the further restriction that not all momenta are zero.36Note that while this entropy is spatially extensive, it is not extensive in momentumspace. That is, it is not true that S(R)/V =∫R ddpf(p).672.5. The extent of entanglement between scales00. 4 6 8 10Figure 2.4: Single-mode entanglement entropy vs magnitude of mode mo-mentum for φ3 field theory in 1+1 (bottom), 2+1 (middle), and 3+1 (top)dimensions. The entropies are normalized by their values at p = 0.the entanglement entropy decreases like 1/p4,1/p3, and 1/p2 respectively.Thus we see that in this case the entanglement of a single mode with therest of the theory declines as power-law of the momentum, even thoughwe found above that the integrated entanglement between modes above andbelow scales µ2 and µ1 only declines logarithmically. The slow decay in thelatter case is arising from the sum over modes.2.5.3 Mutual information between individual modesIt is also interesting to investigate the mutual information between two spe-cific field theory momenta. In the large volume limit, the natural quantityto consider is the mutual information between degrees of freedom in an in-finitesimal range ddp around some momentum p and degrees of freedom inan infinitesimal range ddq around some momentum q. Starting from the ba-sic formula (2.22), only contributions from the second term in curly bracketssurvive the large volume limit. These involve matrix elements between thevacuum state and states where one particle is excited in each of the regionsddp and ddq, and the remaining particles lie outside these regions. The re-682.5. The extent of entanglement between scalessulting mutual information is proportional to ddp and ddq and to spatialvolume, so we have:I(~p, ~q)/V = ddp ddq I(~p, ~q); . (2.57)For φn scalar field theory in d + 1 dimensions, the leading contributionto I isI(~p, ~q) = −λ2 log(λ2)12n(2pi)d(n−1)∫{p3,...,pn}n∏i=3ddpiδd(p+ q + p3 + · · ·+ pn)ωpωqω3 · · ·ωn(ωp + · · ·+ ωn)2+O(λ2)(2.58)where the integral is over distinct sets of n−2 momenta. For φ3 theory, thisisI(~p, ~q) = −λ2 log(λ2)18(2pi)2d1ωpωqωp+q(ωp + ωq + ωp+q)2+O(λ2) (2.59)Thus, the mutual information is enhanced when ~p, ~q or (~p + ~q) are nearzero, and for fixed p, the mutual information falls off as 1/|q|4 for large q.While this expression gives the formal leading order result in any number ofdimensions, we will see below that it should only be trusted as an accurateapproximation to the exact result for d ≤ 4 space dimensions.2.5.4 Convergence and validity of leading order expressions.As for the entanglement entropy S(µ) considered in the previous section, theintegrals in the leading order contributions to the mutual information andentanglement entropy of single modes can contain UV divergences. As weargued in Sec. 2.4.4, such divergences indicate a breakdown of perturbationtheory for the quantity in question. In this subsection, we classify the scalarfield theories for which the perturbative calculation of single-mode quantitiess1(p) and I(p, q) gives sensible results.We begin with the expression (2.58) for the single mode mutual infor-mation of φn scalar field theory in d+ 1 spacetime dimensions. Naively, thiswill converge (i.e. there are enough powers of momenta being integrated692.5. The extent of entanglement between scalesover in the denominator) ifd < 1 +3n− 3. (2.60)Thus, we have convergence in any dimension for n = 3 for d ≤ 3 for n = 4,for d ≤ 2 for n = 5, and only for d = 1 for any higher n.Since higher order interactions (i.e. interactions with more powers of thefield) are more likely to lead to divergences, we should be concerned thatsuch higher order interactions generated in the quantum effective action willproduce divergences at higher orders in perturbation theory. In φ3 theory weget an effective φn vertex at order λn from a one loop diagram. As a functionof the external momenta, this scales like 1/p2n−d−1 as these momenta aretaken large. The contribution to I(p, q) from such a vertex will naively beconvergent ifd < 5 +3n− 1. (2.61)This is satisfied for any n so long as d ≤ 5, but leads to a divergence in 6 andhigher space dimensions. Thus, it appears that I(p, q) can be computed inperturbation theory for φ3 theory in d ≤ 5 (the same dimensions for whichthe theory is renormalizible).For φ4 theory, the effective action contains effective φ2n interactionscoming from one-loop diagrams at order λn. These scale with externalmomenta like 1/p2n−d−1. The contribution to I(p, q) from such a vertex willnaively be convergent ifd < 3 +12n− 1,which is satisfied for any n as long as d ≤ 3. Thus, it appears that I(p, q) isa well defined quantity for φ4 theory in d ≤ 3 (again, the same dimensionsfor which the theory is renormalizible).An almost identical analysis shows that the mutual information betweendegrees of freedom in any two finite region of momentum space convergeswhenever I(p, q) converges. Note that it would be incorrect to supposefrom the considerations above that the leading order I(p, q) is necessarilywell defined for every renormalizable theory. For example, according to702.6. Entanglement in the Thirring modelTheory Dimensions where I(A,B) converges Dimensions where S(A) convergesφ3 d ≤ 5 d ≤ 3φ4 d ≤ 3 d ≤ 2φ5 d ≤ 2 d = 1φn≥6 d = 1 d = 1Table 2.1: Spatial dimensions where momentum space mutual informationand entanglement entropy converge. The results apply for any boundedregions A and B in momentum space.(2.61), the leading order I(p, q) diverges for the renormalizible φ6 theory in3 dimensions.We can also ask when the entanglement of a single mode (or a finiteregion of momentum space) with the rest of the field theory is well defined.For φn theory, we find convergence ford < 1 +3n− 2(2.62)This result extends to entanglement entropy of any finite region of momen-tum space.2.6 Entanglement in the Thirring modelIn this section we wish to explore the entanglement structure of a 1+1dimensional fermionic interacting field theory, known as the Thirring model.The reason is that powerful techniques have been developed to study such atheory. In fact this theory is known to be equivalent to a scalar field theory[11]. It is an example of a more general principle known as bosonization[18, 34], we will briefly review this concept and use it to obtain the exactground state in the massless Thirring model, which we will then use todiscuss the entanglement entropy of the theory.712.6. Entanglement in the Thirring model2.6.1 The correspondenceThe Lagrangian for the massless Thirring model is:LF = −iΨ¯γµ∂µΨ +g2JµJµ, (2.63)where Jµ = Ψ¯γµΨ. Here γµ are the usual gamma matrices in 1 + 1 dimen-sions.Bosonization gives us a correspondence between correlation functions ina free bosonic theory and a free fermionic theory [18, 34]. At the level of fieldsthis correspondence can be written in terms of bilinears in the Fermi theoryand scalars in the bosonic theory, in particular we have: Jµ ∼= −µν 1√pi∂νφ.Thus adding the term J2 in the fermionic theory should be equivalent toadding the term 1pi (∂φ)2. Thus this interaction simply rescales the free bosonaction to:LB =12g˜ (∂φ)2 , (2.64)where g˜ = 1 + g/pi.2.6.2 The ground stateBoson ground stateNow if we wish to find the exact ground state of the Thirring model whatwe need to do is find the exact ground state of the free bosonic model and“refermionize” it.The free boson ground state |Ω〉 can be written in path integral form bygoing to position basis:〈f(x)|Ω〉 =∫ φ(0,x)=f(x)−∞Dφ exp(−g˜∫12(∂φ)2). (2.65)(2.66)Change the path integration variables from φ → φs + δφ where φs is a722.6. Entanglement in the Thirring modelsolution of the equations of motion with boundary condition φs(0, x) = f(x).Then using ∂2φs = 0 and integrating the action by parts with δφ(0, x) = 0it can be seen that the path integral factorizes into a piece that depends onf(x) and one that does not.〈f(x)|Ω〉 =∫ φ(0,x)=f(x)−∞Dφ exp(−g˜∫12(∂φ)2), (2.67)=∫ δφ(0,x)=0−∞Dδφ exp(−g˜(∫12(∂φs + ∂δφ)2)), (2.68)=∫ δφ(0,x)=0−∞Dδφ exp(−g˜(∫12(∂δφ)2 +∫12(∂φs)2)),(2.69)=∫ δφ(0,x)=0−∞Dδφ exp(−g˜∫12(∂δφ)2)exp(g˜∫dx12(f(x)∂tφs(0, x))),(2.70)= C exp(g˜∫dx12(f(x)∂tφs(0, x)))(2.71)So we just need to find φs. This is easy to do once we have the Green’sfunction (satisfies the delta function equation and vanishes on the boundary)since we have:φs(t, x) =∫dt′dx′φs(t′, x′)∂′2G(t, x; t′, x′) =∫ ′ (∂′i(φ′s∂′iG)− ∂′iφ′s∂′iG),(2.72)=∫ ′boundarydAi(φ′s∂′iG)−∫ ′ (−∂′2φ′sG+ ∂′i(∂′iφ′sG)), (2.73)=∫dx′φs(t′ = 0, x′)∂t′G(t′, x′; t, x)|t′=0 (2.74)where the plus sign in the last line comes from the fact that dAi is the out-ward normal, which points in the +t direction. In general we only know thesolution to the translation invariant equation ∂2g(t−t′, x−x′) = δ(t−t′, x−x′) and we use this to satisfy the boundary condition for G(t, x; t′, x′). De-732.6. Entanglement in the Thirring modelmandingG vanish at the boundary can be achieved by settingG(t, x; t′, x′) =−g(t+ t′, x−x′) + g(t− t′, x−x′) (this satisfies the Green’s equation as wellsince t and t′ must always be negative, thus we only get one delta functionwhen we take the Laplacian).Thus we can swap derivatives by t, x with negative derivatives by t′, x′,and since G satisfies Laplace’s equation (on the boundary the δ functiondoes not contribute) we can swap second derivatives by t, t′ by negativesecond derivatives by x, x′. Then our vacuum state Eq. (2.71) becomes:ln〈f(x)|Ω〉 = g˜∫dx12(f(x)∂tφs(0, x)) (2.75)= g˜∫dxdx′12(f(x)f(x′)∂t∂t′G(t, t′;x, x′)|t=0,t′=0)(2.76)= g˜∫dxdx′12(f(x)f(x′)∂t∂t′(g(t− t′;x− x′)− g(t+ t′, x− x′))|t=0,t′=0)(2.77)= g˜∫dxdx′12(f(x)f(x′)∂2t(−g(t− t′;x− x′)− g(t+ t′, x− x′))|t=0,t′=0)(2.78)= g˜∫dxdx′12(f(x)f(x′)∂2x2g(0;x− x′))(2.79)= −g˜∫dxdx′(∂xf(x)∂x′f(x′)g(0;x− x′))(2.80)For flat D = 2 space the Green’s function is just a logarithm of thedistance 37. Correctly normalizedg(t− t′, x− x′) = −12piln√(t− t′)2 + (x− x′)2. (2.81)2.6.3 Thirring ground stateUsing the correspondence between bosons and fermions mentioned at thebeginning we can now go back to the fermion variables via Jµ ∼= −µν 1√pi∂νφ.Thus −∂xf(x) = −∂xφ(x) =√piJ0 =√piρ(x) =√piΨ†(x)Ψ(x).37The electrostatic potential in 2D is ∼ log |r|.742.6. Entanglement in the Thirring modelSo using these relations in Eq. ( 2.80) with (2.81) we finally have thatthe ground state in fermionic variables isΩ[Ψ,Ψ†] = exp(−g˜∫dxdx′(Ψ†(x)Ψ(x)pig(0, x− x′)Ψ†(x′)Ψ(x′)))(2.82)= exp(g˜2∫dxdx′(Ψ†(x)Ψ(x) ln |x− x′|Ψ†(x′)Ψ(x′)))(2.83)Two interpretations of this ground state will be explored:• Ω[Ψ,Ψ†] gives a probability of Grassman configurations of bilinears,to be interpreted in the path integral sense over the grassman fieldsΨ.• Ω[Ψ,Ψ†] = Ω[ρ(x)] gives the probability of the bosonic configurationρ(x) = Ψ†(x)Ψ(x). The probability has meaning on its own, and thepath integral should be taken over the bosonic fields ρ(x).We will investigate both of these possibilities and the extent to which theydiffer.Note that for the bose kinetic term to have the correct sign we haveg˜ > 0. Then from Eq. (2.83) configurations with two well separated lumpsare more probable than two lumps side by side.2.6.4 ChecksIn this section we check that the ground state obtained is reasonable. Thefirst quantized ground states that we will find from (2.83) have been exploredin [36, 37].Particles in R2To check that 2.83 describes the ground state let us check that, in first quan-tization, g˜ = 1 does indeed produce the ground state for the free theory. Todo this go to fixed particle number subspace. Then J0 = Ψ†Ψ = ρ and for n752.6. Entanglement in the Thirring modelparticles we have ρ =∑ni=1 δ(x−xi). We can think about this as integratingall configurations in (2.83) with the constraint δ(Ψ†(x)Ψ(x)−∑δ(x− xi)).This givesΩn =∏i<j|xi − xj |g˜. (2.84)Particles at the same position do not appear. This would have been trueeven with a UV regularization of the Green’s function due to the Grass-man statistics, implementing the Pauli exclusion principle. In the bosonicinterpretation this is nothing but the removal of the self energy.This state has come out with the wrong statistics. This is what weshould expect though since in first quantized form one must impose thecorrect statistics by hand. Adding the proper statistics we have:Ωn =∏i<j|xi − xj |g˜sgn(xi − xj). (2.85)So now we search for a first quantized Hamiltonian for which this wavefunction is an eigenfunction. Computing the kinetic term we find∑i−∂2∂x2iΩn = −∑i<j2g˜(g˜ − 1)(xi − xj)2Ωn. (2.86)Thus choosing the 1/r2 potential V (xi− xj) = 1(xi−xj)2 we have that Ωnis an eigenfunction ofH = p2 + 2g˜(g˜ − 1)∑i<jV (xi − xj), (2.87)with energy E = 0. Since H is positive for g˜ ≥ 1, Ωn is the ground state.Thus we now see that for g˜ = 1, Ωn becomes the ground state for the freeHamiltonian.762.6. Entanglement in the Thirring modelParticles in R× S1To investigate the theory on the cylinder one simply makes the conformaltransformation z = x− it→ exp iz. Since the free bosonic theory is confor-mally invariant we simply have G(x− y)→ G(exp(ix)− exp(iy)) and so theground state becomes:ΩS1[Ψ,Ψ†] = exp(g˜2∫dxdy(Ψ†(x)Ψ(x) ln | exp(ix)− exp(iy)|Ψ†(y)Ψ(y))).