TT¯, the Entanglement Wedge Cross Section, and the Breakdown of The

PHYSICAL REVIEW D 102, 045009 (2020)

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

† Meseret Asrat 1,* and Jonah Kudler-Flam 2, 1Enrico Fermi Institute, University of Chicago, Illinois 60637, USA 2Kadanoff Center for Theoretical Physics, University of Chicago, Illinois 60637, USA

(Received 30 May 2020; accepted 28 July 2020; published 7 August 2020)

We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operator TT¯ and its closely related but nonetheless distinct single-trace counterpart. For holographic conformal field theories, these deformations can be interpreted as modifications of bulk physics in the ultraviolet region of anti-de Sitter space. Consequently, we can use the Ryu-Takayanagi formula and its generalizations to mixed state entanglement measures to test highly nontrivial consistency conditions. In general, the agreement between bulk and boundary quantities requires the equivalence of partition functions on manifolds of arbitrary genus. For the single-trace deformation, which is dual to an asymptotically linear dilaton geometry, we find that the mutual information and reflected entropy diverge for disjoint intervals when the separation distance approaches a minimum, finite value that depends solely on the deformation parameter. This implies that the mutual information fails to serve as a geometric regulator which is related to the breakdown of the split property at the inverse Hagedorn temperature. In contrast, for the double-trace deformation, which is dual to anti-de Sitter space with a finite radial cutoff, we find all divergences to disappear including the standard quantum field theory ultraviolet divergence that is generically seen as disjoint intervals become adjacent. We furthermore compute reflected entropy in conformal perturbation theory. While we find formally similar behavior between bulk and boundary computations, we find quantitatively distinct results. We comment on the interpretation of these disagreements and the physics that must be altered to restore consistency. We also briefly discuss the TJ¯ and JT¯ deformations.

DOI: 10.1103/PhysRevD.102.045009

I. INTRODUCTION flow to the same fixed point. Such deformations are usually difficult to understand with the infrared degrees of freedom, The renormalization group is a fundamental concept in computational uncontrollable, and intractable. many-body physics and quantum field theory. It has been A surprising and deeply consequential breakthrough central to our understanding of the validity of particle phy- came when Zamolodchikov and Smirnov showed that sics, the emergence of macroscopic phenomena, and the a particular class of irrelevant deformations of two- space of quantum field theories. Starting from a fixed point dimensional quantum field theories, in particular theories of the renormalization group (a conformal field theory), one obtained by deforming via the determinant of the stress- may perturb the action by a relevant operator, an interaction energy tensor, are generally under control and “solvable” in that becomes increasingly important as one flows to the IR. the sense that the energy spectrum, partition functions, and Relevant deformations compose the minority of terms that (sometimes) the classical action may be computed exactly are available to add to the action, though their flows through [1–4]. Moreover, the UV descriptions of these theories are theory space are generally well understood. On the contrary, not local quantum field theories. The scale where locality the vast majority of possible deformations are irrelevant, breaks down is determined by the dimensionful defor- describing the (generally infinite) UV directions that may mation parameter, μ. The understanding of this “TT¯” deformation has seen tremendous progress including its *[email protected] reinterpretations as coupling the seed quantum field theory † [email protected] (QFT) to Jackiw-Teitelboim gravity [5] or a flat, random metric [6]. Furthermore, analogous irrelevant deformations Published by the American Physical Society under the terms of have been put forward for theories with Uð1Þ currents that the Creative Commons Attribution 4.0 International license. ¯ ¯ Further distribution of this work must maintain attribution to break Lorentz symmetry, the TJ and JT deformations [7]. the author(s) and the published article’s title, journal citation, Due to the great success of the AdS3=CFT2 correspon- and DOI. Funded by SCOAP3. dence [8,9], a natural question to ask is how these

