Information flows in strongly coupled ABJM theory

JHEP01(2019)232 Published for SISSA by Springer Received:December 16, 2018 Accepted:January 23, 2019 Published:January 31, 2019 Information flows in strongly coupled ABJM theory Vijay Balasubramanian,a,b Niko Jokela,c,d Arttu P¨onnic,d and Alfonso V. Ramalloe,f aDavid Rittenhouse Laboratory, University of Pennsylvania, PA 19104 Philadelphia, U.S.A. bTheoretische Natuurkunde, Vrije Universiteit Brussel and International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium cDepartment of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland dHelsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland eDepartamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain fInstituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: We use holographic methods to characterize the RG flow of quantum information in a Chern-Simons theory coupled to massive fermions. First, we use entanglement entropy and mutual information between strips to derive the dimension of the RG-driving operator and a monotonic c-function. We then display a scaling regime where, unlike in a CFT, the mutual information between strips changes non-monotonically with strip width, vanishing in both IR and UV but rising to a maximum at intermediate scales. The associated information transitions also contribute to non-monotonicity in the conditional mutual information which characterizes the independence of neighboring strips after conditioning on a third. Finally, we construct a measure of extensivity which tests to what extent information that region A shares with regions B and C is additive. In general, mutual information is super-extensive in holographic theories, and we might expect superextensivity to be maximized in CFTs since they are scale-free. Surprisingly, our massive theory is more super-extensive than a CFT in a range of scales near the UV limit, although it is less super-extensive than a CFT at all lower scales. Our analysis requires the full tendimensional dual gravity background, and the extremal surfaces computing entanglement entropy explore all of these dimensions. Keywords: AdS-CFT Correspondence, D-branes ArXiv ePrint: 1811.09500 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP01(2019)232 JHEP01(2019)232 Contents 1 Introduction 1 2 Background solution 2 2.1 Holographic Callan-Symanzik equation 6 3 Entanglement entropy of strips 9 3.1 NfN: the few flavor limit 10 3.1.1 Turning point outside the cavity (x∗≥1) 10 3.1.2 Turning point inside the cavity (x∗<1) 12 3.2 Dimension of the RG driving operator 13 4 The flow of mutual information 17 4.1 Mutual information transitions 17 4.2 c-functions: flow of the number of degrees of freedom 20 4.3 Flow of extensivity 24 4.4 Flow of conditional mutual information 27 5 Discussion 30 A Details of the background 31 A.1 Expansion in flavor 32 B UV expansion of the entanglement entropy 33 1 Introduction Interactions in a quantum field theory (QFT) cause quantum information to be non-locally distributed across space through the phenomenon of entanglement in the wavefunction. It is of great interest to understand how this structure of shared information changes with energy scale, or equivalently, during renormalization group (RG) flow. Essentially the only tool we have to study this question in strongly coupled field theories is holography. Even in these settings there is a key challenge: there are very few exact solutions in holographic gravity that correspond to non-trivial renormalization group flows in QFT. Many “phenomenological” holographic treatments of RG flow are constructed so that ad-dimensional field theory is described by a (d+ 1)-dimensional theory of gravity in which the extra dimension represents QFT scale. Gravitational solutions can be easier to obtain in this simplified setting. However, in cases where an actual gauge-gravity duality has been constructed in string theory, there are always the ten dimensions of string theory in the gravitational description, with internal symmetries of the QFT represented by the – 1 – JHEP01(2019)232 structure of the extra dimensions beyond the radial direction which represents QFT scale. In such settings which are fully grounded in string theory, RG flows of QFT are described as complete ten-dimensional gravitational backgrounds where the “internal” and “AdS” dimensions interact, mix and sometimes exchange roles. Finding exact gravitational solutions in this complete setting is difficult. Our goal in this paper is to examine the RG flow of quantum information in a strongly coupled QFT running between non-trivial interacting fixed points, in a scenario where we have the powerful lever of a complete ten-dimensional classical gravitational dual. To this end, we focus on the ABJM theory [1] which is the (2+1)-dimensional analogue of the 4d maximally supersymmetric Yang-Mills theory. This 3d Chern-Simons gauge theory has a symmetry group U(N)k×U(N)−kwith levels (k, −k) coupled to matter fields in the bifundamental representation of the gauge group. When Nand kare large, the model has a dual holographic description as an AdS4×CP3geometry with fluxes in Type IIA supergravity. The ABJM theory can be generalized by adding fundamental quarks transforming in the (N, 1) and (1, N) representations of the gauge group [2,3]. In the gravity dual, these quarks are incorporated by adding D6-branes wrapping an RP3inside the internal CP3and extended along AdS4. When the number of flavors Nfis sufficiently small, the field theory can be treated in a quenched approximation where the loops of fundamentals are suppressed. In the same limit, the D6-branes in the gravity dual can be treated as probes in the AdS4×CP3 background. When Nfis larger so that the field theory is not quenched, the D6-branes in the gravity description will backreact on the geometry. Constructing the backreacted supergravity solution is a difficult task in general. However, in the Veneziano limit [4] in which both the number of colors Nand flavors Nfare taken to be large with Nf/N fixed, we can employ a systematic perturbative approximation [5]. Using this technique one can find the complete gravity duals. In our context, these generalized ABJM geometries were found in [6–8] for massless flavors and in [9] for massive flavors, with a recent extension [10] to a non-commutative geometry. This is not what some call a massive deformation of the ABJM theory [11,12]; entanglement flows in that context are studied in [13,14]. The extremal surfaces that holographically compute entanglement in our setting will explore all 10 dimensions of the dual gravity theory (see previous examples in, e.g., [15,16]). 