(2.88)Then going to the n particle subspace again we haveΩS1n =∏i<j| exp(ixi)− exp(ixj)|g˜, (2.89)This product is nothing but a Vandermonde determinant, and so forg˜ = 1 we have:ΩS1n = |det(exp(ikixj))|, (2.90)→ exp(iϕ) det(exp(ikixj)). (2.91)with ki = 0, 1, .., n−1, which is the quantization we would expect for modeson a circle. In going from line 1 to line 2 we have imposed the correct Fermistatistics, and noted that in so doing there is a phase ambiguity. This phasecan depend on the coordinates, so long as it doesn’t mess up the statistics.In fact choosing ϕ = −p∑ni=1 xi will shift the momenta ki by −p38. With38This is easily seen by writingdet(exp(ikixj)) =∑σsgn(σ) exp(i∑jkjxσj ) (2.92)where σ is a given permutation. Then the shift on the momenta follows from writingϕ = −p∑ni=1 xi = −p∑ni=1 xσi .772.6. Entanglement in the Thirring modelsuch a choice the free Hamiltonian acting on this state gives:−n∑i=1∂2∂2xiΩS1n =n∑i=1(ki − p)2ΩS1n . (2.93)Thus ΩS1n is a free eigenstate, and it is in fact the ground state since theproper choice of phase will give it minimal energy. The proper choice is ofcourse the center of mass frame, achieved when p = (N − 1)/ ScalingIn what follows we will use some simple scaling arguments to find the entan-glement entropy for the massless Thirring ground state at any coupling. Theresult is the same whether we use the bosonic or Grassman interpretationof (2.83).Position space IConsider how an interval of length L is entangled with its complement. Wesplit up the fields Ψ(x) = Ψ(x)< + Ψ(x)> into pieces inside (<) and outside(>) L. Then the reduced density matrix is:ρL({Ψ<1 ,Ψ†<1 }, {Ψ<2 ,Ψ†<2 }) = (2.94)∫DΨ(x)>†DΨ(x)> Ω[{Ψ<1 ,Ψ†<1 }, {Ψ>,Ψ†>}]Ω[{Ψ<2 ,Ψ†<2 }, {Ψ>,Ψ†>}].(2.95)ThenTr ρnL =∫ n∏i=1DΨi(x)>†DΨi(x)>ρL(1, 2)ρL(2, 3)...ρL(n, 1). (2.96)To give finite results this should be regulated in the UV by a latticespacing a. The measure splits up into degrees of freedom living on each site.Let us also put the system in a box of size B (B >> L).782.6. Entanglement in the Thirring modelNow consider the effect of local field rescalings (note we are not makinga conformal transformation, as we are leaving the position fixed). UnderΨ> →√kΨ> we have ρL → k(B−L)/aρkL (where we have indicated that thenew density matrix has some dependence on k). Under Ψ< →√kΨ< wehave Tr ρnL → k(nL)/a Tr (ρkL)n. Thus under both rescalings Ψ→√kΨ, wesimply haveTr ρnL = knB/a Tr(ρkL)n, (2.97)where B/a is simply the total number of sites. Now using Eq. (1.41) theoverall factor disappears and we haveTr ρL ln ρL = Tr ρkL ln ρkL. (2.98)If we choose the arbitrary parameter k to be 1/g˜1/2 then the RHS of thislast equation becomes independent of g˜. This means that the entanglemententropy for the Thirring model with arbitrary coupling g is the same as thefree theory with g = 0 (g˜ = 1). The entanglement entropy for the free theoryhas been well studied and the result is (for B →∞)S = Sfree =c6lnLa. (2.99)Position Space IIAnother way to get this same result is as follows. Birrell and Davies [7]have shown that the conformal anomaly in curved space is independent ofthe coupling so that< Tµµ >= −124piR. (2.100)Thus the Thirring model is conformal invariant and so we must haveS =c(g)6lnLa, (2.101)where we have noted that the only possible way that the dimensionless cou-792.6. Entanglement in the Thirring modelpling g can now enter is through the central charge. We know, however,that the central charge counts some physical degrees of freedom in the sys-tem. Thus starting from the free theory and adiabatically turning on thecoupling, the physical degrees of freedom cannot continuously change, andso we are left with the same result as before:S =c6lnLa. (2.102)Momentum spaceThe calculation of momentum space entanglement entropy is basically thesame as in position space. Consider the entanglement between modes in aninterval of length µ in momentum space and its complement. The Fouriertransform of the ground state (2.83) is denoted: 39Ω[ψk, ψ†k] = exp(g˜2∫dp2piG(p)|∫dk2piψ†k+pψk|2). (2.105)Again splitting fields into those with support inside the µ interval andthose outside, ψ = ψ< +ψ> the reduced density matrix for modes inside is:ρµ({ψ<1k, ψ†<1k }, {ψ<2k, ψ†<2k }) =∫Dψ>†k Dψ>k Ω[{ψ<1k, ψ†<1k }, {ψ>k , ψ†>k }]Ω[{ψ<2k, ψ†<2k }, {ψ>k , ψ†>k }](2.106)Impose a UV cutoff in momenta denoted Λ and put the system in a boxof size V (Λ >> 1/V ).Now under ψ> →√kψ> we have ρµ → k(Λ−µ)V/piρkµ . Under ψ< →√kψ< we have Tr ρnµ → k(nµV )/pi Tr (ρkµ)n. Thus under both rescalings39In what follows the Fourier transform requires regularization. In the IR this can beaccomplished by adding a mass term. For small mass one usesln (m∆xy) ≈ K0 (m∆xy) (2.103)=∫dp1√p2 +m2eip∆xy , (2.104)K0 being the modified Bessel function of the second kind.802.6. Entanglement in the Thirring modelψ →√kψ, we simply haveTr ρnµ = knΛV/pi Tr(ρkµ)n(2.107)Again using Eq. (1.41) the overall factor disappears and we haveTr ρµ ln ρµ = Tr ρkµ ln ρkµ. (2.108)So again if we choose k to be 1/g˜1/2 the entanglement entropy at anycoupling is the same as in the free theory, which must vanish. Thus theentanglement between modes in the massless interacting Thirring modelmust vanish identically.2.6.6 Direct calculationGrassman wayIn this section we will directly calculate the momentum-space entanglementof the ground state. We put the system in a box of length L with anti-periodic boundary conditions for the fermions. We will consider for simplic-ity the single mode entanglement.We regulate the Green’s function in the IR by adding a small mass term.Then since our definition of the entanglement is independent of normaliza-tion, we multiply by the infinite constant 40:Ω→det−1/2(Gp)Ω (2.109)=∫Duq exp(−1L∑qGq|uq|2)exp(g˜2L3∑pGp|∑kψ†k+pψk|2)(2.110)=∫Duq exp−1L∑qGq|uq|2 −√2g˜L2∑k,mψ†kGk−muk−mψm (2.111)40The mass also removes the zero mode in the determinant, det−1 Gp =∏p√p2 +m2 =∏n√(2pi/L)2 n2 +m2.812.6. Entanglement in the Thirring modelwhere in going from line 2 to 3 we have made the shift uq → uq+√g˜/2L2∑k ψ†k+qψk(k is half-integer moded and q is integer).Then one finds for the reduced density matrix:ρ(ψ<L , ψ<R) =∫Dψ†>k Dψ>k Ω[ψ<L , ψ>]Ω[ψ<R , ψ>] (2.112)=∫DuS det(∑SM>>S)exp−1L∑q,SGq| (uS)q |2 exp(−√2g˜L2ψ†<S M<<SS′ψ<S′)(2.113)∑S fS = fL + fRDuS = DuLDuR(MS)k,m = Gk−m (uS)k−mM<<SS′ =M<<LL M<<LRM<<RL M<<RR=M<<L +M< >L(∑SM>>S)−1M><L M< >L(∑SM>>S)−1M><RM< >R(∑SM>>S)−1M><L M<<R +M< >R(∑SM>>S)−1M><R(2.114)Now we wish to compute Tr ρn. However, this is a fermionic pathintegral which can require non trivial boundary conditions, as in the stan-dard case of the thermal partition function where the Fermi fields are anti-periodic. The boundary conditions in position space have been investigatedin [10, 25] where it was found that there is an extra minus sign coming inbetween each cut. This comes from considering the way spinors rotate. Inmomentum space we will consider both periodic and anti-periodic boundaryconditions, i.e. ψiR = eipiαψi+1L , periodic and anti-periodic being recoveredfor α = 0, 1. We will find that the result is independent of our choice ofboundary conditions. Note that we cannot consider ψi†R = eipiαψi+1L sincethese fields do not transform in the same way.822.6. Entanglement in the Thirring modelUsing these boundary conditions we arrive at:Tr ρn =∫ n∏iDuiS exp−1L∑q,SGq|(uiS)q |2det(∑S(M i)>>S)detn(M(ujS)),(2.115)≡∫ ∏iDU i detn(M(ujS)), (2.116)wheredetn(M(uiS))= detMnRR +M1LL eipiαM1LR 0 · · · 0 e−2ipiαMnRLe−ipiαM1RL M1RR +M2LL eipiαM2LR 0 · · · 00 e−ipiαM2RL M2RR +M3LL. . .... 0. . . . . .......0e2ipiαMnLR 0 · · · Mn−1RR +MnLL,(2.117)and where we have defined{MiSS′ ≡M<<SS′(uiL, uiR). (2.118)Notice the extra factor of e2ipiα that comes in between the first and lastentry, due to the fermion statistics.Now consider the parity of d(uiL, uiR) ≡ det(∑S(M i)>>S)in Eq. (2.115).With a cutoff Λ = (2pi/L)Nmax such that |k| ≤ Λ then832.6. Entanglement in the Thirring modeldim(M i)>>S =2 dim (Λ)− dim (µ) , (2.119)=2Nmax − nµ, (2.120)with µ denoting the length of the interval we are entangling with the rest ofthe system. This implies thatd(−uiL,−uiR) = (−1)nµ d(uiL, uiR). (2.121)We are considering the single mode entanglement and so the determinantis odd, d(−uiL,−uiR) = −d(uiL, uiR).It then follows thatTr ρn =∫ ∏iDU i detn(M(ujS)), (2.122)=∫ ∏iDU i(2∏iMiRR +[ei(n+1)αpi∏iMiLR + c.c.]), (2.123)=2(∫DU jMjRR)n+[ei(n+1)αpi(∫DU jMjLR)n+ c.c.], (2.124)=2an +[ei(n+1)αpibn + c.c.], (2.125)which gives for the entropyS = ln (a+ cos(2piα)b)−a ln a+ cos(2piα)b ln b− (1 + α)pi sin(2piα)ba+ cos(2piα)b+ ln 2.(2.126)This actually only depends on the ratio r = b/a, and for the usualfermion boundary conditions where α = 0, 1 we find:S = ln (1 + r)−r ln r1 + r+ ln 2. (2.127)Thus we find S is bounded between ln 2 (for r → ∞ or r → 0) and ln 4842.6. Entanglement in the Thirring model(for r = 1), which is the maximum entropy for a single mode of a relativisticfermion.Bosonic wayIn this approach we treat ρ(x) as the fundamental field variable. Whetherρ is a density matrix or fermion density should be clear by its argument.ThenΩ[ρ(x)] = exp(g˜2∫dxdx′(ρ(x) ln |x− x′|ρ(x′))). (2.128)This is diagonal in k-spaceΩ[ρ(k)] = exp(g˜2∫dk2piG(k)ρ(k)ρ∗(k)), (2.129)so that the integration over large modes k simply produces an overall nor-malization, which can be dropped. The reduced density matrix isρ[ρ<L (k), ρ<R(k)] = expg˜2∫dk2pi(ρ<L (k) ρ<R(k))G(k) 00 G(k)ρ<L (−k)ρ<R(−k) .(2.130)Being diagonal, the traces in the replica trick simply factorizeZn = (Z1)n , (2.131)and so we simply haveS = 0. (2.132)This is the result we argued for in the scaling section. These consider-ations suggest that the Thirring ground state 2.83 makes sense in terms ofdensity profiles rather than Grassman configurations.852.7. Comments2.7 CommentsWe have obtained a number of results for the scaling of entanglement en-tropies and mutual informations with the upper bound on a momentuminterval. At a technical level, these results all follow from the density ofmodes (measure) in the integrals over momenta, and the energy denomina-tors in the interactions. These are the same ingredients that lead to thedecoupling property of local quantum field theories. Indeed, decoupling isusually understood simply as the power law suppression of higher dimen-sion operators in a low energy effective theory. This suppression means thathigh momentum degrees of freedom have weak effects on the dynamics oflow-momentum degrees of freedom other than renormalizing the interactionstrengths and wavefunctions. Our study of entanglement between degrees offreedom with different momenta, and the resulting entanglement entropiesand mutual informations, refines this understanding of the influence betweenmomentum scales.In more detail, “decoupling” between UV and IR physics implies thatstarting from a generic action SΛ(gI) at scale Λ, that depends on an infinitenumber of parameters gI , the Wilsonian effective action at a much lowerscale µ will be very close to some action SµW (gi) in a family parameterizedby a small number of physical parameters gi. In other words, the operationof integrating out degrees of freedom to successively lower scales results ina flow in the space of effective actions that converges to a low dimensionalsubspace at scales µ Λ. Now, according to (2.5), our effective action SµWat scale µ completely determines the reduced density matrix ρ(µ) for thedegrees of freedom with |p| < µ. Thus we conclude that for the groundstate of a generic field theory defined at scale Λ, the reduced density matrixfor the degrees of freedom below some much lower scale µ will be veryclose to some family of density matrices ρ(µ, gi) that depend on a smallnumber of physical parameters gi. Consequently, knowing the state of thelow-momemtum degrees of freedom tells us relatively little about the detailsof the state at much higher scales.The paucity of information about UV physics contained in the low-862.7. Commentsmomentum density matrix should be reflected in some of the measures ofquantum information we have