2470-0010=2020=102(4)=045009(23) 045009-1 Published by the American Physical Society MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020) deformations of two-dimensional conformal field theories conjectures and to understand the novel structure of their are viewed or interpreted from the bulk gravity (string) dual (nonlocal) field theories. An illuminating observable is theory side. Thus far, gauge-gravity duality has mainly only the entanglement entropy of subregions. Understanding been successful for theories that are asymptotically anti-de entanglement structure has been central in characterizing Sitter, corresponding to field theories that are controlled many-body systems such as gapped, critical, topologically by a UV fixed point. Thus, one may hope that these ordered, and holographic systems [22–32]. Presumably, the irrelevant deformations, which alter the UV physics, may entanglement structure of these TT¯ deformed theories will guide us to understand holography for quantum gravity elucidate their properties. In particular, entanglement theories with different asymptotics, in particular asymp- entropy has played a key role in our understanding of totically flat spacetimes. There have been two1 explicit the renormalization group, providing c-functions in two proposals for how the asymptotics may be altered under the and three dimensions that have clear information theoretic irrelevant deformations. meaning regarding the number of degrees of freedom at The first proposal is what we refer to as “cutoff AdS” each scale [33–35]. Generalized entropic c-functions in which has a surprisingly simple description [11]. Rather arbitrary dimensions for holographic theories were shown than the standard Dirichlet asymptotic boundary conditions to hold in Ref. [36]. of the Gubser-Klebanov-Polyakov-Witten (GKPW) diction- Several studies have been conducted for entanglement ary [12,13], one places Dirichlet boundary conditions at a entropy in holographic TT¯ deformed theories [37–49]. finite radial cutoff. An important ingredient is that the While one can nonperturbatively compute the entropy deformation parameter is sign definite in the direction where using the Ryu-Takayanagi formula [29,30], in general, there always exist complex energies in the spectrum, a one must use techniques from conformal perturbation puzzling feature whose physical implications must be theory to compute the entropy from the field theory side. addressed. This conjecture has passed several tests such as One exception to this is for the entropy of very specific matching the energy spectrum, thermodynamics, and signal configurations such that the replica trick may be computed propagation speeds [11]. However, in order to match bulk nonperturbatively. For example, for an entangling surface and boundary two-point functions, one must add an addi- consisting of two antipodal points on a sphere, it is shown tional double-trace deformation to the boundary theory [14]. that the cutoff AdS and boundary computations precisely The other holographic proposal regards a closely related agree [37]. Impressively, this technique has been general- but distinct deformation of the conformal field theory, a ¯ ized to the less symmetric case of a finite interval [38]. “single-trace” TT [15–17]. The following is meant by There are many reasons why the entanglement structure of single trace. We consider a symmetric orbifold theory TT¯, TJ¯, and JT¯ deformed theories deserves further attention, MN 2 =SN (where N is an integer). Rather than deforming though the following three are our main motivations: the theory by the stress tensor of the full theory, we deform (1) Thus far, only von Neumann entropy has been M each block, , and take the symmetric product of the studied which is only a reasonable measure of resulting theories. This proposal has the deformation entanglement for pure states. This is severely limit- parameter be sign definite opposite to “cutoff AdS.” The ing as there is deep structure in mixed state entan- energies of states (on a cylinder) are real. The bulk glement and multipartite entanglement, particularly description of this deformation is a truly marginal defor- in holographic systems. For this reason, we consider mation of the world-sheet string theory. This affects the mixed state correlation measures such as the mutual bulk theory by changing the asymptotics to linear dilaton. information and measures dual to the entanglement Such asymptotics are quite close to asymptotically flat wedge cross section, a bulk geometric object distinct spacetimes though the high-energy density of states exhib- from the Ryu-Takayanagi surface [50–54]. We re- its novel Hagedorn growth (S ∝ E). Analogous to Ref. [7], view the relevant holographic conjectures in the there are generalizations to the single-trace versions of the following subsection. ¯ ¯ TJ and JT deformations whose proposed holographic (2) The central input into the holographic dictionary is duals have warped AdS as asymptotics [20]. Also, see the equivalence of bulk and boundary partition generalizations in Ref. [21]. functions Both of these proposals are fascinating and potentially quite important for our understanding of holography in Z ½B¼Z ½∂B; ð1:1Þ generic spacetimes. It is important to both test these gravity QFT where B is an arbitrary bulk manifold and ∂B is its 1 There is actually a third, recent proposal [10] motivated by (asymptotic) boundary. This is the main ingredient ’ Cardy s random geometry interpretation. We will return to this in in the derivation of the Ryu-Takayanagi formula the discussion section. 2CFTs in the moduli space of this theory describe the long [55]. For the agreements in Refs. [37,38], (1.1) B string sector of string theory on AdS3 [18,19]. M is the only must hold for a solid sphere. Such an conformal field theory of a single long string. equivalence of partition functions is quite unique.