2 Background solution The maximally supersymmetric three-dimensional U(N)k×U(N)−kChern-Simons theory coupled to massive bifundamental quarks should be conformal in the UV, at scales well above the quark masses, and in the IR, at scales well below the quark masses. Thus, we expect a dual description in Type IIA supergravity that tends to AdS4times a compact factor both near infinity and in the deep interior. The relevant solution, interpolating smoothly between two AdS4spacetimes, was found in [9]. The string frame metric in this solution is: ds2 10 =h(x)−1/2dx2 1,2+h(x)1/2e2g(x)dx2 x2+q(x)ds2 S4+ds2 S2 f,(2.1) – 2 – JHEP01(2019)232 where ds2 1,2in the Minkowski metric in 2 + 1 dimensions, ds2 S4is the standard metric of a unit four-sphere, and S2 fis a fibered two-sphere (see below). We are just presenting the metric here and not the fluxes, since the latter do not play a role in our analyses. The radial coordinate xin (2.1) is related to a canonical coordinate rin which the metric is asymptotically ds2=−r2dt2+dr2/r2+. . . as follows dr =egdx x.(2.2) Through this relation, the IR (r= 0) and the UV (r=∞) correspond to x= 0 and x=∞, respectively. As the geometry is not globally AdS, the dilaton is also not a constant (see below), but in the IR and UV the geometry tends towards AdS4in Poincar´e coordinates times a compact factor. Thus x= 0 is the Poincar´e horizon. The function e2g(x), the local radius of the internal S2, quantifies the relative squashing between the internal and external manifolds. The radius of S4is given by the function e2f:= q e2g. The function q(x) measures the relative squashing between the S2fiber and the S4base on the internal manifold. A convenient representation of the metric for S4is ds2 S4=4 (1 + ξ2)2"dξ2+ξ2 3 X i=1 (ωi)2#,(2.3) where ωiare the SU(2) left-invariant one-forms which satisfy dωi=1 2ijkωj∧ωkand 0≤ξ < ∞. The metric of the fibered two-sphere S2can be written in terms of one-forms E1, E2: ds2 S2 f=E12+E22(2.4) E1=dθ +ξ2 1 + ξ2sin ϕω1−cos ϕω2(2.5) E2= sin θdϕ −ξ2 1 + ξ2ω3+ξ2 1 + ξ2cos θcos ϕω1+ sin ϕω2,(2.6) where 0 ≤θ≤π, 0≤ϕ < 2π. The massive quarks in the field theory require the inclusion of D6-branes that span the AdS4part of the geometry and wrap a part of the internal manifold. The D6-brane charge density is measured by violation of the Bianchi identity of the RR two-form F2, and this turns out to be proportional to η−1 [9] where η(x) = 1 + ˆ1−1 x2Θ(x−1) ,(2.7) in terms of the Heaviside function Θ(z). Thus, ηdictates the distribution of smeared flavor D6-branes that act as sources for the SUGRA equations of motion. Notice that η= 1 for x < 1, meaning that the D6-branes do not span the whole radial coordinate but end above the Poincar´e horizon. Thus, in the following, we call the region x≤1 the “cavity”. The parameter ˆencodes backreaction of the massive quarks and relates to the number Nfof D6-branes as: ˆ=3Nf 4k=3 4 Nf Nλ , (2.8) – 3 – JHEP01(2019)232 where the ’t Hooft coupling is related to the Chern-Simons level kand the number of colors Nas λ=N/k. The metric ansatz (2.1) is guaranteed to preserve N= 1 supersymmetry in three dimensions [6,9], if the following master equation is satisfied W00 + 4η0+W0+ 4ηW0+ 10η 3W−W0+ 4η+ 6 x(W0+ 4η)= 0 (2.9) for Wdefined as W(x) = 4 kh1/4e2f−g−φ.(2.10) Then, up to freedom in integration constants, the functions f,g,h, and the dilaton φ can be constructed from W(see appendix A). Moreover, the squashing function qcan be written in terms of the master function Wand its derivative as: q=3W x(W0+ 4η).(2.11) The quark mass mqencoded in this gravitational background can be computed by evaluating the Nambu-Goto action of a fundamental string stretched along the holographic direction between x= 0 and x= 1 at fixed values of the other spacelike coordinates. This gives mq=rq/2πα0, where α0is the Regge slope (which we will set to 1), with rqgiven by: rq=Z1 0 dxeg x.(2.12) Following (2.2), rqis just the canonical rcoordinate of the tip of the flavor D6-branes. The IR limit. In the region x<1 (the “cavity” region) there are no D6-branes (see (2.7)). Thus, the solution in this region should solve the matter-free equations with Nf→0, ˆ→0, and η→1. In this limit, W= 2xgives one solution to the master equation. This choice reproduces the original matter-free ABJM solution after transforming to the canonical coordinate r[9]. In this case the squashing function qin (2.11) becomes equal to one and the internal metric in (2.1) is the standard Fubini-Study metric of CP3giving a 10d metric ds2 10,=0 =L2 ABJM ds2 AdS4+ 4 L2 ABJM ds2 CP3,(2.13) where ds2 AdS4and ds2 CP3are respectively the AdS4and CP3metrics. The former, in Poincar´e coordinates, is given by: ds2 AdS4=r2dx2 1,2+dr2 r2.(2.14) In (2.13)LABJM is the radius of the AdS4part of the metric and in string units is L4 ABJM = 2π2N k,(2.15) where Nand kare two integers which correspond, in the gauge theory dual, to the rank of the gauge groups and the Chern-Simons level, respectively. For our purpose of solving the full equations with matter, the pure ABJM solution does not suffice in the x < 1 region. Rather we need a solution that will match appropriately – 4 – JHEP01(2019)232 onto the solution in the x > 1 region where the D6-branes backreact on the geometry. Fortunately, in the limit ˆ→0, there is a one-parameter family of analytic solutions to (2.9): Wrunning =4(1 + 4γx)x 1 + √1+4γx ,(2.16) where γis a constant. These “running solutions” reduce to ABJM in the deep IR (Wrunning(x→0) →2x). But as xincreases, the running solution flows away from the ABJM fixed point and eventually asymptotes at x→ ∞ to a metric describing a resolved Ricci flat cone. With a finite number of quark flavors and hence ˆ > 