discussed. Specifically, it seems likely thatthere is a connection between the decoupling behavior of field theories andthe power-law fall off in mutual information observed in Sec. 2.5. It wouldbe interesting to make this connection precise.Relation to AdS/CFT: In the context of gauge-theory / gravity dual-ity (the AdS/CFT correspondence) [82], there is now evidence that certainmeasures of entanglement in quantum field theory carry geometrical infor-mation about the dual spacetime. For field theories with a weakly curveddual gravity description, Ryu and Takayanagi have proposed 1.86 that theentanglement entropy for a spatial region A is proportional to the area ofthe minimal surface A˜ in the bulk space whose boundary coincides with theboundary of A,S(A) =Area(A˜)4GN.Given the holographic interpretation of position-space entanglement en-tropy, it is natural to ask whether the momentum-space quantities consid-ered in this chapter have some simple dual geometrical interpretation forfield theories with gravity duals. As an example, the quantity S(µ) mea-sures the entanglement between degrees of freedom above and below thescale µ. Since energy/momentum scale in holographic field theories corre-sponds to radial position in the dual geometry, we might guess that S(µ)/Vis related to the area (per unit field theory volume) of a surface separatingthe IR region r < r(µ) of the dual geometry from the UV region r > r(µ).For the dual geometry to a translation-invariant field theory state, this areafunction is a well-defined observable.41 However, we currently have no wayto check whether this or some similar observable corresponds to momentum-space entanglement entropy, since we cannot calculate this entropy for anystrongly coupled field theory with a gravity dual.4241If the spatial part of the dual metric is dr2 +f(r)dx2, the area of the surface at radiusr(µ) per unit field theory volume is proportional to a power of f(r(µ)).42S(µ) will probably not always correspond in a simple way to the specific area ob-servable mentioned, since that area would be well-defined even for gravity duals of 0+1872.7. CommentsRelation to DMRG and MERA: Here we have explored various as-pects of entanglement in quantum field theory and the connection to renor-malization theory. In the condensed matter literature, the ideas of entangle-ment and renormalization have come together previously in various schemesfor approximating the ground state of many-body systems [64, 65]. Whilethe focus and details of that work are rather different from the present dis-cussion, it may be useful to briefly review those ideas here.Consider a quantum many-body system described by some lattice ofdegrees of freedom, for which the Hilbert space decomposes as a tensorproduct of Hilbert spaces for the individual sites. The dimension of the fullHilbert space is dN where d is the dimension of the individual Hilbert spacesand N is the number of sites. A general state (and in particular, the exactground state of the system for a given Hamiltonian) can be representedexactly by a tensor T i1···iN that gives the coefficient of the basis state |i1〉 ⊗· · · ⊗ |iN 〉.A general numerical determination of the ground state is impractical dueto the large number dN of independent coefficients. For certain systems, usu-ally in 1+1 dimensions, an efficient variational approach to approximatingthe ground state is to consider tensors T that can be decomposed into con-tractions of lower-rank tensors. For example, the “Matrix Product State”(MPS) decomposition corresponds toT i1···iN = (M1)i1a1a2(M2)i2a2a3 · · · (MN )iNaNa1 .In practice, one uses this decomposition as a variational ansatz, varying theindividual matrices M i to arrive at the best approximation to the groundstate. If the dimension of the matrices M i is large enough, any tensor T canbe represented in this way, so the variational method gives an exact result.However, for a wide class of systems, it has been found that the groundstate can be well approximated by matrices of much lower dimension. Inthis case, the matrix product ansatz represents a truncation of the Hilbertdimensional field theories, for which there is no way to divide up the degrees of freedom byspatial momentum, and therefore no way to define S(µ). Of course such low-dimensionalgauge/gravity dualities (e.g. AdS2/CFT1) also have many other special features [62, 63]).882.7. Commentsspace to a subspace of lower dimension, and in cases where it is effective,the true ground state is close to the ground state in this subspace.It turns out that the success of this method is related to the entanglementproperties of the ground state. The optimal method of truncating to a lower-dimensional Hilbert space is to retain as much of the entanglement entropyfor the various subsystems (blocks of sites) as possible.43 The procedureworks most efficiently (i.e. for smallest matrices M) when there is limitedentanglement between the subsystems corresponding to blocks of sites. Forsystems with a highly entangled ground state, the method is much lessefficient.Another approach that is more successful in cases with long-range entan-glement is the “Multiscale Entanglement Renormalization Ansatz” (MERA[65]). In this approach the tensor T is represented by an iterative proce-dure. The tensor is first written in terms of a “disentangled” tensor T˜ usingunitary matrices U :T i1···i2N(n) = (U(n)1 )i1i2j1j2 · · · (U(n)N )i2N−1i2Nj2N−1j2NT˜ j1···j2N(n)and then T˜ is represented in terms of a lower rank tensor using “projectors”P :T˜ j1···j2N(n) = (P(n)1 )j1j2I1· · · (P (n)N )j2N−1j2NINT I1···IN(n+1) .The latter step can be understood as a “coarse-graining” of the system,though the dimension of the index space I is not necessarily the same asthat of the original index space i. The original tensor T(1) is thus representedby the individual matrices (U (n)i )iji′j′ and (P(n)i )ijI which are the variationalparameters used to approximate the ground state.The introduction of the U matrices is motivated by the observation thatcoarse graining works most efficiently when there is little entanglement be-tween the adjacent blocks. The unitary matrices U can remove short-range43This idea arose first in the “Density Matrix Renormalization Group” (DMRG) [64] aniterative renormalization procedure on the state of the system that truncates the Hilbertspace in each step while retaining as much entanglement entropy as possible. The DMRGis now understood to give results equivalent to this MPS variational method.892.7. Commentsentanglement between adjacent blocks before coarse graining. In this way,the matrices U(n) encode the entanglement between sites at the nth level,which corresponds in the original picture to blocks of 2n sites. Thus, inthe MERA representation of a ground state the unitary matrices U encodeentanglement at different scales. This information is certainly related to thescale-dependent entanglement entropies considered in this chapter, thoughthe MERA entanglements would seem to be more closely related to posi-tion space entanglement. Also, the original MERA applies only to discretesystems, though an extension to continuum quantum field theories has beenrecently proposed in [66]. An interesting connection between MERA andthe AdS/CFT proposal above has been given in [67].90Chapter 3Momentum-spaceentanglement for interactingfermions at finite density3.1 IntroductionThe physics of finite density systems of interacting fermions plays a crucialrole in our understanding of a wide variety of condensed matter systems,from ordinary materials to nuclear matter in a neutron star. To understandeven crude macroscopic properties of these systems, quantum mechanics(e.g. the Pauli Exclusion Principle) is essential. Nevertheless, our theo-retical investigations often focus on classical observables such as thermody-namic quantities, correlation functions, and response functions, since thesequantities are simpler to access via experiment. In this chapter, we insteadintroduce and investigate some intrinsically quantum observables in simpleexamples of finite-density fermion systems.One of the key features that distinguishes quantum systems from clas-sical systems is the possibility of entanglement between different degreesof freedom. This entanglement can be quantified: given any subset A ofdegrees of freedom, the von Neumann entropy SA = − tr(ρA log ρA) of thedensity matrix ρA = trA¯ |Ψ〉〈Ψ| provides a measure of the entanglement be-tween A and the rest of the system A¯. This is known as the entanglemententropy of the subsystem A (for a review, see [68]). Entanglement entropyand related observables have been studied extensively in quantum field the-ory and many-body systems over the past several years (see, for example913.1. Introduction[69–71, 73]), but these studies typically choose A to be the subset of degreesof freedom inside a particular spatial region. In this work, we study theentanglement entropy for even simpler subsystems: we take A a subset ofmomentum space, focusing on the simplest possible subset consisting of asingle field theory mode (i.e. a single allowed momentum). In the context ofspin systems, similar momentum-space entanglement observables have beenstudied for example in [72].Momentum-space entanglement entropy in quantum field theory sys-tems was investigated in Chapter 2. There, it was emphasized that en-tanglement entropy in momentum space vanishes in the ground state ofnon-interacting systems and remains finite in the continuum limit, in con-trast to the position-space entanglement entropy which is non-zero for non-interacting systems and diverges in the continuum limit. The work [74]showed further that in the presence of weak interactions, the momentum-space entanglement entropy can often be computed in perturbation theory,as we review in Sec. 2. The goal of this chapter is to carry out such per-turbative calculations of the entanglement entropy for the simplest possibleinteracting finite-density fermion systems, a gas of free non-relativistic or rel-ativistic fermions in one dimension perturbed by a four-fermion (two-body)interaction.We focus on two quantities in particular: the single-mode entanglemententropy S(p), and the mutual information I(p, q) = S(p) + S(q) − S(p, q)that measures entanglement and correlations between two individual modesat distinct momenta. In the non-relativistic case, for either lattice fermions(Sec. 2) or continuum fermions (Sec. 3), we show that the leading perturba-tive expression for S(p) diverges logarithmically in |pF − p| as p approachesthe Fermi momentum from above or below. The logarithmic divergencealso indicates a breakdown of perturbation theory when the momentum iswithin e−c/λ2of the Fermi momentum, where c is some λ-independent con-stant. For continuum fermions, we also calculate the leading perturbativecontribution to the mutual information between any two modes, both in thenon-relativistic case and (in Sec. 5) for Dirac fermions with a (ψ¯ψ)2 interac-tion. This quantity shows discontinuities when either of the momenta cross923.1. Introductionthe Fermi surface and is largest when both momenta are near the Fermipoint.The renormalization group picture of such interacting fermion systems(see [75] for a review) suggests that the low-energy physics of the systems weconsider should be described by a scale-invariant Luttinger liquid system.Luttinger liquids correspond to stable renormalization group (RG) fixed-points, so the low-energy physics should be largely insensitive to the detailsof physics for modes far from the Fermi surface. As a check of this, we showthat the behavior of the entanglement entropy for modes near the Fermi-surface is the same in a theory with a cutoff ||k| − kF | < Λ. As discussedin [74], there may be a direct connection between the behavior of systemsunder renormalization group flows and the momentum-dependence of en-tanglement observable; investigating this further is an interesting questionfor future work.The results in this chapter represent a preliminary exploration of momentum-space entanglement observables in the simplest interacting fermion systems.The fact that our results show a striking momentum dependence for thesingle-mode entanglement with a sharp peak at the Fermi point gives newinsight into the entanglement structure of the ground state away from thefree limit and suggests that perhaps single-mode entanglement can be usedmore generally as a new signature of the Fermi surface for interacting sys-tems. The results motivate the study of this observable for