045009-2 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020) Z 1 ffiffiffi On the contrary, the replica tricks used for the mixed d p SA ¼ d x g: ð1:2Þ state correlation measures that we study implicitly ðdÞ γ 4G A require the equivalence of partition functions for ∂B N arbitrary genus Riemann surfaces. This is a signifi- Here, γA is the extremal surface (with respect to the cantly stronger check of the holographic dualities. integrand) homologous to boundary region A. Important Surprisingly, we find that calculations on the two for our discussion is how this formula must be modified sides of the duality do not agree. when the dilaton is not trivial. In Ref. [30], it was posited (3) While entanglement in local quantum field theory that in the case in which the dilaton is not constant, has been studied extensively, entanglement in non- ¯ Z qffiffiffiffiffiffiffi local theories is much less understood. The TT 1 d −2Φ ðsÞ deformation induces a flow that leads to nonlocality SA ¼ d xe g ; ð1:3Þ 4 ðdÞ γ at the UV scale. This provides us a rare tractable GN A testing ground for studying information theoretic aspects of renormalization group flows and certain where Φ is the dilaton and gðsÞ is the metric in string frame. concepts in algebraic quantum field theory, such as We provide a simple derivation of this modification in nuclearity and the split property, which we review Appendix A. later in the Introduction. Various generalizations of the Ryu-Takayanagi formula The rest of the paper is organized as follows: In Sec. II, have played important roles in recent years, such as a we introduce the asymptotically linear dilaton background covariant description [32], quantum corrections [59,60], proposed to be dual to the single-trace deformation. and Rényi entropies [61], all of which have been derived We generalize the results for von Neumann entropy in under mild assumptions [55,59,61–63]. Refs. [42,46] to finite temperature states and compute the A large generalization was proposed in the context mutual information for disjoint intervals. Then, we compute of entanglement of purification (EoP), a mixed state the entanglement wedge cross section for disjoint intervals. correlation measure that reduces to the von Neumann We find that both quantities are UV divergent even when the entropy for pure states [64]. Motivated by information- intervals are only a finite distance away from each other theoretic inequalities and tensor network models of holog- determined by the nonlocality scale of the deformed theory. raphy, this was conjectured to be dual to the area of the Such divergences at finite distances signal breakdowns of the entanglement wedge cross section [50,51]. The entangle- geometric regulators of Refs. [54,56] that are used e.g., to ment wedge of boundary region A, ΞA, is the bulk c isolate -functions, a consequence of the split property of codimension-one region whose boundary is A ∪ γA. The quantum field theory [57,58] ceasing to hold. Furthermore, entanglement wedge cross section of A ∪ B, EWðA∶BÞ,is both the mutual information and reflected entropy are the extremal surface in ΞA∪B separating regions A and B monotonically increasing with the deformation parameter. (see Fig. 1 for a depiction of EW for disjoint intervals). In Sec. III, we do the same thing but for AdS with a hard While well motivated and certainly plausible, the EoP ¼ finite radial cutoff at both zero and finite temperature. In EW conjecture is unlikely to be proven using known sharp contrast to the previous section’sresults,wefindthe methods due to the large optimization procedure in the quantities to be UV finite even when the intervals are definition of EoP.3 The entanglement wedge cross section brought arbitrarily close together. Here, they are monoton- was later conjectured to be dual to logarithmic negativity ically decreasing with (the absolute value of) the deformation [52], a well-known mixed state entanglement measure that parameter. In Sec. IV, we perturbatively (in the deformation is only sensitive to purely quantum correlations [67–69]. parameters) compute the reflected entropy for disjoint An important aspect of this proposal is that EW must intervals. We find the first order corrections from the JT¯ ¯ backreact nontrivially on the geometry akin to the Rényi and TJ deformations to vanish due to twist fields being entropies [61]. This conjecture was later derived for 1 ¯ uncharged under the Uð Þ symmetry. For double-trace TT, AdS3=CFT2 in Ref. [70]. A similar quantity, the “odd we find formally similar, but numerically distinct, results to entropy,” was conjectured to be dual to EW without back- the holographic calculations, showing tension in the holo- reaction in Ref. [53]. Finally, in Ref. [54], it was shown graphic proposal. In Sec. V, we discuss physical implications that EW was equal to half of the “reflected entropy,” which and open questions. Finally, in the Appendices, we collect is the von Neumann entropy of a particular (not minimal) various derivations and formulas. purification ρ → jρ1=2i. This was derived under mild assumptions in generic dimensions. A. Holographic entanglement and mixed-state Due to the fact that the reflected entropy conjecture correlation measures does not require difficult backreaction in the bulk (like The Ryu-Takayanagi formula has become a standard in high-energy physics and its relation to quantum informa- 3Related optimized correlation measures were argued to be tion theory [29,30]: dual to EW in Refs. [65,66].