0, the running solution will apply in the “cavity” x < 1 which is not penetrated by the D6-branes and hence is locally described by an Nf= 0 solution. We will match the running solution in the x < 1 region onto the appropriate metric in the x > 1 region. The UV limit. To gain control over the x > 1 region, we first consider the asymptotic UV limit x→ ∞. In this limit, the D6-brane distribution tends to a constant since η→1 + ˆ≡η0. Then one solution of the master equation (2.9) is W(x→ ∞)≈q0(η0+q0) 2−q0 x , (2.17) where the squashing function qis also a constant q0=3+3η0−p9η2 0−2η0+ 9/2. The resulting ten-dimensional asymptotic geometry is of the type AdS4×M6, where M6 is a squashed version of CP3with constant squashing factors. Indeed, the metric takes the form ds2 10 ≈L2 0ds2 AdS4+L2 0 b2hq0ds2 S4+ds2 S2 fi,(2.18) where L0is constant and given by L4 0= 128π2N k (2 −q0)q3 0 (η0+q0) (q0+ 1)5,(2.19) and bis another constant, which can be written in terms of the asymptotic UV squashing q0as b=2q0 q0+ 1 .(2.20) The parameter bwill play an important role in what follows. Its interpretation is clear from (2.18): it represents the relative squashing of the CP3part of the asymptotic metric with respect to the AdS4part. In the unflavored case ˆ= 0 we have q0=b= 1. In general q0and bgrow with ˆand reach their maximal values q0= 5/3 and b= 5/4 when ˆ→ ∞. In fact, (2.17) solves the master equation for all xif ηis constant everywhere, and corresponds to the limit of massless quarks (mq∝rq→0) that backreact in the AdS description. This case is discussed in depth in [6–8,17], and gives the desired asymptotics because at energies far above the mass scale of the quarks they should act as if they are massless. Holographically, this means that the full solution should asymptote to (2.17). – 5 – JHEP01(2019)232 The full solution. The solution to the master equation that we are interested in has two pieces. Inside the cavity (x≤1) the master function Wis given by (2.16). At the boundary of the cavity x= 1 we need to glue this solution continuously to another one which asymptotes to (2.17). This can be obtained numerically by the standard shooting technique starting at x= 1, with boundary conditions W(1) and W0(1) given by the running solution (2.16). Shooting is done by varying the constant γin the running solution for x < 1 until the desired asymptotic solution for W(x→ ∞) is obtained by integrating outward with boundary conditions that impose continuous Wand W0at x= 1. A generic value of γwill provide a solution for which W∼x3/2as x→ ∞, i.e., not asymptotically AdS4but rather a G2cone. For a given ˆthere exists a unique γ=γ(ˆ) such that W∼x, as x→ ∞, i.e., such that the solution asymptotes to (2.17). Such a piecewise metric then describes an interpolation between two AdS4spacetimes with different radii. These solutions were constructed numerically in [9] and as a power series in ˆin [18]. In addition to the numerical solutions, we will also use this power series, which is reviewed in appendix A. 2.1 Holographic Callan-Symanzik equation The solution given above is a holographic description of the renormalization group flow of a strongly interacting theory of matter and gauge fields. The AdS4geometries at x→0 and x→ ∞ indicate that the flow runs between different conformal fixed points. We can characterize the flow in terms of a Callan-Symanzik equation for the quark mass, extending the discussion in [17]. The natural mass scale in a top-down holographic setup such as ours involving flavor and color branes is the distance between these two sets of branes. This distance is related to the mass of fundamental quarks in the field theory that are dual to the open strings ending on the flavor brane [19]. In our model, the flavor branes are D6-branes embedded supersymmetrically in the background. The quark mass is obtained by evaluating the Nambu-Goto action for a string extended from the origin and ending on the branes [20]. To evaluate the quark mass, consider a probe D6-brane embedded along the four AdS4 directions and wrapping a three-dimensional cycle inside the compact internal manifold. The precise form of this cycle for a supersymmetric embedding can be determined by using kappa symmetry. It was shown in [6] that this requires that the pullbacks to the worldvolume of the SU(2) one-forms ω1and ω2of (2.3) must vanish. Extending the brane along the coordinate ϕof the S2fiber, the induced metric on the brane worldvolume takes the form: ds2 7=1 ph(r)dx2 1,2+ph(r)(1 + e2g(r)θ0(r)2)dr2 +e2g(r)ph(r)q(r)dα2+qsin2αdβ2+ sin2θ(dψ + cos αdβ)2,(2.21) where, for convenience, we are using the canonical AdS4coordinate rintroduced in (2.2). The angular coordinates α,β, and ψare related to ξ,ϕ, and to the one used to parameterize the pullback of ω3(see [6] for details). The kappa symmetry condition then reduces to the – 6 – JHEP01(2019)232 following first-order differential equation for θ=θ(r): eg(r)dθ dr = cot θ . (2.22) This equation can be integrated exactly after a change of variable to a new radial coordinate u: dr eg(r)=du u→u= exp Zr e−g(r0)dr.(2.23) Then, the embedding function θis simply cos θ=R0 u,(2.24) where R0is an integration constant. Thus, for our SUSY embedding ucos θis constant. In order to easily describe the corresponding seven-dimensional surface, it is convenient to introduce two new coordinates (R, ρ) as R=ucos θ;ρ=usin θ . (2.25) In these coordinates the (r, θ) sector of the ten-dimensional metric takes the form ph(r)dr2+e2g(u)dθ2=ph(u)e2g(u) u2dR2+dρ2,(2.26) where u2=R2+ρ2. For supersymmetric embeddings R=R0= constant, which leads to the induced worldvolume metric ds2 7=1 √hdx2 1,2+√h e2g pR2 0+ρ2dρ2 +e2g√hqdα2+qsin2αdβ2+ sin2θdψ2+ cos αdβ2,(2.27) where all functions depend on ρthrough the combination pR2 0+ρ2. Therefore, ρplays the role of the holographic coordinate on the worldvolume. Actually, in the UV region ρ≈u→ ∞ we have that eg≈r/b and (2.23) can be integrated as r∼ρ1 b. Since ris the canonical AdS4coordinate we should identify the energy scale with ρ1 b. Consider now a fundamental string located at the point ρ=ρ∗. The string extends from R= 0 to its boundary value R=R0, where it intersects with the D6-brane. The running mass of the corresponding dual (valence) quark is equal to the Nambu-Goto action of this string per unit time. The induced metric on the string worldsheet extended in (t, R) at ρ=ρ∗is given by ds2 2=−dt2 √h+√h e2gdR2 R2+ρ2 ∗ .