more stronglyinteracting systems and for systems in higher dimensions. As a particularapplication, it may be interesting to study the the entanglement betweenmodes in trapped cold-atom systems. In this case, the experimental systemis such that the interaction strengths can be varied in a controlled way, andthe occupation of particular modes can be observed experimentally. Thus,it may be possible to experimentally investigate observables similar to thosestudied in this chapter.4444We thank Kirk Madison for discussions on this point.933.2. Momentum-space entanglement in perturbation theory3.2 Momentum-space entanglement inperturbation theoryIn this section, we review the general result found in Chap. 2 for entangle-ment entropy at leading order in perturbation theory and its application tothe calculation of various measures of entanglement for subsystems corre-sponding to subsets of modes in momentum space. Although this chapteris concerned with a perturbative calculation of mode entanglement we em-phasize that the observable can be defined via a path integral for generalfield theories [74] and in particular does not assume that the theory underconsideration includes particle or quasiparticle states.General result for entanglement entropy in perturbation theoryConsider any quantum system with Hilbert space H = HA⊗HB and Hamil-tonian H = HA ⊗ 1 + 1 ⊗ HB + λHAB. Letting |n〉 and |N〉 be energyeigenstates of HA and HB respectively, a state |n〉 ⊗ |N〉 (an energy eigen-state of the λ = 0 Hamiltonian, e.g. the vacuum state) has no entanglementbetween the subsystems. Turning on the interaction, the perturbed eigen-state may be calculated using ordinary quantum-mechanical perturbationtheory. From this, we can compute the density matrix for subsystem A andthe associated entanglement entropy SA. As shown in ([74]), the leadingorder perturbative expression in the non-degenerate case is 45SA = −λ2 log(λ2)∑n′ 6=n,N ′ 6=N|〈n′, N ′|HAB|n,N〉|2(En + E˜N − En′ − E˜N ′)2+O(λ2) . (3.1)This involves a sum over matrix elements of the interaction Hamiltonianbetween the original state and states for which both subsystems A and Bhave been changed. 46 In Appendix B.1, we present a simple application of45Note that this perturbative result is perfectly valid for systems with degeneracy, solong as the state about which we are perturbing is itself non-degenerate.46In cases where the coupling λ is dimensionful there is a corresponding dimensionfulquantity entering the log which serves to make it dimensionless. The details of thisquantity affect the result only at O(λ2) so we do not write it explicitly.943.2. Momentum-space entanglement in perturbation theorythis result, and as a demonstration that this leading perturbative expressionis reliable in a case where the exact answer is known.Entanglement entropy for a region of momentum space inquantum field theoryFor quantum field theory, we can start at finite volume so that the theHilbert space has a discrete Fock-space decomposition H = ⊗pHp and theunperturbed Hamiltonian is a sum of termsH0 =∑pEp,αa†p,αap,α ,each acting on a single factor of the tensor product. Here, α labels thespecies of particle if there is more than one, and Ep,α is the energy of aparticle of species α with momentum p. This may include the contributionof a chemical potential added to give a ground state with finite density|Ψi〉 =∏ia†pi,αi |0〉 . (3.2)We can take A to be some subset of the allowed momenta for the theory,i.e. all modes with a particular set of allowed wavelengths, and consider theentanglement entropy of the modes in region A for the state (3.2). Takingthe large volume limit with the region A of momentum space fixed, we find[74] that the formula (3.1) gives an entanglement entropy for modes in theregion A that is extensive (i.e. proportional to spatial volume), withSA/V = −λ2 log(λ2)∑f∫ ∗∏ ddpa(2pi)d(2pi)dδ(pf − pi)|Mfi|2(Ef − Ei)2+O(λ2) .(3.3)Here, the matrix element Mfi is defined by〈Ψf |HI |Ψi〉 = (2pi)dδ(pf − pi)Mfi(p1, . . . , pN ) (3.4)where |Ψf 〉 are occupation number basis elements of the form a†p1,α1 · · · a†pn,αn |0〉.953.2. Momentum-space entanglement in perturbation theoryThe sum and integral are over the possible states |Ψf 〉 appearing in the ma-trix element. Specifically, the sum is over the possible number of particles ofeach type present in the state |Ψf 〉, while the integral is over the momentapa of the particles that have been added/removed from the initial state |Ψi〉to produce |Ψf 〉, with the constraint that we have added/removed at leastone particle with momentum in the region A and at least one particle withmomentum in the complementary region A¯.Single-mode entanglement entropy in quantum field theoryIn the case where A corresponds to a single mode, the entanglement entropyis finite and volume-independent in the large volume limit (since we are nolonger keeping the momentum-space volume of the the region A fixed in thelimit). We findS(p) = −λ2 log(λ2)∑f∫ ∗∏ ddpa(2pi)d(2pi)dδ(pf − pi)|Mfi|2(Ef − Ei)2+O(λ2) .(3.5)where now in the sum/integrals over the basis state |Ψf 〉 the constraint isthat the occupation number of the mode with momentum p is different thanfor |Ψi〉, and the occupation number of at least one mode with some othermomentum has changed.For the special case of spinless fermions, the Hilbert space associatedwith a single mode is only two-dimensional, and this allows us to give anexplicit result for the order λ2 terms in the entanglement entropy in termsof the order λ2 log(λ2) piece. From equation (18) in [74], we see that forS(p) = −λ2 log(λ2)a+O(λ2),we must haveS(p) = −λ2 log(λ2)a+ λ2a(1− log(a)) +O(λ3),Below, we will write explicitly only the leading O(λ2 log(λ2)) terms.963.3. Entanglement entropy for lattice fermions with nearest neighbor interactionsMutual information between modes in quantum field theoryFinally, we will consider the mutual information between two modes withmomentum p and q. Letting Ap and Aq represent subsets of momentumspace corresponding to infinitesimal volumes ddp and ddq about momenta pand q, we find that the mutual information I(Ap, Aq) ≡ S(Ap) + S(Aq) −S(Ap ∪ Aq) is proportional to volume in the large volume limit, and alsoproportional to ddp and ddq. If we define I(p, q) byI(Ap, Aq)/V =ddp(2pi)dddq(2pi)dI(p, q) , (3.6)then we findI(p, q) = −λ2 log(λ2)∑f∫ ∗∏ ddpa(2pi)d(2pi)dδ(pf − pi)|Mfi|2(Ef − Ei)2+O(λ2) ,(3.7)where now the state |Ψf 〉 in the matrix element is required to differ fromthe state |Ψi〉 in the occupation numbers of modes p and q and at least oneother mode.3.3 Entanglement entropy for lattice fermionswith nearest neighbor interactionsWe now study the entanglement between modes in several models of fermionsat finite density, starting with a systems of spinless fermions on a lattice inone spatial dimension with nearest neighbor interactions.We choose a Hamiltonian47H = −12∑j(ψ†j+1ψj + ψ†jψj+1) + λ∑j(ψ†jψj −12)(ψ†j+1ψj+1 −12)47Here, we have rescaled H to be dimensionless. To restore the physical dimensions,we can multiply by 1/(a2m), where a is the lattice spacing and m is the effective particlemass, so that excitations with momentum p  1/a about the unfilled state have energyp2/(2m).973.3. Entanglement entropy for lattice fermions with nearest neighbor interactionsfor which the ground state has half-filling and an exact particle-hole sym-metry [75]. We define the momentum-space modes ψp byψp =√a∑jψjeijpa ψj =√a∫ pia−piadp2piψpe−ijpasuch that{ψp, ψ†q} = (2pi) δ 2pia(p− q) ,a delta function with period 2pi/a, where a is the lattice spacing.The Hamiltonian becomesH =∫ pia−piadp2piψ†pψp(− cos(pa)) +λ4− λ∫ pia−piadp2piψ†pψp + λHI (3.8)whereHI = −a8pi3∫ pia−piadPdQdpdqδ(P +Q− p− q)ψ†Pψ†Qψpψqei(Q−q)a . (3.9)The third term in (3.8) represents an adjustment to the chemical potentialsuch that the state remains at half-filling in the presence of the interaction.For λ = 0, the mode energy − cos(pa) is negative for |p| < pi/(2a), sothe ground state is|Ψi〉 =∏|p|< pi2aψ†p|0〉 .We can now calculate the entanglement entropy of a single mode with mo-mentum k/a using (3.5). In this case, all states |Ψf 〉 for which the matrixelement (3.4) is nonzero have the same number of particles as |Ψi〉, with twoparticles removed inside the Fermi surface (at momenta p and q), and twoparticles added outside the Fermi surface (at momenta P and Q). Denotingsuch a state by |Ψi;P,Q, p, q〉, we have〈Ψi;P,Q, p, q|HI |Ψi〉 = −a(2pi)δ(P+Q−p−q)(ei(Q−q)a−ei(Q−p)a−ei(P−q)a+ei(P−p)a)983.3. Entanglement entropy for lattice fermions with nearest neighbor interactionsso that the squared matrix element appearing in (3.5) is|M(P,Q, p, q)|2 = a2|(eiQa − eiPa)(eiqa − eipa)|2 .When the momentum k/a is inside the Fermi surface, we can take p = k/aand then integrate over all q inside the Fermi surface and (P,Q) outsidethe Fermi surface. Performing the Q integral using the delta function setsQ = k/a + q − P in the integrand and the remaining integral is over allpossible q inside the Fermi surface and P outside the Fermi surface suchthat Q = k/a + q − P is also outside the Fermi surface. The result is thatthe leading perturbative contribution to the entanglement entropy isSin(k) = −14pi2λ2 log(λ2)∫R1dqdP4(cos(P − k)− cos(P − q))2(cos(q) + cos(k)− cos(P )− cos(k + q − P ))2,where the region of integration isR1 = {−k ≤ q ≤pi2;−pi+k + q2≤ P ≤ −pi2}∪{−pi2≤ q ≤ −k;pi2≤ P ≤k + q2+pi} ,and we have absorbed a factor of a into the integration variables.Similarly, when the momentum k/a is outside the Fermi surface, we cantake P = k/a, and integrate over Q outside the Fermi surface and (p, q)inside the Fermi surface. The result isSout(k) = −14pi2λ2 log(λ2)∫R2dQdp4(cos(p− k)− cos(p−Q))2(cos(Q) + cos(k)− cos(p)− cos(k +Q− p))2,whereR2 = {pi2≤ Q ≤ pi;−pi2≤ p ≤Q+ k2− pi} ∪ {−pi ≤ Q ≤ −k;−pi2≤ p ≤k +Q2}∪{−k ≤ Q ≤ −pi2;k +Q2≤ p ≤pi2} .It is straightforward to show that Sout(k) = Sin(pi/2−k); thus, the entangle-ment entropy is exactly symmetric about the Fermi surface, a consequenceof particle-hole symmetry.993.3. Entanglement entropy for lattice fermions with nearest neighbor interactions012345–3 –2 –1 1 2 3Figure 3.1: Leading perturbative contribution to single mode entanglemententropy S(k) as a function of mode momentum for lattice fermions at halffilling. This diverges logarithmically at the Fermi points k = ±pi/2.The function S(k) is plotted in Fig. 3.1. We see that this leading per-turbative expression diverges at the Fermi momentum; the divergence islogarithmic in |k − kF | and remains if we include a cutoff |k − kF | < restricting to momenta near the Fermi surface. In this case, we findS(k) = −14pi2λ2 log(λ2) log(|k − kF |)+ C() +O(k − kF ),where C is a momentum-independent constant of order 2.Since the Hilbert space for a single mode is two-dimensional, the exactmode entanglement entropy is bounded by log(2). Thus, the divergence inour leading perturbative expression indicates a breakdown in perturbationtheory when the momentum is taken too close to the Fermi surface. Specifi-cally, we expect that the perturbative result is reliable only if it is much lessthan one. This requires that |k − kF |  e−1/λ2.In the exponentially small region |k − kF | . e−1/λ2non-perturbativeeffects are important. One expects on general grounds that an exact cal-culation would smooth out the singularity in this region, giving a finiteentanglement entropy. Outside this exponentially tiny region, our results1003.4. Entanglement entropy for continuum non-relativistic fermionsare reliable and demonstrate that modes become sharply entangled with therest of the system as the mode momentum approaches the Fermi surface.3.4 Entanglement entropy for continuumnon-relativistic fermionsWe now consider the continuum limit of the previous model, obtained bytaking the lattice spacing to zero and adjusting the chemical potential sothat states up to some fixed momentum (independent of a) remain occupied.Restoring the overall factor of 1/(a2m) in the Hamiltonian, we can rewritethe interaction (3.11) asHI = −132pi3am∫ pia−piadPdQdpdqδ(P+Q−p−q)ψ†Pψ†Qψpψq(eiQa−eiPa)(eiqa−eipa) .(3.10)In the limit a→ 0, this givesHI =a32pi3m∫ pia−piadPdQdpdqδ(P +Q− p− q)ψ†Pψ†Qψpψq(Q− P )(q − p) .(3.11)Rescaling λ → λ/(am) so that the Hamiltonian is independent of a in thelimit, we finally obtainH = H0 + λHIwhereH0 =∫ ∞−∞dp2piψ†pψp(p22m− µ) (3.12)andHI =132pi3m2∫ ∞−∞dPdQdpdqδ(P +Q− p− q)ψ†Pψ†Qψpψq(Q− P )(q − p) .