045009-3 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

FIG. 1. There are two phases of the entanglement wedge for disjoint intervals. In the connected regime (left), the Ryu-Takayanagi surfaces (red) stretch between the boundary two subregions (purple and orange). This phase manifestly has nontrivial mutual information and entanglement wedge cross section (green). Alternatively, when the intervals are sufficiently distant (d>d), the disconnected regime (right) dominates and the entanglement wedge becomes the union of the two individual entanglement wedges. This phase has manifest zero mutual information and entanglement wedge cross section. negativity) and it has a convincing derivation, in this paper, To gain further intuition on EW and its relation/distinct to we focus on computing this quantity. However, due to its mutual information, we plot the two quantities for disjoint recent introduction in the literature, little is known about its intervals (½x1;x2 and ½x3;x4) in the vacuum of a conformal 4 properties i.e., what is it really measuring? Thus, we find it field theory in Fig. 2. Both quantities only depend on the conceptually useful to consider it as a proxy for logarithmic conformally invariant cross ratio negativity, whose quantum information theoretic properties are well understood. Precisely, in holographic theories, the x21x43 x ¼ : ð1:4Þ negativity is equal to half of the Rényi reflected entropy at x31x42 Rényi index 1=2. Here, we collect some properties of EW and the reflected Much of this work is understanding how Fig. 2 changes entropy between two subsystems, denoted AðAiÞ and BðBiÞ once we turn on the irrelevant deformations. below [50,54]. (1) Reduction to von Neumann entropy: when comput- B. Review of the split property of QFT ing E for a bipartition of a global pure state, E W W and geometric regulators reduces to the standard Ryu-Takayanagi surface. Likewise, the reflected entropy reduces to twice the In this subsection, we provide a minimal review of the von Neumann entropy. nuclearity condition and the split property of local quantum (2) Upper bound: EWðA∶BÞ is always bounded from field theory. The interested reader may consult Ref. [76] for above by the entropies SðAÞ and SðBÞ and the further details. inequality is only saturated for pure states. Similarly Unlike finite dimensional quantum systems, in quantum reflected entropy always is bounded by twice the field theory, the Hilbert space does not admit a tensor entropies. (3) Lower bound: E ðA∶BÞ is always larger than half W 1.0 of the mutual information IðA∶BÞ. Likewise, this holds for reflected entropy in all quantum sys- 0.8 tems SRðA∶BÞ >IðA∶BÞ. 0.6 (4) Monotonicity: EW is monotonic under inclusions ≤ ∪ i.e., EWðA; BÞ EWðA; B CÞ. This makes it a rea- 0.4 sonable correlation measure. While this has been proven for the integer Rényi reflected entropies, it 0.2 has yet to be proven in the von Neumann limit, though it is suspected to hold. 0.2 0.4 0.6 0.8 1.0 (5) Strong superadditivity: EW obeys strong super- FIG. 2. The entanglement wedge cross section and half the additivity EWðA1 ∪ A2;B1 ∪ B2Þ ≥ EWðA1;B1Þþ mutual information per central charge are plotted as a function of EWðA2;B2Þ. Interestingly, there are counterexam- ples to this property for the reflected entropy of the conformally invariant cross ratio (1.4) in the vacuum state of a 2D CFT. The bound E >I=2 is manifest. Notably, E classically correlated finite dimensional systems. W W discontinuously jumps to zero at x ¼ 1=2. At this point, the mutual information is continuous but its first derivative is 4See discussion and analysis of this question in discontinuous. These analytic breakdowns are thought to be Refs. [54,71–75]. artifacts of the c → ∞ limit.