(2.28) The running quark mass is then mq=1 2π α03/2ZR0 0p−det g2dR =1 2πα03/2ZR0 0 eg(u)dR pR2+ρ2 ∗ .(2.29) – 7 – JHEP01(2019)232 We want to find an evolution equation for mqwith respect to the energy scale Λ. As argued above, we should identify the energy scale with ρ1/b ∗. Therefore, we define: Λ := ρ1/b ∗.(2.30) We can work out the evolution equation for mq, which after a straightforward calculation boils down to: 2πα03/2∂mq ∂log Λ =bρ∗ ∂mq ∂ρ∗ =bZR0 0pR2+ρ2 ∗ R∂RegdR −bR0eg√R2 0+Λ2b pR2 0+ Λ2b.(2.31) The UV-limit of the above equations can be worked out easily. Indeed, in this limit the function gfor ρ=ρ∗is given by eg≈(R2+ρ2 ∗)1 2b b,(2.32) and we infer from (2.29) that the running mass is given by mq≈1 2π α03/2bZR0 0 (R2+ρ2 ∗)1 2b−1 2≈Λ1−bR0 2π α03/2b,(2.33) where, in the last step, we took the limit in which Λ is large. Moreover, plugging (2.32) into (2.31) we arrive at ∂mq ∂log Λ =mq−1 2πα03/2 R0 (R2 0+ Λ2b)b−1 2b .(2.34) Furthermore, in the deep UV limit in which Λ is large, we can neglect the R2 0in the denominator and using (2.33) we conclude that the second term in (2.34) is just −b mq. We finally obtain the evolution equation for the quark mass mq: ∂mq ∂log Λ =−γmmq,(2.35) where we introduced the anomalous dimension of the quark mass γm=b−1.(2.36) The result (2.35) is precisely the Callan-Symanzik equation for the effective mass which we were looking for. We see in this way that the gravitational background describes a renormalization group flow of the full theory. Notice that γm= 0 for the unflavored case (and thus there is no running of mqwhen there are no dynamical flavors) and γm→1/4 as ˆ→ ∞. – 8 – JHEP01(2019)232 UV-behavior of the entanglement entropy, providing a cross-check on the formulas derived in the last section.1 Consider the holographic dual of any three-dimensional field theory near a UV fixed point. The four-dimensional part of the bulk metric must asymptote to AdS4and can be written near the boundary as ds2 4=L2 UV z2dz2 f(z)−dt2+dx2+dy2,(3.31) where f(z) = 1 + O(zα) and the boundary lies at z= 0. Departures from AdS4, and thus the effects of RG flow, are encoded in the subleading behavior of f(z). We now assume that the RG flow is driven by a relevant scalar perturbation Owith a dimension 1/2<∆<3. This is dual holographically to a scalar field φwith mass m2= ∆(∆ −3). The bulk action will then be S4=1 16πG(4) NZd4x√−g4R+6 L2 UV −1 2∂µφ∂µφ−1 2m2φ2+. . ..(3.32) When φ(z, ~x) = 0 the corresponding background solution reduces to AdS4. The equation of motion for the scalar gives the near boundary behavior φ(z, ~x) = z3−∆(φ0(~x) + O(z2)) (3.33) and corresponds to a perturbation of the UV CFT by S3=S(UV) 3+Zd3x φ0(~x)O(~x).(3.34) Let us focus on a uniform perturbation φ0(~x) = const. = φ0. In this case the Einstein equations give the backreaction of the scalar field on the metric f(z) = 1 + 3−∆ 4φ2 0z2(3−∆) .(3.35) Given the metric functions near the UV this can be used to find the operator dimension ∆. The effect of RG flow is also encoded in the UV behavior of the entanglement entropy. Consider the entanglement entropy of a strip in a theory with a gravity dual (3.31). For narrow strips the holographic entanglement entropy in the Einstein frame is S=1 4G(4) NZ√g2,(3.36) where ds2 2=L2 UV z2z02 f(z)+ 1(dx1)2+ (dx2)2(3.37) 1Our treatment partly follows [22]. – 15 – JHEP01(2019)232 is the induced metric on a bulk surface anchored to the strip and the prime denotes derivation with respect to x1. Explicitly, this gives S=LyL2 UV 2G(4) NZl/2 0 dx z2sz02 f(z)+ 1 ,(3.38) where Lyand lare the length and the width of the strip, respectively. To find the minimal surface we evaluate the Euler-Lagrange equations for z(x), z0=√f(z)√z4 ∗−z4 z2, which, when inserted in (3.36), yields S=LyL2 UV 2G(4) NZz∗ 0 z2 ∗dz z2p(z4 ∗−z4)f(z)(3.39) =LyL2 UV 2G(4) NZz∗ 0 z2 ∗ z2pz4 ∗−z4−z2 ∗z4−2∆ 2pz4 ∗−z4 3−∆ 4φ2 0+. . .!dz . (3.40) In addition to the usual divergence of areas of surfaces anchored to asymptotically AdS boundaries, there is a divergence coming from the scalar field-dependent term when 5/2≤ ∆<3. Subtracting these divergences, Sdiv = lim →0 LyL2 UV 2G(4) N1 +5−2∆ 2(5 −∆) 3−∆ 4φ2 0,(3.41) the regulated entanglement entropy Sreg =S−Sdiv reads Sreg =LyL2 UV 2G(4) N −√2π3/2 Γ1 42 1 z∗−√πΓ5 4−∆ 2 8Γ 7 4−∆ 2 3−∆ 4φ2 0z5−2∆ ∗+. . .!.(3.42) To work out the turning point as a function of strip width z∗(l), we evaluate l= 2 Zz∗ 0 z2dz pf(z)pz4 ∗−z4= 2 Zz∗ 0 z2 pz4 ∗−z4−z2+2(3−∆) pz4 ∗−z4+. . .!dz (3.43) =2√2π3/2 Γ1 42z∗−√πΓ9 4−∆ 2 4Γ 11 4−∆ 2 3−∆ 4φ2 0z7−2∆ ∗+. . . . (3.44) This relation can be reverted to give z∗=Γ1 42 2√2π3/2l+2−14+3∆π−23/2+3∆Γ1 416−4∆ Γ9 4−∆ 2 Γ11 4−∆ 2 3−∆ 4φ2 0l7−2∆ +. . . . (3.45) Finally, the entanglement entropy is Sreg =LyL2 UV 2G(4) N −4π3 Γ1 4 1 l−2−32/2+3∆π−7+3∆Γ1 410−4∆ Γ5 4−∆ 2 Γ11 4−∆ 2 3−∆ 4φ2 0l5−2∆ +. . . !. (3.46) – 16 – JHEP01(2019)232 In our massive ABJM setting we can identify f(z) = L2 0z2 ph(z)= 1 −κ2b 2h2−4g2 2b−1(rqz)2b+. . . (3.47) leading to ∆=3−b= 2 −γm,(3.48) for the dimension of the leading operator driving the RG flow. Here we used the expression for the mass anomalous dimension (2.36) in the last step. We can likewise use (3.46) to derive the RG-driving operator dimension from the UV-behavior of the entanglement entropy. It turns out that the leading correction to the entanglement entropy behaves as δSreg ABJM ∼l−1+2b(details in appendix B) which correctly implies (3.48), ∆ = 3 −b. This confirms the validity of our analytical expressions for the entanglement entropy. 