(3.13)1013.4. Entanglement entropy for continuum non-relativistic fermions02468101214161820–4 –2 2 4Figure 3.2: Leading perturbative contribution to single mode entangle-ment entropy S(pFk) as a function of mode momentum (as a fraction ofthe Fermi momentum) for weakly interacting continuum non-relativisticfermions. This diverges logarithmically at the Fermi points.We now follow the same steps as in the previous section to obtain resultsfor the single mode entanglement entropy. For 0 < k < 1, we findS(pFk)λ2 log(λ2)= −p2F4pi2m2{∫ 1−kdq∫ −1−∞dP +∫ −k−1dq∫ ∞1dP}(k − q)2(2P − k − q)2(P − k)2(P − q)2.For 1 < k < 3, we findS(pFk)λ2 log(λ2)= −p2F4pi2m2{∫ −1−kdQ∫ 1k+Q2dp+∫ −k−2−kdQ∫ k+Q2−1dp}(k −Q)2(2p− k −Q)2(p− k)2(p−Q)2.For k > 3, we haveS(pFk)λ2 log(λ2)= −p2F4pi2m2{∫ 2−k−kdQ∫ 1k+Q2dp+∫ −k−2−kdQ∫ k+Q2−1dp}(k −Q)2(2p− k −Q)2(p− k)2(p−Q)2.In each case, we have rescaled the integration variables by a factor of pF(the Fermi momentum) to make them dimensionless. All of these integrals1023.4. Entanglement entropy for continuum non-relativistic fermionsmay be evaluated analytically to obtainS(pFk) = −p2F4pi2m2λ2 log(λ2)f(k)wheref(k) =−(k + 1)2 log(1− k)− (1− k)2 log(1 + k) + k2(143 + 2 log(2))+ 23 + 2 log(2)0 < k < 1−(k + 1)2 log(k − 1) + (k + 1)2(73 + log(2))− 343 (k + 1) +283 −83(k+1)1 < k < 3,163(k2−1) 3 < k .(3.14)The results for k < 0 are obtained using S(−k) = S(k).The single-mode entanglement entropy S(pFk) is plotted in figure 3.2. It diverges logarithmically at k = ±1 with the behavior on either sidedescribed byf(k) = −4 log |1− k|+163sgn(1− k) + 4 log(2) +O(1− k).At k = 3, the function f(k) and its first, second, and third derivatives areall continuous, with a discontinuity appearing only in the fourth derivative.3.4.1 Mutual information between modesWe can also look at the entanglement structure in more detail by calculatingthe mutual information between individual modes with momenta pFk andpF l. Specifically, we calculate the function I(pFk, pF l) defined in (3.6).The mutual information I(pFk, pF l) satisfiesI(pFk, pF l) = I(pF l, pFk) = I(−pFk,−pF l) ,so we can restrict to the region {k > 0, |l| < k} and find I for the othervalues using the symmetries. For each choice of k and l, the integral in (3.7)is over distinct pairs of momenta such that together with the momenta pFk1033.4. Entanglement entropy for continuum non-relativistic fermionsk1ABC DEE11I(k,l) = I(l,k)I(k,l) = I(-k,-l)I(k,l) = I(-l,-k)Figure 3.3: Regions of (k, l) space with different behaviors for I(k, l). Re-sults for the unshaded regions may be obtained from the results for theshaded regions using the indicated symmetries of I.and pF l, we have two momenta inside the Fermi surface and two momentaoutside the Fermi surface (otherwise the matrix element in (3.7) vanishes).Performing one integral using the delta function, we are left with a singleintegral in each case. For the various regions depicted in figure 3.3, we findthe following results:• Region A: {0 < k < 1, |l| < k}I(pFk, pF l) = −λ2 log(λ2)pFm2∫ −1−∞dP2pi(k − l)2(2P − k − l)2(P − k)2(P − l)2= −λ2 log(λ2)pF2pim2{2(k − l) log(1 + k1 + l)+(k − l)2(k + l + 2)(k + 1)(l + 1)}• Region B: {l < −1,−l < k < 2− l}I(pFk, pF l) = −λ2 log(λ2)pFm2∫ 1k+l2dp2pi(k − l)2(2p− k − l)2(p− k)2(p− l)2= −λ2 log(λ2)pF2pim2{2(l − k) log(1− lk − 1)+(k − l)2(k + l − 2)(k − 1)(l − 1)}1043.5. Entanglement entropy for relativistic fermions• Region C: {−1 < l < 1, 1 < k < 2 + l}I(pFk, pF l) = −λ2 log(λ2)pFm2∫ k−1−l−1dp2pi(p− l)2(2k − p− l)2(k − p)2(k − l)2= −λ2 log(λ2)pF2pim2{(k − l)3(k + 1)(l + 1)+13(k + 1)3 − (l + 1)3(k − l)2− 2(k − l)}• Region D: {−1 < l < 1, k > 2 + l}I(pFk, pF l) = −λ2 log(λ2)pFm2∫ 1−1dp2pi(p− l)2(2k − p− l)2(k − p)2(k − l)2= −λ2 log(λ2)pF2pim2{13(k + 1)3 − (k − 1)3(k − l)2+2(k − l)2k2 − 1− 4}Region E: {k > 1, l > 1} ∪ {k > 3, 2− k < l < −1}I(pFk, pF l) = 0 .The function I(pFk, pF l) is plotted in figure 3.4, with plots for specific valuesof k given in figure 3.5. We see that this leading-order contribution to I isgenerally discontinuous as one momentum crosses ±pF and diverges whenboth momenta approach one of the Fermi points, unless the two momentaare equal.3.5 Entanglement entropy for relativisticfermionsIn this section, we consider Dirac fermions with a four-fermion interaction,described by the actionS =∫d2x{iψ¯γµ∂µψ −mψ¯ψ − λ˜ψ¯ψψ¯ψ}.This reduces exactly to the non-relativistic model in the previous section(with λ = λ˜/2) if we take chemical potential µ = m + µ˜ with µ˜  m andconsider observables related to energy scales small compared to m so that1053.5. Entanglement entropy for relativistic fermions–3 –2 –10 1 23–3–2–101231020Figure 3.4: Mutual information I(pFk, pF l) between individual modes fornon-relativistic fermions with Fermi momentum pF .1063.5. Entanglement entropy for relativistic fermions00.–2 –1 1 2–2 –1 1–2 –1 1 2–2 –1 1 2Figure 3.5: Mutual information I(pFk, pF l) vs l for non-relativistic fermionswith k = 0 (top left), k = 0.75 (top right), k = 0.95 (bottom left), andk = 1.25 (bottom right). The overall scale for I is arbitrary1073.5. Entanglement entropy for relativistic fermionsthe particle number is fixed and antiparticles decouple.Here, we consider general values of the chemical potential µ and thecorresponding ground state |µ〉 for which all particle states with energy lessthan µ are occupied (and none of the antiparticle states are occupied).The field may be expanded as usual in terms of creation and annihilationoperators asψ(x) =∫dk2pi1√2ωk(ak uk e−ik·x + b†k vk eik·x). (3.15)where ak and bk respectively annihilate a particle and antiparticle with mo-mentum k.For the calculation of any of the entanglement observables in Sec. 3.2,the non-vanishing matrix elements are those between the ground state andstates obtained by1. adding two particles and two antiparticles, corresponding to the oper-ator combinations a†b†a†b† in the expansion of∫dx (ψ¯ψ)2;2. adding two particles and removing two particles, corresponding toa†aa†a; or3. adding two particles, removing one particle and adding an antiparticle,corresponding to a†aa†b† and a†b†a†a.Particles can only be added outside the Fermi surface and can only be re-moved inside the surface; antiparticles can be added anywhere, but cannotbe removed without annihilating the ground state.Using (A.1), we can write the relevant terms in the interaction Hamilto-nian in terms of the momentum-space creation and annihilation operatorsasH1 =∫ {∏ dki2pi√2ωi}u¯(k1)v(k2)u¯(k3)v(k4)a†k1b†k2a†k3b†k4(2pi)δ(k1 + k2 + k3 + k4)H2 = 2∫ {∏ dki2pi√2ωi}u¯(k1)v(k2)u¯(k3)u(k4)a†k1b†k2a†k3ak4(2pi)δ(k1 + k2 + k3 − k4)H3 =∫ {∏ dki2pi√2ωi}u¯(k1)u(k2)u¯(k3)u(k4)a†k1a†k3ak2ak4(2pi)δ(k1 + k3 − k2 − k4)1083.5. Entanglement entropy for relativistic fermionsUsing these, we can calculate the matrix elements appearing in the calcu-lation of entanglement observables. As an example, consider the matrixelement of H1 between the ground state |µ〉 and a state |µ, p1p2; p3p4〉 wheretwo particles with momenta p1 and p2 and two antiparticles with momentap3 and p4 have been added to the ground state. We find〈µ, p1p3; p2p4|H1|µ〉 =(2pi)δ(∑pi)4√ω1ω2ω3ω42(u¯(p1)v(p2)u¯(p3)v(p4)−u¯(p1)v(p4)u¯(p3)v(p2))(3.16)so that from (3.4), we get|Mp1p3;p2p4 |2(∆E)2=14ω1ω2ω3ω4|u¯(p1)v(p2)u¯(p3)v(p4)− u¯(p1)v(p4)u¯(p3)v(p2)|2(ω1 + ω2 + ω3 + ω4)2=(p1 · p3 −m2)(p2 · p4 −m2)ω1ω2ω3ω4(ω1 + ω2 + ω3 + ω4)2≡ J1(p1, p2, p3, p4)Similarly, we can calculate the matrix element of H2 between the groundstate and the state |µ, p1p3p¯2; p4〉 where two particles and an antiparticlehave been added with momenta p1, p3 and p4 respectively, and a particlewith momentum p2 has been removed. We find|Mp1p3p¯2;p4 |2(∆E)2=14ω1ω2ω3ω4|u¯(p1)v(p4)u¯(p3)u(p2)− u¯(p3)v(p4)u¯(p1)u(p2)|2(ω1 + ω3 − ω2 + ω4)2=(p1 · p3 −m2)(p2 · p4 +m2)ω1ω2ω3ω4(ω1 + ω3 − ω2 + ω4)2≡ J2(p1, p2, p3, p4) .Finally, we have|Mp1p3p¯2p¯4 |2(∆E)2=14ω1ω2ω3ω4|u¯(p1)u(p4)u¯(p3)u(p2)− u¯(p3)u(p4)u¯(p1)u(p2)|2(ω1 + ω3 − ω2 − ω4)2=(p1 · p3 −m2)(p2 · p4 −m2)ω1ω2ω3ω4(ω1 + ω3 − ω2 − ω4)2≡ J3(p1, p2, p3, p4)for the matrix element ofH3 between the ground state and the state |µ, p1p3p¯2p¯4〉1093.5. Entanglement entropy for relativistic fermionswhere particles have been added with momenta p1 and p3 and particles withmomenta p2 and p4 have been removed.Single-mode entanglementTo calculate the entanglement entropy for a single mode with momentum p,we use the expression (3.5), taking the sum over the three types of final statesdiscussed above. However, we find that the integrals in the terms involvingH1 and H2 diverge. Thus, as noted in Chapter 2 for the case withoutchemical potential, the leading perturbative expression for the single-modeentanglement entropy in this model is ill-defined. Since the exact answer isnecessarily less than log(2), this must indicate a breakdown of perturbationtheory, as discussed in more detail in section 2.4.4. Thus, for this model, wefocus on the mutual information between modes, which can be computed inperturbation theory.Mutual information between modesTo calculate the mutual information between modes, we use (3.7). In eachcase, I(p, q) is calculated using matrix elements for which the occupationnumber of the particle modes with momenta p and q have been changedrelative to the ground state and for which occupation numbers for two otherparticles or antiparticles (with momenta P and Q) have been changed.When p and q are both inside the Fermi surface, we have:I(p, q) = −λ2 log(λ2)12pi[12∫>dP∫>dQ δ(P +Q− p− q)J3(P, p,Q, q)],where∫> and∫< indicates that the integration variable ranges outside andinside the Fermi surface, respectively.When p is inside, and q is outside:I(p, q) = −λ2 log(λ2)12pi[∫>dP∫dQ δ(P +Q− p+ q)J2(q, p, P,Q)+∫>dP∫<dQ δ(P −Q− p+ q)J3(q, p, P,Q)].1103.5. Entanglement entropy for relativistic fermions00.–20 –10 10 20–20 –10 10 2000.–20 –10 10 20–20 –10 10 20Figure 3.6: Mutual information I(p, q) vs q for relativistic fermions withµ = 10m and p = 0 (top left), p = 7.5 (top right), p = 9.5 (bottom left),and p = 12.5 (bottom right). Momenta are given in units of m, so the Fermipoints are at ±10. The overall scale for I is arbitrary.This also covers the case when p is outside and q is inside, since I(p, q) =I(q, p).Finally, when both p and q are outside:I(p, q) = −λ2 log(λ2)12pi[12∫dP∫dQ δ(P +Q+ p+ q)J1(p, P, q,Q)+∫<dP∫dQ δ(−P +Q+ p+ q)J2(p, P, q,Q)+12∫<dP∫<dQ δ(−P −Q+ p+ q)J3(p, P, q,Q)].All these integrals are straightforward to perform numerically. We findbehaviour qualitatively similar to the non-relativistic case, with discontinu-ities at the Fermi momenta and the largest mutual information when both1113.5. Entanglement entropy for relativistic fermions–10 –50 510–10–505100.10.2Figure 3.7: Mutual information I(p, q) as a function of p and q for relativisticfermions with µ = 0. Momenta are given in units of m.modes are close to the Fermi point. As an example, in figure 3.6 the mu-tual information I(p, q) at µ = 10m is plotted as a function of q for severalfixed values of p. These may be compared with the non-relativistic resultsin figure 3.5.For comparison, we plot in Fig. 3.7 the mutual information I(p, q) vs qfor several values of p in the case with zero chemical potential, where theunperturbed ground state is the Fock-space vacuum. Here, we find smoothbehavior with the largest mutual information between pairs of momenta ofopposite sign and momenta of order the mass scale. The mutual informationfalls off as 1/q for fixed p.1123.6. Discussion3.6 DiscussionIn this chapter we have investigated the entanglement between modes inboth finite density and relativistic weakly interacting fermi systems. In par-ticular the entanglement entropy S(p) is an observable which measures thedegree of entanglement between a degree of freedom of a given wavelength(corresponding to p) and the rest of the system. It should be stressed thatwhile a correlation function in momentum space also measures classical cor-relations, S(p) is a measure of purely quantum mechanical correlations.At finite density we found a sharp amplification in quantum correlationsas we approach the Fermi surface: degrees of freedom near the fermi surfaceare the most strongly entangled with the rest of the system. This is inline with the intuition that fermi surface physics dominates the low energybehaviour.This interesting structure for the quantum correlations in the pertur-bative regime motivates the study of these observables in more stronglycoupled theories, where the Luttinger model is sure to be important.113Chapter 4Gravitational DynamicsFrom Entanglement“Thermodynamics”4.1 IntroductionSince the first connections between gravity and thermodynamics were real-ized in the study of black hole physics [76–78], various attempts have beenmade to derive Einstein’s equations from the