045009-4 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020) factorization. The algebras of local observables in a νβ;r ¼ inf Tr½Tβ;r; ð1:9Þ subsystem of quantum field theory are generically type Tβ;r III von Neumann algebras. The partial trace of a state is thus ill defined and consequently, the von Neumann which must be bounded as entropy is always infinite. Even with these setbacks, one crdβ−n would like to be able to consider states localized in νβ;r . d is the number of spatial dimensions. It Kr, in quantum field theory are generically of type III, we has been shown that if (1.10) is satisfied, then the split can consider two disjoint causal diamonds, K1 and K2 with property holds as long as K1 and K2 are disjoint, though ¯ the causal complement of K2, K2, subsuming K1. The split they may be arbitrarily close [78]. property then asserts that there exists a type I factor, N , The nuclearity index is closely related to the thermody- such that namic partition function. In certain theories, the partition function can diverge at a finite temperature, β , called the ¯ H AðK1Þ ⊂ N ⊂ AðK2Þ: ð1:5Þ Hagedorn temperature. This temperature governs the high- energy density of states. In these theories, the nuclearity Type I von Neumann algebras admit traces and are condition is not satisfied and the split property fails to isomorphic to the set of bounded operators in some hold when K1 and K2 are at a finite distance from one ¯ Hilbert space, so entropies may be well defined. To connect another determined by βH [defined in (2.10)]. TT deformed with our general intuition regarding tensor factorization of conformal field theories exhibit Hagedorn thermodynam- Hilbert spaces, we note that the split property equivalently ics, so it is natural to investigate how this split property says that there exists an isomorphism between AðK1Þ ∨ breaks down in these theories. This provides an opportunity AðK2Þ and the tensor factorization AðK1Þ ⊗ AðK2Þ.Itis where we have a concrete example to test the dynamical important to note that this mapping is not unique thence N breakdown of (1.5). is not unique. As previously mentioned, the von Neumann entropies There is, however, a “canonical” choice for the split, associated to subregions in quantum field theory are N ψ , that always exists and was first discussed in Ref. [58]. generically divergent. It is thus desirable to regulate the This is defined via the relation entropy to extract quantities physically meaningful regarding the theory or specific state. Of particular interest are N ψ ¼ AðK1Þ ∨ Jψ AðK1ÞJψ ; ð1:6Þ the quantities extracted from the von Neumann entropy that provide entropic c-functions [33,34]. In particular, the von where Jψ is the modular conjugation operator of Tomita- Neumann entropy for a ball-shaped region, A, has a generic Takesaki theory associated to the state ψ and algebra expansion including the following terms: A K1 ∨ A K2 . It is the von Neumann entropy of the ð Þ ð Þ ( h i state in N ψ that is associated to the reflected entropy. d−1 L ð−1Þ2 4c0 log A ;d∈ 2Z ϵUV This is why the reflected entropy is so useful for us in SA ⊃ ð1:11Þ d−1 understanding if/how the split property holds in a given ð−1Þ2 2πc0;d∈ 2Z þ 1; theory for a given split distance, s. For more comprehen- ϵ sive discussion on this point, we direct the reader to where LA is the radius of region A and UV is the ultraviolet Refs. [54,75,77]. cutoff. The constants c0 are universal and independent of The split property has been shown to follow from the the details of the UV regularization. The definition of c0 nuclearity condition which we describe now. Consider can be made unambiguous if the UV regulator is chosen the set appropriately as to allow the divergent terms in the entropy to be expressible as “geometric,” i.e., integrals over the N −βHAð1Þ Ω β;r ¼ e ðKrÞj i; ð1:7Þ entangling surface [35,79]. The validity of the split property hints that the mutual information and reflected entropy where jΩi is the vacuum state, β > 0, H is the Hamiltonian, of nearby, but disjoint, regions are natural geometric ð1Þ and A ðKrÞ is the set of operators in AðKrÞ with norm regulators [54,56]. Both of these quantities may be well less than or equal to 1. This set is called nuclear if there defined without assuming a tensor factorization. The exists a positive, trace class operator, Tβ;r, such that regularization scheme is imposed by considering region A and its causal complement with the region at their ð1Þ N β;r ⊂ Tβ;rH ; ð1:8Þ interface of width ϵ excised (see Fig. 3). Then, one should take the limit of LA=ϵ → 0, though it is important that ð1Þ ϵ ≫ ϵ where H are the elements of H with norm less than or during this limit, one keeps UV. Then, in the final equal to 1. From here, we can define the nuclearity index as expression, all constants, e.g., c0, are physical and