4 The flow of mutual information The mutual information between two entangling regions Aand Bis defined as I(A, B) = S(A) + S(B)−S(A∪B),(4.1) and characterizes the information that is shared between these domains. The flow of mutual information with the size of the entangling regions characterizes how information is shared across space at different scales. We will characterize this flow for the mutual information between strips in four ways: (a) in terms of a “phase transition” defined by the separation distance between strips of fixed width at which the mutual information vanishes, (b) in terms of a c-function obtained from the mutual information that counts the degrees of freedom that are active at the scale of the strips, (c) in terms of a measure of “extensivity” that characterizes the additivity of the mutual information between a region Aand two other regions Band C, and (d) in terms of mutual information between regions Aand B conditioned on knowledge of a third region C. 4.1 Mutual information transitions In theories with a holographic dual there is typically a scale-dependent phase transition in the mutual information: Iis finite when two regions of fixed sizes are sufficiently close, but vanishes when the separation grows to a certain critical size. Holographically, this transition occurs when the minimal bulk surface that approaches A∪Bon the boundary fragments into the union of minimal surfaces for Aand B(see figure 3). In the few flavor limit we will find an analytic expression for the critical separation at which the phase transition occurs in the mutual information between strips. The critical separation will be a function of the scale ldefined by the strips themselves. To this end, let us consider a system of two parallel strips of width lseparated by a distance s. The transition between the two configurations in figure 3happens when their entropies are equal: 2Sreg(l) = Sreg(2l+s) + Sreg(s).(4.2) – 17 – JHEP01(2019)232 l ls A B bdry bulk (A) l ls A B bdry bulk (B) Figure 3. Bulk extremal surfaces (blue) that can contribute to the holographic mutual information between boundary strips Aand Bof length l. (A) When the separation sof the strips is large the minimal area surface bounded by A∪Bis the union of the minimal surfaces for Aand B separately. Thus the mutual information (4.1) vanishes because S(A)+S(B) = S(A∪B). (B) When the separation sof the strips is small, the minimal area surface bounded by A∪Bis the union the pictured surfaces. Thus the mutual information (4.1) is non-vanishing because S(A) + S(B)> S(A∪B). If the theory is conformal (either because there are no quarks, or when they are massless) the text shows that this mutual information phase transition occurs when s/l|ˆ=0 = (√5−1)/2 = 1/ϕ [23,24] where ϕis the golden ratio (√5 + 1)/2. Substituting the ˆexpansion Sreg =Sreg,0+ ˆSreg,ˆin (4.2) we find 2(Sreg,0(l) + ˆSreg,ˆ(l)) =Sreg,0(2l+s) + ˆSreg,ˆ(2l+l/ϕ) + Sreg,0(s) + ˆSreg,ˆ(l/ϕ),(4.3) where we have introduced the golden ratio ϕ= (√5 + 1)/2: 1 ϕ=√5−1 2.(4.4) Solving for s(l) gives s l=1 ϕ+ ˆ3 (ϕ+ 2) Γ 1 44 40√2π3 λ2l N2Ly2Sreg,ˆ(l)−Sreg,ˆ(2l+l/ϕ)−Sreg,ˆ(l/ϕ)+O(2),(4.5) where the entropies of individual configurations are given either by (3.21) or by (3.30) depending on whether their lengths are smaller or larger than lcrit given in (3.22). Close to the CFT fixed points in the UV and IR we find that UV : s l≈1 ϕ+ ˆΓ1 48 384(2ϕ−1)π7rql √λ2 ,2l+l/ϕ < lcrit (4.6) IR : s l≈1 ϕ+ ˆ32√2(5/2−ϕ)π6 15Γ 1 44rql √λ−3 ,rql √λ→ ∞ .(4.7) In the UV limit the bulk minimal surfaces remain in the region x > x∗= 1, while in the IR limit all the surfaces penetrate to x<x∗. To leading order in ˆ,s(l) coincides with the result for a conformal theory dual to AdS4:s(l)|ˆ=0/l = 1/ϕ = (√5−1)/2≈0.618 [23,24].2It also tends to this value in the 2Interestingly, for the symmetric case of mequally separated strips of equal width l, as also studied in [23,24], the critical separation l/s =(2 m)+q(2 m)2+4 2is the so-called “metallic mean”. – 18 – JHEP01(2019)232 0 5 10 15 20 25 30 l rq λ 0.618 0.620 0.622 0.624 0.626 0.628 critical s l Figure 4. Critical values of s/l for ˆ= 0.1, at which the mutual information between strips of width lseparated by a spacing sdrops to zero. The dashed horizontal orange line denotes the critical value in a CFT. The numerical results (red dots) closely match the analytical results (blue curve; eq. (4.5)). Higher order in ˆcorrections to (4.5) should produce an even closer match. The vertical red dashed line is lqrq/√λwhere lqis the width of the strip whose bulk minimal surface just touches the cavity. conformal limits well above and below the mass scale of the quarks (l→0 and l→ ∞). At intermediate scales l, the mutual information transition occurs at a separation sbetween strips that is greater than in a conformal field theory (figure 4), and is maximized at a scale where the bulk minimal surface almost reaches the “cavity” associated to the quark mass scale. Interestingly, and surprisingly, this suggests that information is more non-locally shared away from the conformal fixed points. The non-monotonic behavior in figure 4also implies that in a range of fixed values of s/l (between the peak and the horizontal dashed line), the mutual information between strips will vanish both for large land for small l. In both these cases the massive ABJM theory is in a phase where the entropy of the union of strips is given holographically by the disconnected surfaces in figure 3A. For these s/l there is also an intermediate range of l(determined by the intersection of a horizontal line of fixed s/l with the critical curve in figure 4) in which the mutual information is non-zero, or equivalently, in which the entropy of the union of strips is given by the connected surfaces in figure 3B. This has a remarkable implication about the organization of quantum information in this theory. Apparently, two strips at a fixed s/l that share mutual information can be mutually disentangled by increasing the strip width at fixed s/l. This sort of purification by expansion does not happen in a CFT where strips with a given s/l will always either share some mutual information or none at all, independently of the value of l. As the number of quarks or, equivalently, ˆ= (3/4)(Nf/N)λ, increases, the peak in figure 4becomes higher (figure 5) and narrows but also acquires a longer tail towards the infrared (large l). The tail means that the deviation from the conformal theory persists – 19 – JHEP01(2019)232 ϵ =9 ϵ =1 ϵ =0.1 5 10 15 20 25 30 l rq λ 0.64 0.66 0.68 0.70 critical s l 0.5 1 5 10 50 ϵ  0.64 0.66 0.68 0.70 peak