thermodynamics of some un-derlying degrees of freedom, starting with Jacobson’s intriguing paper [79](see also [80, 81]). With the AdS/CFT correspondence [82, 83], the underly-ing degrees of freedom for certain theories of gravity with AdS asymptoticshave been explicitly identified as the degrees of freedom of a conformal fieldtheory. It is thus interesting to ask whether the Einstein’s equations in thegravitational theory can be derived from some thermodynamic relations forthe CFT degrees of freedom.In this note, following [84–88] we demonstrate that at least to linearorder in perturbations around pure AdS, Einstein’s equations do follow froma relation dE = dS closely related to the First Law of Thermodynamics, butwhere the entropy S is the entanglement entropy of a spatial region in thefield theory, and E is a certain energy associated with this region. A keypoint is that dS and dE can be defined and the relation dS = dE shown tohold for arbitrary perturbations around the vacuum state; thus, the relationis more general than the ordinary first law which applies only in situationsof thermodynamic equilibrium.1144.1. IntroductionThe specific relation we employ, which we write asδSA = δEhypA (4.1)was derived recently by Blanco, Casini, Hung, and Myers in [88]. Here Arepresents a ball-shaped spatial region, δSA represents the change in entan-glement entropy of the region A relative to the vacuum state, and δEhypArepresents the “hyperbolic” energy of the perturbed state in the region A,the expectation value of an operator which maps to the Hamiltonian of theCFT on hyperbolic space times time under a conformal transformation thattakes the domain of dependence of the region A to Hd × time. We reviewthe derivation of this relation in the next section.For holographic conformal field theories, each side of (4.1) has an inter-pretation in the dual gravity theory. Assuming that the perturbed state |Ψ〉corresponds to some weakly-curved classical spacetime, the entanglemententropy SA may be calculated (at the leading order in the 1/N to whichwe work) via the Ryu-Takayanagi proposal [84] and its covariant general-ization [85] as the area of an extremal surface in the bulk, as we review inSec. 4.3.1. In Sec. 4.3.2, we recall that the energy δEA can be calculatedfrom the asymptotic behavior of the metric. Thus, the field theory relationδSA = δEhypA translates to a constraint on the dual geometry.In section 4, we show that this constraint is precisely that the bulk metriccorresponding to |Ψ〉 must satisfy Einstein’s equations to linear order in theperturbation around pure AdS (the geometry corresponding to the CFTvacuum state). That solutions of Einstein’s equations satisfy δSA = δEhypAhas already been shown in [88] (see also the related earlier work [87, 89, 90]).For completeness, we provide an alternate demonstration of this in section4.2. In section 4.3, we go the other direction, showing that any perturbationto pure AdS satisfying δSA = δEhypA must satisfy Einstein’s equations. Thisrequires more than simply reversing the arguments of section 4.2 (or of[88]). In particular, demanding that δSA = δEhypA for all ball-shaped spatialregions A in a particular Lorentz frame only places mild constraints onthe metric, determining the combination Hxx + Hyy in terms of the other1154.2. Entropy-energy relationcomponents. It is only when we demand that δSA = δEhypA in an arbitraryLorentz frame (i.e. for ball-shaped regions on arbitrary spatial slices) thatthe full set of linearized Einstein’s equations is implied.In Appendix C.1, we give an alternative proof that Einstein’s equationsimply δSA = δEhypA that is perhaps more straightforward, but assumes thatthe metric is analytic.We conclude in section 5 with a discussion.4.2 Entropy-energy relationIn this section, we review the relation δSA = δEhypA , derived by Blanco,Casini, Hung, and Myers in [88] as a special case of an inequality thatfollows from the positivity of relative entropy.General expression for variation of the entanglement entropyConsider a CFT on Rd,1 in some state |Ψ〉. Choosing a spatial region A,define ρA to be the reduced density matrix associated with this region forthe state |Ψ〉,ρA = trA¯ |Ψ〉〈Ψ| .From this, we can define the modular Hamiltonian HA byρA = e−HA .For general states, this modular Hamiltonian is not related to the usualHamiltonian, and cannot be written as the integral of a local density. Wenow consider an arbitrary variation of the state |Ψ〉. The change in entan-glement entropy SA for the region A is given byδSA = δ(− tr(ρA log ρA))= − tr(δρA log ρA)= tr(δρAHA)= δ〈HA〉1164.2. Entropy-energy relationwhere we have used the fact that tr(δρA) = 0, a consequence of assumingthat the density matrix has a fixed normalization. In the last line, HA isthe original modular Hamiltonian associated with the density matrix ρA forthe original state. Thus, we have the general relationδSA = δ〈HA〉 , (4.2)valid in any spatial region A for arbitrary perturbations of an arbitrary state.“Thermodynamic” relation for perturbations around the vacuumstateWe now specialize to the case where |Ψ〉 is the vacuum state, and the re-gion A is a ball of radius R. In this case, the domain of dependence ofthe ball-shaped region48 can be mapped by a conformal transformation tohyperbolic space times time. As shown in [86], such a transformation mapsthe vacuum density matrix for the region A to the thermal density matrixe−βHhyp for the hyperbolic space theory, where the temperature is related tothe hyperbolic space curvature radius RH by β = 2piRhyp. In this case Hhypis the integral of the local operator T 00hyp over hyperbolic space. Mappingback to the ball-shaped region of Minkowski space, it follows [86] that themodular Hamiltonian can be written asHvacA = 2pi∫AddxR2 − r22RT 00where T 00 is the energy density operator for the CFT and r is a radialcoordinate centered at the center of the ball.In this case, we haveδ〈HA〉 = 2pi∫AddxR2 − r22RδT 00 ≡ δEhypA , (4.3)i.e. the variation in the expectation value of the vacuum modular Hamilto-48The domain of dependence of A is the set of points p for which all inextensible causalcurves passing through p also pass through A.1174.3. Gravitational implications of dS = dE in holographic theoriesnian HvacA under a small perturbation away from the vacuum state is equalto the change in the “hyperbolic” energy of the region. Thus, the generalrelation (4.2) givesδSA = δEhypA , (4.4)reminiscent of the First Law of Thermodynamics. We emphasize howeverthat the entanglement entropy SA can be defined for any state, in contrastto the usual thermodynamic entropy which applies to equilibrium states.Thus, (4.4) represents a much more general result.4.3 Gravitational implications of dS = dE inholographic theoriesLet us now consider the case of a holographic conformal field theory onMinkowski space, whose states correspond to asymptotically AdS spacetimesin some quantum theory of gravity. In this case, each side of the relationδSA = δEhypA has a straightforward gravitational interpretation. As wereview below, the left side may be calculated using the Ryu-Takayanagiproposal [84, 85], while the right side can be calculated from the asymptoticform of the metric. The equality of these quantities represents a constrainton the gravitational dynamics implied by the dual field theory. In the nextsection, we show that this constraint is precisely equivalent to Einstein’sequations linearized about AdS.4.3.1 Gravitational calculation of dSAccording to the Ryu-Takayanagi proposal [84] and its covariant generaliza-tion [85], the entanglement entropy SA for a state with a geometrical gravitydual is proportional49 to the area of the extremal co-dimension two surfaceA˜ in the bulk whose boundary coincides with the boundary of the region A49Here, we are working to leading order in 1/N . See the discussion section for commentson 1/N corrections.1184.3. Gravitational implications of dS = dE in holographic theorieson the AdS boundary,SA =Area(A˜)4GN.The surface A˜ is an extremum of the area functionalA(G,Xext) =∫ddσ√gwhereg = det(gab) = det(Gµν)dXµdσadXνdσb.Starting from pure AdS, with metric50ds2 = G0µνdxµdxν =1z2(−dt2 + dz2 + d~x2) (4.5)the extremal surface ending on the spatial boundary sphere of radius R isdescribed by the spacetime surface~x2 + z2 = R2 . (4.6)We now consider a small variationGµν = G0µν + δGµν . (4.7)In this case, the extremal surface changes, and the new area isA(G0 + δG,X0ext + δX)where the variation δX will be of order δG. Since the original surface wasextremal, we haveA(G0, X0ext + δX) = A(G0, X0ext) +O(δX2) .Thus, the variation of the surface gives rise to changes in the area that startat order δG2. To find the order δG variation of the area, we need only50Throughout this chapter, we set the AdS radius to one.1194.3. Gravitational implications of dS = dE in holographic theoriesevaluateA(G0 + δG,X0ext)−A(G,X0ext)expanded to linear order in δG. We find thatδA =∫ddσ12√g0gab0 δgab , (4.8)where we have used lower-case letters to represent pullbacks to the extremalsurface. Thus, for field theory state |Ψ〉 close to the vacuum state withdual geometry described by (4.7), the change in the entanglement entropyfor region A relative to the vacuum state is given by an integral of themetric perturbation over the original extremal surface A˜. Using the explicitmetric (4.5) and parameterizing the extremal surface (4.6) by the boundarycoordinates xi, we have finally thatδS =R8GN∫ddx(δij −1R2xixj)Hij . (4.9)4.3.2 Gravitational calculation of dEGeneral asymptotically AdS spacetimes with a Minkowski space boundarygeometry may by described using Fefferman-Graham coordinates by a metricds2 =1z2(dz2 + dxµdxµ + zdHµν(x, z)dxµdxν) . (4.10)where pure AdS, dual to the CFT vacuum, corresponds to Hµν = 0. Withthis parametrization, the expectation value tµν of the field theory stress-energy tensor is simply related to the asymptotic metric by [91, 92]tµν(x) =d+ 116piGNHµν(z = 0, x) .Thus, we may write the change in the hyperbolic energy (4.3) relative to thevacuum state asδEhypA =d+ 116GN∫AddxR2 − r2RδH00(0, x) . (4.11)1204.4. Derivation of linearized Einstein’s equations from dE = dSThis is an integral of the boundary value of H over the region A.4.4 Derivation of linearized Einstein’s equationsfrom dE = dSWe are now ready to demonstrate that using the holographic dictionary re-viewed in the previous section, the CFT relation δSA = δEhypA is equivalentto the constraint that the metric corresponding to the perturbed CFT sat-isfies Einstein’s equations to linear order. For clarity, we focus on the caseof 2+1 dimensional conformal field theories, corresponding to gravitationaltheories with four non-compact dimensions. However, the result can also beproven for general higher-dimensional theories.Using the results (4.9) and (4.11), the CFT relation δSA = δEhypA impliesthat for a disk of any radius R centered at any point (x0, y0) on the boundary,the integralδSˆ =∫DRdxdy{Hxx(√R2 − x2 − y2, t, x+ x0, y + y0)(R2 − x2)+Hyy(√R2 − x2 − y2, t, x+ x0, y + y0)(R2 − y2)−2Hxy(√R2 − x2 − y2, t, x+ x0, y + y0)xy}(4.12)over the bulk extremal surface must equal the integralδEˆ =32∫DRdxdy(R2 − x2 − y2)Htt(0, t, x+ x0, y + y0) (4.13)over the z = 0 surface, where we have absorbed a factor of 1/8GNR todefine δSˆ(R, x0, y0) and δEˆ(R, x0, y0) (we drop the hats from now on). Wewill now show that this equality is true for all disks in all Lorentz frames ifand only if the bulk metric satisfies Einstein’s equations to linear order inH. As shown in [88], these are equivalent to the set of equationsHαα = 0 ∂µHµν = 01z4∂z{z4∂zHµν}+∂2Hµν = 0 (4.14)that arise by plugging the Fefferman-Graham form of the metric (4.10) into1214.4. Derivation of linearized Einstein’s equations from dE = dSthe zz, zµ, and µν components of Einstein’s equationsWµν = Rµν −12gµνR− 3gµν = 0 ,respectively and using the fact that H is regular at z = 0. In (4.14), the lastequation is equivalent to saying that each component of z3H must satisfythe Laplace equation on the AdS background.4.4.1 Proof that δS = δE for solutions of Einstein’sequationsWe begin by showing that solutions of the linearized Einstein’s equationsobey the equality δS = δE. This has already been checked in Sec. 3.1 of [88]by demonstrating the result for a complete basis of solutions to the equations(4.14). In this section, we offer an alternative proof that does not requireusing an explicit basis of solutions. A third proof that is perhaps morestraightforward but assumes a series expansion of H is given in AppendixC.1.Using