height (A) (B) Figure 5. (A) Numerical analysis of the critical value of s/l (s= strip separation, l= strip width) at which the mutual information transition occurs for varying numbers of flavors: (3/4)(Nf/N)λ= ˆ= 0.1,1,9. As Nfincreases, the peak deviation from the conformal value of s/l grows, and occurs further in the UV (smaller strip widths l). The vertical dashed lines are lqrq/√λwhere lqis the width of the strip whose bulk minimal surface just touches the cavity for each ˆ. (B) Peak deviation of the critical s/l as a function of the number of flavors. The numerical evidence confirms that in the absence of flavors (ˆ→0) we return to the conformal result s/l = 1/ϕ where ϕis the golden ratio, and suggests that in the limit of many flavors (ˆ→ ∞) the deviation from the conformal value saturates. deeper into the infrared. When the ˆ > 1, the transition occurs at an s/l far from the conformal value even at scales well into the infrared. Of course in the deep infrared limit the theory approaches a conformal fixed point (an AdS4geometry in the bulk description) and so the transition s/l eventually returns to the conformal value. Meanwhile the scale lat which there is the greatest deviation from the conformal result (i.e., the peak of the bump in figure 4) moves further into the UV as Nfincreases (figure 6). At the same time this scale approaches more closely the strip width lqfor which the corresponding minimal surface just touches the cavity. It would be interesting to study the ˆ→ ∞ limit to see whether the peak height approaches a finite value as suggested by the numerical analysis in figure 5and whether the peak location coincides precisely with lq. 4.2 c-functions: flow of the number of degrees of freedom In quantum field theory one seeks to define a c-function that counts the number of degrees of freedom available as a function of the scale of measurement l. Liu and Mezei [25,26] proposed that a c-function can be defined in terms of the derivative of the entanglement entropy with respect to scale.3In our context, where we are computing the entanglement 3A spatial integral of a similar quantity, called the differential entropy, was shown to reproduce the areas of closed surfaces in AdS space [27,28], see also [29]. – 20 – JHEP01(2019)232 lpeak lq 0.1 0.5 1 5 10 50 ϵ  1 2 3 4 5 l rq λ 0.1 0.5 1 5 10 50 ϵ  0.050 0.075 0.100 0.125 0.150 lq-lpeak lq (A) (B) Figure 6. Numerical analysis of the scale lpeak at which the critical value of s/l maximally deviates from the conformal value. (A) Comparison of lpeakrq/√λwith lqrq/√λas a function of the number of flavors (ˆ= (3/4)(Nf/N)λ). Here lqis the strip width at which the corresponding bulk minimal surface just touches the cavity; i.e., this is the scale set by the quark masses. The absolute difference between these scales shrinks in the many flavor limit. (B) The relative difference between these scales also shrinks in the many flavor limit, but may be approaching a finite value. of strips, we can write this as F(l) = l2 Ly ∂Sreg ∂l .(4.8) where lis the strip width and Lyis the strip length which can be taken to infinity.4F(l) is UV finite and interpolates in a renormalizable QFT between fixed, scale-independent values at the UV and IR fixed points. We can compute F(l) analytically using our explicit expressions for entanglement entropy as a function of strip width. Plotting the results (figure 7) indeed shows that F(l) in our massive ABJM theory approaches a constant value in the UV (small lrq/√λ) and decreases monotonically to another constant in the IR (large lrq/√λ), consistently with the interpretation that F(l) counts the degrees of freedom at a scale l. Similar statement also holds for disk regions in the same geometry [9]. In our model Fis extremized at the fixed points of the flow, i.e., the first derivative ∂F/∂l will vanish in the large and small llimits. Since Fis both continuous and monotonic, the first non-vanishing derivatives should be negative in the UV (lrq/√λ→0) and positive 4In more detail, following [26], the entanglement entropy of the strip will have the general behavior S∼Ly(divergent + finite). The divergence comes from short distance modes straddling the edges of the strip and so will be proportional to Ly, but will not depend on the strip width l. Thus we can extract the finite part by computing l∂S/∂l. However, this quantity will diverge with the length of the strip so we should divide by Lyto get a finite entanglement per unit length as (l/Ly)∂S/∂l. We can make this quantity dimensionless by multiplying by l, to define F= (l2/Ly)∂S/∂l. At the conformal points there is no scale, so Fshould be a constant, independently of the strip width l, but away from conformality Fwill be a dimensionless combination of the scales in the theory. – 21 – JHEP01(2019)232 0 5 10 15 20 25 30 l rq λ 0.340 0.345 0.350 0.355 0.360 0.365 0.370 λ ℱ N2 Figure 7.Fmonotonically interpolates between constant values at the UV (lrq/√λ→0) and IR (lrq/√λ→ ∞) conformal fixed points. The vertical dashed line is at lqrq/√λwhere lqis the width of the strip whose bulk minimal surface just touches the cavity. The points are from the full numerical solution and the continuous curve is from our analytic results in the few flavor limit (small ˆ). Here ˆ= 0.1. in the IR (lrq/√λ→ ∞). Thus, we expect the second derivative F00(l) to change sign at some intermediate scale. In the few flavor limit (small ˆ= (3/4)(Nf/N)λ) we can show explicitly that ∂2F(l) ∂l2=     −ˆΓ(1 4)4 72√2π4 N2r2 q λ3/2, l →l− crit ˆΓ(1 4)2 12π7/2 N2r2 q λ3/2l lcrit −1−1/2, l →l+ crit . (4.9) Notice that since we are working at first order in ˆ,lcrit in (4.9) can be interpreted as lq, i.e., as the width of the strip whose bulk minimal surface just touches the cavity in the gravity solution. In other words, F00(l) changes sign at a scale corresponding to the mass of the fundamental quarks. Note also that the second derivative is discontinuous across the cavity. This is expected, as the Einstein equation of our background solution has a discontinuity in the source at the location of the cavity where the energy-momentum tensor is abruptly turned on. One can refine the smearing procedure [30] by using embedded D6-branes with a distribution of tip positions that are not delta-function peaked at x∗. This would lead to a continuous topological transition in the geometry at the scale rqwith smooth second derivatives of the background metric and thus of F(l). Casini et al. [31] proposed an alternative method of computing a c-function for threedimensional QFTs from the mutual information I(A+, A−) between the interior of a smaller circle (A−) and the exterior of a larger circle (A+) with an annulus of width δbetween them. The advantage of working with mutual information is that this quantity is UV-finite, and is thus well-defined without any regularization unlike entanglement entropy. Since we are working with strips, not circular regions, we consider the mutual information between – 22 – JHEP01(2019)232 δ δl AB BC C bdry bulk Figure 8. Strip configuration we consider in the definition of the c-function. We have a strip of width lbordered by two thin strips of width δ. We consider the mutual information between the center strip (red) and the exterior (orange). a strip of width l(region A) and the remainder of the space (region B) outside a boundary region of width δon either side of the strip (region C) — see figure 8. We will take δ→0 in the end. We then define our candidate c-function to be c(l) = lim δ→0 l2 Ly ∂I(l, δ) ∂l ,(4.10) where the mutual information is given by I(A, B) = S(A) + S(B)−S(A∪B), where S(X) is the entanglement entropy of region X. In fact, we can show that c(l)=2F(l).(4.11) To this end, consider a QFT partitioned into three regions — A, characterized by length scale l; C, bordering Acharacterized by a size δl; and B, which is the rest of the space. We now write our first candidate c-function (4.8) in terms of the differential operator Ll= (l2/Ly)∂/∂l as F(l) = LlS(A).(4.12) Our second candidate c-function is c(l) = lim δ→0LlI(A, B).(4.13) The equality c= 2Fthen follows for pure states: c(l) = lim δ→0LlS(A) + lim δ→0LlS(B)−lim δ→0LlS(A∪B) (4.14) =LlS(A) + lim δ→0LlS(B) | {z } =LlS(A) −lim δ→0LlS(C) | {z } =0 = 2F(l).(4.15) Here we used the fact that Cis the complement of A∪Band so S(C) = S(A∪B) for pure states. As δvanishes, so does the “bulk” contribution to S(C), so that S(C) becomes equal to its UV-divergent piece. The latter is proportional to the length of the boundary between Aand Band does not depend on l. Thus LlS(C) vanishes in the δ→0 limit. – 23 – JHEP01(2019)232 Likewise, LlS(B)→LlS(A) as δ→0, since Bis the complement of Ain this limit, and S(A) = S(¯ A) for pure states. Note that, as discussed earlier, the differential operator Ll removes the divergences in the strip entanglement entropy so that these arguments are well-defined. We stated this proof for strips, but a similar argument holds for compact A. 4.3 Flow of extensivity Consider three entangling regions A,B, and Cand their tripartite information I3(A, B, C) = I(A, B) + I(A, C)−I(A, B ∪C).(4.16) Despite appearances, this quantity is symmetric between A,B, and Cas can be verified by expressing the mutual informations in terms of entanglement entropies. I3is a measure of extensivity of mutual information in the sense that I3= 0 implies that the information that Ashares with B∪Cis the sum of the information shared with Band Cseparately. In a general quantum field theory I3can take either sign depending on the choice of state and entangling regions [32], but it was shown in [33] that in holographic theories I3≤0. This means that the information that region Ashares with Band Cis in general super-extensive in a holographic theory — there is information in intrinsically 3-party entanglement that cannot be uncovered just from the 2-party entanglement. To investigate how the extensivity of mutual information varies with scale we define the ratio e=I(A, B ∪C) I(A, B) + I(A, C),(4.17) so that e= 1 in an extensive theory, e > 1 in a super-extensive theory, and e < 1 in a sub-extensive theory. The extensivity eis only well defined when the denominator is nonvanishing, i.e., if the region Ashares at least some mutual information with at least one of Bor Cseparately.5As we have seen, in some entangling region configurations the mutual information can be identically zero. If that happens for I(A, B) and I(A, C) while I(A, B ∪ C) remains positive then ediverges, representing maximal super-extensivity, and this can certainly happen in quantum theories.6Below we will first determine configurations where e is finite for a CFT, and then ask how the massive ABJM theory behaves in these situations. Consider three strips A,B, and Cof equal width lthat are separated by distance s with Ain the middle. We can study how the extensivity of entanglement changes with scale by varying lwhile keeping s/l fixed (figure 9). In the absence of massive quarks the theory is conformal, and so the extensivity eis constant under this scaling variation. Applying the standard formulas for the entanglement entropy of intervals in a (2 + 1)-dimensional CFT (see, e.g., [21] or the first term on the right hand side of eq. (B.13)) one finds a simple result: e(s/l) = 1 2·3(s/l)3+ 8(s/l)2+ (s/l)−6 2(s/l)3+ 5(s/l)2+ (s/l)−3,for a CFT .(4.18) 5Note that the mutual information is a non-negative quantity so that the two terms in the denominator cannot cancel each other. 