the equations (4.14), we have:∂2tHtt = ∂2t (Hxx +Hyy)⇒ ∂t(∂xHxt + ∂yHyt) = ∂2t (Hxx +Hyy)⇒ ∂2xHxx + ∂2yHyy + 2∂x∂yHxy = ∂2t (Hxx +Hyy)⇒ ∂2xHxx + ∂2yHyy + 2∂x∂yHxy = (∂2x + ∂2y)(Hxx +Hyy) +1z4∂z(z4∂z(Hxx +Hyy))⇒ 2∂x∂yHxy = ∂2yHxx + ∂2xHyy +1z4∂z(z4∂z(Hxx +Hyy))(4.15)We would like to use the last equation to eliminate Hxy from (4.12). How-ever, we have Hxy rather than ∂x∂yHxy in (4.12). To make progress, webegin by differentiating δS by x0 and y0 (the coordinates of the center ofthe boundary disk). This gives∂x0∂y0δS =∫DRdxdy{∂x∂yHxx(√R2 − x2 − y2, t, x+ x0, y + y0)(R2 − x2)+∂x∂yHyy(√R2 − x2 − y2, t, x+ x0, y + y0)(R2 − y2)1224.4. Derivation of linearized Einstein’s equations from dE = dS−2∂x∂yHxy(√R2 − x2 − y2, t, x+ x0, y + y0)xy}(4.16)Now, using (4.15), we have∂x0∂y0δS =∫DRdxdy{∂x∂yHxx(R2 − x2) + ∂x∂yHyy(R2 − y2)−xy(∂2yHxx + ∂2xHyy +1z4∂z(z4∂z(Hxx +Hyy)))}(4.17)It is straightforward to check that this expression is equal to the integralover the extremal surface of an exact form dA, where A is defined for all(x, y, z, t) asA =(−xz∂zHxx − 3xHxx + z2∂xHyy)dx+(z2∂yHxx − yz∂zHyy − 3yHyy)dy+ (−yz∂yHxx − xz∂xHyy) dz . (4.18)By Stokes theorem, this equals the integral of A over the boundary of theextremal surface, so we have∂x0∂y0δS =∫∂DRA= −3∫∂DR(xHxxdx+ yHyydy)where we have used the fact that all other terms in A vanish for z = 0.Similarly, we find that ∂x0∂y0δE may be written as∂x0∂y0δE =32∫DRdxdy∂x0∂y0Htt(0, t, x+ x0, y + y0)(R2 − x2 − y2)=32∫DRdxdy∂x∂y(Hxx(0, t, x+ x0, y + y0) +Hyy(0, t, x+ x0, y + y0))(R2 − x2 − y2)=32∫DRdAˆ ,where we can chooseAˆ =(−2xHxx + (R2 − x2 − y2)∂xHyy)dx+(−2yHyy + (R2 − x2 − y2)∂yHxx)dy .1234.4. Derivation of linearized Einstein’s equations from dE = dSAgain, using Stokes theorem, this reduces to the integral of (3/2)Aˆ over theboundary, so∂x0∂y0δE =32∫∂DRAˆ= −3∫∂DR(xHxxdx+ yHyydy)= ∂x0∂y0δSWe conclude that for any H satisfying Einstein’s equations,δS(x0, y0, R;H)− δE(x0, y0, R;H) = Cx(x0, R;H) + Cy(y0, R;H) ,where Cx and Cy are some functionals linear in H that do not depend ony0 or x0 respectively. Now, consider the class of functions H that vanish forsufficiently large x20 + y20 at the time t = 0 where we evaluate δS and δE.In this case, fixing any x0 and taking y0 → ∞ or fixing any y0 and takingx0 →∞, the left side must vanish. For this to be true on the right side, bothCx and Cy must be constant (as functions of x0 and y0), with Cx +Cy = 0.Thus, the right side vanishes for any H that vanishes as x20 + y20 →∞. Butmore general H can be written as linear combinations of such functions,and since the right side is a linear functional in H, it must vanish for all H.This completes the argument that δSA = δEhypA for solutions of Einstein’sequations.4.4.2 Proof that δS = δE implies the linearized Einstein’sequationsIn this section, we go the other direction to show that the relation δS = δEimplies that the metric satisfies Einstein’s equations to linear order, i.e. thatthe equivalence of (4.12) and (4.13) implies the relations (4.14).Given the boundary stress tensor tµν , let HEEµν be the correspondingmetric perturbation that follows from Einstein’s equations, i.e. the solutionof (4.14) satisfying HEEµν (0, t, x, y) = (16piGN/3)tµν . We will show thatthere is no other H with these boundary conditions for which δS = δE in1244.4. Derivation of linearized Einstein’s equations from dE = dSall frames of reference.Suppose there were another H for which δS = δE for all disk shapedregions in all Lorentz frames. Then the difference ∆ = H − HEE mustsatisfy∆µν(z = 0, t, x, y) = 0 , (4.19)and0 =∫DRdxdy{∆xx(√R2 − x2 − y2, x+ x0, y + y0)(1−x2R2)+∆yy(√R2 − x2 − y2, x+ x0, y + y0)(1−y2R2)−2∆xy(√R2 − x2 − y2, x+ x0, y + y0)xyR2}(4.20)for arbitrary R, x0, and y0, and in an arbitrary Lorentz frame.Let us first see the consequences of demanding this result in a fixedframe. To begin, we note that (4.20) may be expanded in powers of R usingthe basic integral∫DRdxdy(R2 − x2 − y2)n2 x2mxy2my = Rn+2mx+2my+2In,mx,my ,whereIn,mx,my =Γ(mx + 12)Γ(my +12)Γ(n2 + 1)Γ(n2 +mx +my + 2). (4.21)Defining∆µν(z, x, y) =∞∑n=0zn∆(n)µν (x, y) (4.22)1254.4. Derivation of linearized Einstein’s equations from dE = dSwe find that (4.20) becomes510 =∑Rn+2mx+2my+2{1(2mx)!(2my)!∂2mxx ∂2myy ∆(n)xx (t, x0, y0)(In,mx,my − In,mx+1,my)+ 1(2mx)!(2my)!∂2mxx ∂2myy ∆(n)yy (t, x0, y0)(In,mx,my − In,mx,my+1)−2 R2 1(2mx+1)!(2my+1)!∂2mx+1x ∂2my+1y ∆(n)xy (t, x0, y0)In,mx+1,my+1}(4.23)The vanishing of the terms at order RN+2 implies that∆(N)xx (t, x0, y0) + ∆(N)yy (t, x0, y0) =∑(mx,my) 6=(0,0)CN,mx,myxx ∂2mxx ∂2myy ∆(N−2mx−2my)xx+CN,mx,myyy ∂2mxx ∂2myy ∆(N−2mx−2my)yy+CN,mx,myxy ∂2mx−1x ∂2my−1y ∆(N−2mx−2my)xy ,where the C coefficients can be read off from (4.23). As examples, the firstfew equations give∆(0)xx (t, x0, y0) + ∆(0)yy (t, x0, y0) = 0∆(1)xx (t, x0, y0) + ∆(1)yy (t, x0, y0) = 0∆(2)xx (t, x0, y0) + ∆(2)yy (t, x0, y0) = −14(∂2y∆(0)xx (t, x0, y0) + ∂2x∆(0)yy (t, x0, y0))−320(∂2x∆(0)xx (t, x0, y0) + ∂2y∆(0)yy (t, x0, y0))+15∂x∂y∆(0)xy (t, x0, y0)∆(3)xx (t, x0, y0) + ∆(3)yy (t, x0, y0) = −16(∂2y∆(1)xx (t, x0, y0) + ∂2x∆(1)yy (t, x0, y0))−16(∂2x∆(1)xx (t, x0, y0) + ∂2y∆(1)yy (t, x0, y0))+19∂x∂y∆(1)xy (t, x0, y0) (4.24)We see that this set of equations completely determines the combination∆xx + ∆yy at each order in z in terms of the lower order terms in the ex-pansion of ∆. However, apart from the constraint (4.19) on the boundarybehavior (equivalent to ∆(0)µν = 0), the remaining elements of ∆µν are com-51Here, we are assuming that the function ∆ is analytic. It would be useful to find aderivation of our result that holds more generally.1264.4. Derivation of linearized Einstein’s equations from dE = dSpletely unconstrained.To constrain ∆µν further, we need to use the requirement that the rela-tion (4.20) should hold in an arbitrary Lorentz frame. Thus, for each choiceof reference frame, we will have equations analogous to (4.24). Specifically,consider a general boostΛ =γ γβx γβyγβx 1 + β2xγ2γ+1 βxβyγ2γ+1γβy βxβyγ2γ+1 1 + β2yγ2γ+1In the equations for a general frame of reference obtained by such a boost,the left sides in (4.24) will be replaced byΛxµΛxν∆µν + ΛyµΛyν∆µν .Up to an overall constant factor, this gives∆ii + 2βi∆it + β2(∆tt −12∆ii) + (βiβj −12δijβ2)∆ij .Now, consider the general version of the second equation in (4.24) (the firstequation already holds by (4.19)). This requires the vanishing of∆(1)ii + 2βi∆(1)it + β2(∆(1)tt −12∆(1)ii ) + (βiβj −12δijβ2)∆(1)ij .For a fixed x0 and y0, this is a polynomial in βi that must vanish for allvalues of βi. Thus, the polynomial must be identically zero. At order β0,this gives∆(1)ii (t, x0, y0) = 0as we had before. At order β, we get∆(1)it (t, x0, y0) = 0 .1274.5. DiscussionAt order β2, this gives∆(1)tt (t, x0, y0) =12∆(1)ii (t, x0, y0) = 0and∆(1)ij (t, x0, y0)−12∆ij∆(1)kk (t, x0, y0) = 0 .Thus, we have ∆(1)µν = 0. We can now continue to analyze the remainingequations in (4.24) in turn. Supposing that we have shown ∆(k)µν = 0 fork < n, the general version of the nth equation in (4.24) requires the vanishingof∆(n)ii + 2βi∆(n)it + β2(∆(n)tt −12∆(n)ii ) + (βiβj −12δijβ2)∆(n)ij ,since the right hand side in (4.24) will be zero. Repeating the analysis above,we conclude that ∆(n)µν = 0. By induction, this holds for all n, so we haveshown that ∆µν = 0, completing the proof.4.5 DiscussionIn this chapter, we have seen that to linear order in perturbations about thevacuum state, the emergence of gravitational dynamics in the theory dualto a holographic CFT is directly related to a general relation satisfied byCFT entanglement entropies on ball-shaped regions. This relation is closelyrelated to the First Law of Thermodynamics, but is more general since itapplies to arbitrary perturbations of the state rather than perturbations forwhich the system remains in thermal equilibrium.While the CFT relation (1) is an exact equivalence, we have made useof this relation only at the leading order in 1/N where the entanglemententropy maps over to the extremal surface area. This corresponds to work-ing in the classical limit in the bulk. According to [93], 1/N correctionsto the CFT entanglement entropy correspond to bulk quantum correctionsincluding the entropy of entanglement of bulk quantum fields across the ex-tremal surface. It will be interesting to understand the implications of theCFT relation (1) beyond the classical level in the bulk, but we leave this for1284.5. Discussionfuture work.The derivations in Sec. 4.4 were written specifically for the case of four-dimensional gravity. However, the proof given in [88] that Einstein’s equa-tions imply δS = δE, and our method of proof in section 4.2 that δS = δEimplies the linearized Einstein’s equations work for general dimensions.52The linearized Einstein’s equations we derived are for the metric com-ponents in the field theory directions and radial direction of the bulk. Anyadditional fields in the gravitational theory, including metric componentsin any compactified directions, are not constrained by the CFT relation wehave considered. At linear order, the equations for these fields decouple fromthe linearized Einstein’s equations for the metric in the non-compact direc-tions. Thus, we can say that the universal relation δS = δE is equivalent tothe universal sector of the linearized bulk equations.Our results do not imply that all holographic theories are dual to gravi-tational theories whose metric perturbations satisfy Einstein’s equations. Inthis chapter, we assumed that entanglement entropies are related to areasvia the usual Ryu-Takayanagi formula, and that the stress-energy tensor inthe dual field theory is related to the asymptotic form of the metric. In moregeneral theories, the entanglement entropy may correspond to a more com-plicated functional of the bulk geometry and the relation between the stresstensor and asymptotic metric may be modified. In these cases, we expectthat the bulk equations will be different, for example involving α′ correc-tions with higher-derivative terms. However, it may be possible followingthe methods in this chapter to derive the linearized version of these moregeneral equations given a particular choice for the holographic entanglemententropy formula and the holographic formula for the stress tensor.It will be interesting to see whether the first non-linear corrections toEinstein’s equations in the bulk are equivalent to some simple property ofentanglement entropies.52Specifically, Eq. (20) becomes 0 =∫DRddx(∆ii − xixj/R2∆ij); expanding this inpowers of R using the generalization of Eq. (21) yields at each order in R an equationthat relates ∆ii(n) to quantities calculated from ∆ at lower orders in n. The steps in theproof are as before.1294.5. DiscussionFinally, we comment on the relation to the work of Jacobson [79], whichpartly motivated our investigations. Jacobson realized that Einstein’s equa-tions could be derived from the assumption that the energy flux through apart of any bulk Rindler horizon gives rise to a proportional local changein area of this horizon. Interpreting the area as an entropy, such a relationlooks like the first law of thermodynamics. However, in Jacobson’s work, itwas not clear why areas of segments of an arbitrary bulk Rindler horizon(not necessarily associated with any black hole) should correspond to anentropy, so the origin of the thermodynamic relation remained mysterious.In our case, the “thermodynamic relation” dS = dE is an exact quantumrelation (i.e. not really thermodynamics) derived to hold for the underlyingfundamental degrees of freedom associated with our gravitational system.Thus, while our final result (in contrast to Jacobson’s work) applies so faronly at the linearized level, the starting point is well understood. In detail,the bulk interpretation of our dS = dE relation is somewhat different thatJacobson’s starting point (the bulk surfaces/horizons we deal with are globalrather than local and the energy has a different interpretation), but the tworelations were similar enough to motivate the question of whether Einstein’sequations could be derived from the first law of [88].130Chapter 5OutlookThere are many interesting questions which are left open in the work of thisthesis. In fact one of the questions asked in the introduction by way of mo-tivation was left unanswered: does momentum-space entanglement entropycorrespond to some simple geometric quantity in the gravity theory? Becauseof the UV/IR connection in AdS/CFT which relates a radial coordinate r toa CFT energy scale µ it is tempting to speculate that the the entanglemententropy between modes above and below some scale µ is related to the areaof a gravitational radial slice at r(µ), i.e.S(µ)?