6See an interesting example in the context of multi-boundary wormholes in [34]. – 24 – JHEP01(2019)232 of cyclic inequalities n X i=1 S(Ai|Ai+1 . . . Ai+k)≥S(A1. . . An),(5.1) where nis the number of regions and n= 2k+ 1. We have numerically checked this family of inequalities in massive ABJM theory for parallel strip configurations up to 9 separate regions. The results in this paper depended on having a fully ten-dimensional gravitational dual to a strongly coupled QFT. In fact, the extremal surfaces computing entanglement entropy explored the full 10 dimensions. Many recent applications of holography to the study of quantum information have treated the gravitational dual as effectively having one extra dimension compared to the field theory. But a fully realized duality in string theory requires 10 dimensions, and, as we have seen here, the details of the full geometry manifest themselves in the structure of the RG flow. Indeed, as discussed in [15] the “internal” and AdS dimensions can interact significantly, and even sometimes completely exchange roles [16]. It would be useful to develop more such examples. Finally, it would be interesting to put the Chern-Simons field theory studied in this paper on a 3-sphere and test the recent proposal for the c-function [38]. Acknowledgments We would like to thank Matt DeCross, Arjun Kar, Esko Keski-Vakkuri, Onkar Parrikar, and G. S´arosi for helpful discussions. A. V. R. is funded by the Spanish grants FPA2014- 52218-P and FPA2017-84436-P by Xunta de Galicia (GRC2013-024), by FEDER and by the Maria de Maeztu Unit of Excellence MDM-2016-0692. V. B. was supported in part by the Simons Foundation (# 385592, V. B.) through the It From Qubit Simons Collaboration, and the US Department of Energy grant FG02-05ER-41367. V. B. also acknowledges the hospitality of the Aspen Center for Physics which is supported by National Science Foundation grant PHY-1607611. This work was initiated during the “Holographic methods for strongly coupled systems” workshop of the Galileo Galilei Institute in Florence. A Details of the background Here we will give the detailed expressions for the different functions appearing in our background (for a derivation see [9]). Inside the cavity, i.e., for x≤1, these functions are analytic and depend on the constant γcharacterizing the running of the master function W(x) in the sourceless region. The functions f,g, and the dilaton φare given by: ef rq =1 + √1+4γ √2 x √1+4γx(1 + √1+4γx)1/2 eg rq =1 + √1+4γ 2 x √1+4γx keφ=1 + p1+4γx 1+4γx(r4 qh)1/4,(A.1) – 31 – JHEP01(2019)232 where rqis the rcoordinate of the tip of the brane (see (2.12)). For x≥1 the master equation (2.9) for W(x) must be integrated numerically (although an analytic solution in powers of ˆcan be obtained, see below). In terms of Wand η, the different functions are: ef rq =(1 + √1+4γ)2 2√1+4γ1/3r3x W0+ 4ηW1/6exp 2 3Zx 1 η(ξ)dξ W(ξ) eg rq =(1 + √1+4γ)2 2√1+4γ1/3x W1/3exp 2 3Zx 1 η(ξ)dξ W(ξ) keφ=(1 + √1+4γ)2 2√1+4γ1/312x(r4 qh)1/4 W1/3(W0+ 4η)exp 2 3Zx 1 η(ξ)dξ W(ξ).(A.2) The constant γappearing in (A.1) and (A.2) is obtained by the shooting technique described in the main text. The dilaton in these two equations has been written in terms of the warp factor h. For x≤1 the function h(x) can be found analytically: k Nr4 qh=π2 2(p1+4γ−1)41 + 1 4γx"α+ 24 log √4γx √1+4γx + 1 +1 2+ 6γx +1 + (1 −6γx)√1+4γx 4γx √1+4γx + 1 γ2x2#,(A.3) whereas for x≥1 it can be written in terms of W(x) as: k Nr4 qh= 4π2r4 qe−gW0+ 4η"Z∞ x ξe−3g(ξ)dξ W(ξ)2#.(A.4) The integration constant αthat appears in the warping his fixed by continuity at x= 1: limx→1−h(x) = limx→1+h(x). However, the expression for αis lengthy, so we do not write it here explicitly. A.1 Expansion in flavor The master equation, and the corresponding functions of the ansatz, can be solved analytically in a power series expansion in the flavor deformation parameter ˆ[18]. If we write: W(x) = X n=0 Wn(x)ˆn= 2x+W1(x)ˆ+W2(x)ˆ2+W3(x)ˆ3+O(ˆ4),(A.5) then we get for the first three functions in (A.5) in the region x≤1: W1(x) = 12 5x2 W2(x) = 8 875 (171 −70 x)x2 W3(x) = 16 197071875 235688 + 135135 35 x−76) x.(A.6) – 32 – JHEP01(2019)232 Moreover, for x≥1 we have: W1(x) = 7x2−2 2x−1 10x3 W2(x) = 13125x8+ 3500x6−3962x4−3360x4log x+ 300x2−35 14000x7 W3(x) = −23 208000x11 +89 184800x9−302 39375x7+8293 98000x5−2120717 8400000x3+37 96x −log(x)10703x4+ 20160x4log(x)−3600x2+ 840 70000x7−15x 32 .(A.7) The constant γobtained from the shooting method can also be expanded in powers of ˆ, with the result: γ=2 ˆ 5+228 ˆ2 875 +18855104 ˆ3 591215625 +O(ˆ4).(A.8) The remaining functions of the ansatz can be obtained by plugging these expansions into (A.2) and (A.4). The expressions found in this way are lengthy and will not be reproduced here. B UV expansion of the entanglement entropy In this section we work out the UV expansion of the strip entanglement entropy. The holographic entanglement entropy on a strip can be computed from equation (3.5). The UV expansion is given by S(l) = S∞(l) + δS(l),(B.1) where S∞(l) is the entanglement entropy computed at the UV fixed point and δS(l) is the first perturbation around the UV. Explicitly, these are given by S∞(l) = −4π2FUV(S3) Γ1 44 1 l(B.2) δS =V6Ly 2G(10) NZ∞ x∗∂L ∂H UV δH +∂L ∂H∗UV δH∗+∂L ∂GUV δGdx . (B.3) The UV expansions of H(x) and G(x) are H(x) = H0x4/b 1 + H2 x2+O(x−4)(B.4) G(x) = G0x−2−2/b 1 + G2 x2+O(x−4),(B.5) where the coefficients are H0=L8 0κ4r4 qq4 0e−4φ0 b12 , G0=L4 0 b2r2 qκ2(B.6) H2= 2(h2+ 4f2+ 2g2−2φ2), G2=h2+ 2g2.(B.7) – 33 – JHEP01(2019)232 Now the UV perturbation δS is given by δS =V6Ly 2G(N) NpG0H0Z∞ x∗ (G2+H2)x4/b + (H2x2−(G2+ 2H2)x2 ∗)x4/b ∗ 2x∗(x4/b −x4/b ∗)3/2x−3+3/bdx =V6Ly 2G(N) N √G0H0x−2+1/b ∗ 2Z∞ 1 (G2+H2)z4/b +H2(z2−2) −G2 (z4/b −1)3/2x−3+3/bdz =V6Ly 2G10 b√G0H0√π 16 x−2+1/b ∗ (2G2+ (1 + 2b)H2)Γ −1 4+b 2 Γ1 4+b 2−H2Γ−1 42 4√2π!.(B.8) We must re-express the above result in terms of the strip width l. The strip width near the UV is l= 2pH∗Z∞ x∗pG(x) pH(x)−H∗ dx =2√G0 x1/b ∗ b√2π3/2 Γ1 44+δl , (B.9) where the first term on the last expression is the UV width and δl is the first correction. The correction is δl =pG0x−2+2/b ∗Z∞ x∗ x4/b(H2x2+ (G2−H2)x2 ∗)−G2x2+4/b ∗ (x4/b −x4/b ∗)3/2x−3−1/bdx =pG0x−2−1/b ∗Z∞ 1 z4/b(G2+H2(z2−1)) −G2 (z4/b −1)3/2z−3−1/bdz =b√G0√π 8x−2−1/b ∗ (2G2+ (1 + 2b)H2)Γ 3 4+b 2 Γ5 4+b 2−H2Γ−1 42 4√2π!.(B.10) At this point we know l(x∗). We still need to revert this relation for use in equation (B.8). x∗as a function of lis x1/b ∗=2√2b√G0π3/2 Γ1 42 1 l+δx1/b ∗(B.11) δx1/b ∗=b1−2bG1/2−b 0Γ1 44b 23+3bπ−1/2+3bl−1+2b (2G2+ (1 + 2b)H2)Γ 3 4+b 2 Γ5 2+b 2−H2Γ−1 42 4√2π!. (B.12) Using these results we conclude that the regulated entanglement entropy on a strip including the first UV correction is S(l) Ly =4π2FUV(S3) Γ1 44−1 l+β(b)l−1+2b+. . .,(B.13) where β(b) = (b−1)(−5+3b)Γ 1 42+4bΓ−1 4+b 2 27/2+4bb(5 −4b)bπ1+5b((2 −b)b/κ)2bΓ5 4+b 2.(B.14) One application of this UV expansion is to the determination of the region in the {ˆ, s/l}plane where the theory shows greater super-extensivity than in a CFT. 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