=Area(r(µ))4GN. (5.1)We have been confined mainly to results in perturbation theory, where ourexpressions are valid for small coupling. To answer this question we wouldrequire a better understanding of the strong coupling behaviour, where thegravity side is classical. There is, however, a consistency check: the area ofa radial slice should be proportional to the field theory volume and we doindeed find that the momentum-space entanglement entropy scales with thevolume in our calculations. 53We have shown how to obtain a reduced density matrix for low momen-tum modes and it would be interesting to investigate how quantities in fieldtheories compare when computed via low momentum effective actions ver-sus the usual low energy Wilsonian effective actions which show up in therenormalization group. Also, as mentioned earlier, there is a possibility thatthe types of quantities we have been discussing can be measured in the lab53This is related to the fact that the number of modes below a scale µ to entangle isproportional to the volume of the system, and so this result should be true outside ofperturbation theory.131Chapter 5. Outlookproviding a possible experimental validation of the results presented.We have been able to give one specific answer to the question: howentanglement in a CFT can affect the gravitational structure of its dual?Changing the amount of energy in a ball in the CFT incurs a change inthe amount of entanglement in that region, and in the dual theory this isprecisely tantamount to the Einstein equations. This was carried to linearorder in the metric, however, and it would be interesting to investigatewhat non-linear contributions to Einstein’s equations correspond to in theCFT, and if in particular these are related to some simple properties ofentanglement.It would be unbearable to end this thesis without pointing out an amus-ing irony which it entails. Einstein’s great achievement was his theory ofGeneral Relativity. This classical theory stood in stark contrast to quan-tum mechanics, which Einstein did not accept as a complete theory of na-ture, and in particular pointed to entanglement as one of its quintessentiallybizarre and counterintuitive features. As we have seen though, holography(AdS/CFT specifically) indicates to us that General Relativity is the wayit is, and in particular Einstein’s equations are obeyed, precisely due to thequantum mechanical phenomenon of entanglement. 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We find thatS(P ) = −λ2 log(λ2)∑n6=0,N 6=0|〈n,N |HAB|0, 0〉|2(E0 + E˜0 − En − E˜N )2+O(λ2)= −λ2 log(λ2)∑n6=0,N 6=0∫ ∞0dττ〈0, 0|HI |n,N〉e(E0,0−En,N )τ 〈n,N |HI |0, 0〉+O(λ2)= −λ2 log(λ2)∑n6=0,N 6=0∫ ∞0dττ〈0, 0|eH0τHIe−H0τ |n,N〉〈n,N |HI |0, 0〉+O(λ2)= −λ2 log(λ2)∫ ∞0dττ〈0, 0|eH0τHIe−H0τΠAHI |0, 0〉+O(λ2)= −λ2 log(λ2)∫ ∞0dττ〈HI(−iτ)ΠAHI(0)〉+O(λ2)= −λ2 log(λ2)∫ ∞0dττ∫d3xd3y〈HI(−iτ, x)ΠAHI(0, y)〉+O(λ2)= −V λ2 log(λ2)∫ ∞0dττ∫d3x〈HI(−iτ, x)ΠAHI(0, 0)〉+O(λ2)Here, we use the standard definition of time-dependent operators in the“interaction picture”:HI(t) ≡ eiH0tHIe−iH0t .The operator Π is projects to intermediate states with at least one par-ticle having momentum in the subset P and at least one particle havingmomentum in the complementary subset of momenta.142A.2. Entanglement entropy in a fermionic systemThe factor of volume in the last line comes from the y integral in theprevious line, which is trivial since the correlator in that line can dependonly on the combination x − y. The entropy per unit volume S(P )/V willhave a finite limit, so that S(P ) is an extensive quantity.A.2 Entanglement entropy in a fermionic systemHere we calculate the entanglement entropy in a fermionic theory with a(ψ¯ψ)2 interaction. Consider for definiteness the renormalizable theory in1+1 dimensions. The fermion fields are expanded asψ(x) =∑p1L121√2ωp(apu(p)e−ipx + b†pv(p)eipx). (A.1)As a straightforward application of (2.20) the entanglement entropy isSµ = −λ2 log(λ2)∑t∗∑p∣∣∣〈{p, t}1, ..., {p, t}4∣∣∣(ψ¯ψ)2∣∣∣ 0〉∣∣∣2(ω1 + ω2 + ω3 + ω4)2 +O(λ2), (A.2)where t indicates the type of fermion (i.e. particle or antiparticle). The starindicates that the sum over momenta is restricted to the set where at leastone momentum is above and at least one momentum is below the scale µ.Substituting the expansion (A.1) into (A.2)Sµ = −λ2 log(λ2)6 · 424L2∗∑pδ∑i pi(∑i ωpi)2∏i ωpi|u¯(p1)v(p2)u¯(p3)v(p4)− u¯(p1)v(p4)u¯(p3)v(p2)|2 ,(A.3)where 6 is the number of ways or choosing 2 particles and 2 antiparticles.Using 1+1 dimensional spinor and gamma matrix identities, and passing tothe infinite volume limit we are left withSµ/L = −λ2 log(λ2)6(2pi)3∫ ∗dp1...dp4δ(∑ipi)(p1 · p3 −m2)(p2 · p4 −m2)(∑i ωpi)2∏i ωpi.143A.2. Entanglement entropy in a fermionic systemIn the region where three momenta are taken to be large this integral divergeslinearly.144Appendix BB.1 Two qubit system exampleIn order to elucidate the essential features of the type of entanglement weconsider in this thesis, this section will be concerned with detailing a simpleand explicit example. As a simple model for a system with multiple fermionicdegrees of freedom, consider a system of two spins with Hamiltonian (unitsare chosen so that ~ = 1):Hf =σ(1)z2⊗ 1 + 1 ⊗σ(2)z2, (B.1)Our notation here for spin up is |u > and for spin down it is |d >. Theseare defined by σz|u >= |u > and σz|d >= −|d >.The ground state has both spins pointing down, |0 >= |d, d > withenergy -1. This state is separable and there is no entanglement between thespins.Now imagine adding some interaction which generates entanglement,Hint = σ(1)x ⊗ σ(2)x , (B.2)so thatH = Hf + λHint. (B.3)We first consider |λ| << 1 so that we may get a perturbative solutionfor the ground state. One can easily see that to O(λ3) this state is simplygiven by |s >= |d, d > −λ2 |u, u > since we have5454In our notation we have σx|d >= |u > and σx|u >= |d >.145B.1. Two qubit system exampleH|s >=Hf |s > +λHint|s >, (B.4)=− |d, d > −λ2|u, u > +λ|u, u > −λ2|d, d >, (B.5)=− (1 +λ22)|s > +O(λ3). (B.6)Thus up to O(λ3) the ground state is proportional to |s > with energy−(1 + λ2/2). The correctly normalized ground state is|Ω >=1√1 + λ24(|d, d > −λ2|u, u >)+O(λ3). (B.7)The corresponding density matrix is thenρ =|Ω >< Ω|, (B.8)=(1−λ24)|d, d >< d, d| −λ2(|u, u >< d, d|+ |d, d >< u, u|) +λ24|u, u >< u, u|+O(λ3).(B.9)Note that to this order Tr ρ = 1 as it should.The reduced density matrix for the first spin (or the second) is obtainedby tracing out the second (or the first),ρ(1) = Tr (2)ρ, (B.10)=(1−λ24)|d >< d|+λ24|u >< u|+O(λ3). (B.11)Thus the density matrix is diagonal to this order and the entanglemententropy is easily calculated:146B.1. Two qubit system exampleS =− Tr ρ(1) log ρ(1), (B.12)=−(1−λ24)log(1−λ24)−λ24logλ24+O(λ3), (B.13)=−14λ2 log λ2 +1 + 2 log 24λ2 +O(λ3 log λ3). (B.14)Now let us calculate the lowest order term in the entanglement entropyof the ground state by using our general formula Eq. (3.1)S = −λ2 log(λ2)∑n6=0,N 6=0|〈n,N |Hint|0〉|2(E0 + E˜0 − En − E˜N)2 +O(λ2). (B.15)Let us quickly recall what is meant in this formula. Consider a Hilbert spacewhich decomposes into the tensor product of two factors H = H(1) ⊗H(2).Then Eq. (B.15) applies to any Hamiltonian which is the sum of free andinteracting pieces H = Hf + λHint, with the free Hamiltonian being givenby a sum acting on each factor separately, i.e.Hf = H(1) ⊗ 1 + 1 ⊗H(2), (B.16)Then the states |n > and |N > are the energy eigenstates of H(1) and H(2)respectively, and the energies En and E˜N their corresponding eigenvalues.Now specializing to our qubit system (Eqs. (B.1) and (B.2)) the freeground state was |d, d >, and thus this state corresponds to |0, 0 > in Eq.(B.15). Then from the free spin hamiltonian Eq. (B.1) we have Ed = E˜d =−1/2 and Eu = E˜u = +1/2. Since the sum in Eq. (B.15) exludes n = d andN = d it reduces to a single term:147B.1. Two qubit system exampleS =− λ2 log(λ2)∣∣∣〈u, u|(σ(1)x ⊗ σ(2)x)|d, d〉∣∣∣2(Ed + E˜d − Eu − E˜u)2 +O(λ2), (B.17)=−14λ2 log(λ2) +O(λ2), (B.18)in agreement with what we found in Eq. (B.14).Now of course this system is exactly solvable and so we can check theseresults.The eigenvalues ofH = Hf+λHint are found to be√1 + λ2, λ,−λ,−√1 + λ2.Thus for any finite λ, −√1 + λ2 is the lowest energy with correspondingeigenvector |d, d > − λ1+√1+λ2|u, u > as one can easily check. Then theexact ground state of this model is simply,|G >=(1 +√1 + λ2)1/221/2 (1 + λ2)1/4(|d, d > −λ1 +√1 + λ2|u, u >). (B.19)Then one finds for the reduced density matrixρ(1) = Tr (2)ρ, (B.20)=1 +√1 + λ22 (1 + λ2)1/2|d >< d|+λ2(1 +√1 + λ2)2 |u >< u| , (B.21)=(a+ 12a)|d >< d|+(a− 12a)|u >< u|. (B.22)where we have defined the constant a(λ) ≡√1 + λ2. Then one arrives atthe following simple result for the entanglement entropy between the twoqubits,S = −a+ 12aloga+ 12a−a− 12aloga− 12a. (B.23)148B.1. Two qubit system exampleThus as we expect, the system is unentangled at λ = 0 (a = 1) and becomesmaximally entangled as λ→∞ (a→∞). Expanding (B.23) to O(λ3 log λ3)we haveS = −14λ2 log λ2 +1 + log 44λ2 +O(λ3 log λ3). (B.24)This is in agreement with Eq. (B.14), and with Eq. (B.18) obtained fromthe general formula (B.15).149Appendix CC.1 Alternative derivation of linearizedEinstein’s equations from δE = δSIn this appendix, we offer an alternative proof that solutions of Einstein’sequations satisfy δSA = δEhypA . This proof replaces δSA = δEhypA with theinfinite set of relations obtained by matching the terms in the power seriesexpansion of this relation in R, the radius of the disk A, as we did in section4.2.C.1.1 Expansion of δE = δS in powers of RTo begin, we expand both (4.12) and (4.13) in powers of R. DefiningHµν(z, x, y) =∞∑n=0znH(n)µν (x, y) (C.1)we haveδE =32∑mx,my=0R2+2mx+2myI2,mx,my∂2mxx ∂2myy H(0)tt (t, x0, y0) (C.2)whileδS =∑Rn+2mx+2my+2{1(2mx)!(2my)!∂2mxx ∂2myy H(n)xx (t, x0, y0)(In,mx,my − In,mx+1,my)+ 1(2mx)!(2my)!∂2mxx ∂2myy H(n)yy (t, x0, y0)(In,mx,my − In,mx,my+1)−2 R2 1(2mx+1)!(2my+1)!∂2mx+1x ∂2my+1y H(n)xy (t, x0, y0)In,mx+1,my+1}(C.3)where I was defined in (4.21).150C.1. Alternative derivation of linearized Einstein’s equations from δE = δSC.1.2 Checking that solutions of Einstein’s equationssatisfy δS = δEUsing these expansions, it is straightforward to verify that any solution ofthe linearized Einstein’s equations (4.14) satisfies δE = δS, as was doneoriginally in [88] and by another alternative approach in section 4.Using the expansion (C.1), the equations (4.14) becomeH(n)tt = H(n)xx +H(n)yy (C.4)∂tH(n)tt = ∂xH(n)tx + ∂yH(n)ty (C.5)∂tH(n)tx = ∂xH(n)xx + ∂yH(n)xy (C.6)∂tH(n)ty = ∂xH(n)xy + ∂yH(n)yy (C.7)H(n)µν =1n(n+ 3)(∂2t − ∂2x − ∂2y)H(n−2)µν n ≥ 2 (C.8)H(1)µν = 0 . (C.9)Starting with (C.4) and then using (C.5), (C.6), (C.7), and finally (C.8), wefind:∂2tH(n)tt = ∂2t (H(n)xx +H(n)yy )⇒ ∂t(∂xH(n)xt + ∂yH(n)yt ) = ∂2t (H(n)xx +H(n)yy )⇒ ∂2xH(n)xx + ∂2yH(n)yy + 2∂x∂yH(n)xy = ∂2t (H(n)xx +H(n)yy )⇒ ∂2xH(n)xx + ∂2yH(n)yy + 2∂x∂yH(n)xy = ∂2t (H(n)xx +H(n)yy )⇒ ∂2xH(n)xx + ∂2yH(n)yy + 2∂x∂yH(n)xy = (∂2x + ∂2y)(H(n)xx +H(n)yy ) + (n+ 2)(n+ 5)(H(n+2)xx +H(n+2)yy )⇒ 2∂x∂yH(n)xy = ∂2yH(n)xx + ∂2xH(n)yy + (n+ 2)(n+ 5)(H(n+2)xx +H(n+2)yy )Using this last equation, we can eliminate H(n)xy from (C.3). This givesδS =∑Rn+2mx+2my+2{ 1(2mx)!(2my)!∂2mxx ∂2myy H(n)xx (t, x0, y0)Cxxn,mx,my+1(2mx)!(2my)!∂2mxx ∂2myy H(n)yy (t, x0, y0)Cyyn,mx,my}(C.10)151C.1. Alternative derivation of linearized Einstein’s equations from δE = δSwhere for n ≥ 2 we haveCxxn,mx,my = In,mx,my − In,mx+1,my −2my2mx + 1In,mx+1,my −n(n+ 3)(2mx + 1)(2my + 1)In−2,mx+1,my+1= 0Cyyn,mx,my = In,mx,my − In,mx,my+1 −2mx2my + 1In,mx,my+1 −n(n+ 3)(2mx + 1)(2my + 1)In−2,mx+1,my+1= 0while for n = 1 and n = 0, we haveCxx1,mx,my = I1,mx,my − I1,mx+1,my −2my2mx + 1I1,mx+1,my =43I3,mx,myCyy1,mx,my = I1,mx,my − I1,mx,my+1 −2mx2my + 1I1,mx,my+1 =43I3,mx,myandCxx0,mx,my = I0,mx,my − I0,mx+1,my −2my2mx + 1I0,mx+1,my =32I2,mx,myCyy0,mx,my = I0,mx,my − I0,mx,my+1 −2mx2my + 1I0,mx,my+1 =32I2,mx,my .In each case, we have made simplifications using the definition (4.21) of I.Using these results together with (C.9), we find that (C.10) simplifies toδS =∑R2mx+2my+21(2mx)!(2my)!∂2mxx ∂2myy (H(0)xx (t, x0, y0) +H(0)yy (t, x0, y0))(32I2,mx,my)=32∑R2mx+2my+21(2mx)!(2my)!∂2mxx ∂2myy H(0)tt (t, x0, y0)I2,mx,my= δEThus, we have verified that δS = δE for linearized solutions of Einstein’sequations, providing an alternate argument to the one in [88].152


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