045009-5 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

FIG. 4. The single-trace TT¯ deformation changes the bulk AdS3 geometry to asymptote in the UV to a linear dilaton regime FIG. 3. The geometric regularization scheme for the von represented in yellow. This smoothly crosses over to the un- Neumann entropy. We take A and B to be disjoint and separated deformed AdS3 regime in the IR. For further generality, we have by ϵ. Then, the regulated “entropy” is SRðA∶BÞ=2 or IðA∶BÞ=2 as placed a black hole in the IR regime which sets the boundary LA=ϵ → 0. theory to finite temperature. The IR region is thus more generally that of the Bañados-Teitelboim-Zanelli (BTZ) black hole [81]. unambiguous. We will see that the ability of taking the ¯ Suppressing internal compact dimensions, the bulk LA=ϵ → 0 limit disappears when we consider TT deformed theories. background induced by this deformation has string frame metric [20,46,80]

TT¯ 2hλ 2hλ− II. SINGLE-TRACE AND THE LINEAR ds2 ¼ dϕ2 þ hdγdγ¯ þ pffiffiffiþ dψdγ¯ þ pffiffiffi dψdγ¯ DILATON BACKGROUND k k 1 Let us begin by reviewing the holographic proposal of þ hf−1dψ 2; ð2:4Þ Refs. [15–17,20]. This involves deforming the world-sheet k conformal field theory by a linear combination of truly where the dilaton, Φ, and Neveu–Schwartz two-form, marginal current-current operators, B, are δL λ − ¯− λ ¯− λ ¯ − 2Φ 2 −2ϕ ws ¼ JSLJSL þ þKJSL þ −KJSL; ð2:1Þ e ¼ gs e h; Bγγ¯ ¼ gγγ¯;Bγψ ¼ gγψ Bψγ¯ ¼ gγψ¯ ; ð2:5Þ where K (K¯) are world-sheet Uð1Þ currents associated to − ¯− left(right)-moving momenta on a unit circle, and JSLðJSLÞ and are bosonic SLð2;RÞ left(right)-moving world-sheet affine −1 −2ϕ λ − 4λ λ −1 −1 4λ λ currents. h ¼ e þ þ −;f¼ h þ þ −: ð2:6Þ It was shown that this deformation in the long string 1 sector is equivalent to deforming M by a general linear ψ is the coordinate on the circle S , ψ ∼ ψ þ 2π. combination of the irrelevant operators, The boundary is located at ϕ ¼þ∞. The coordinates γ and γ¯ are δL −μ ¯ − μ ¯ − μ ¯ ¼ TT þJT −JT; ð2:2Þ lsγ ¼ t þ x; lsγ¯ ¼ −t þ x; ð2:7Þ ¯ pffiffiffiffi where J and J are left- and right-moving U(1) currents, α0 ¯ where ls ¼ is the intrinsic string length. respectively, and T and T are the holomorphic and anti- We restrict to the parameter range where the energy holomorphic stress tensor components, respectively, of the M spectrum was shown to be real and that there are no closed conformal field theory . With our normalization, the timelike curves [20], coupling constants of the spacetime and world-sheet 5 deformations are related as 2 λ ðλ þ λ−Þ − þ > 0: ð2:8Þ rffiffiffiffiffiffiffi 4πα0 32α0 6α0λ 4 3α0 μ μ λ ¼ ; ¼ ; ð2:3Þ λ λ δ cπ π c Furthermore, we take þ ¼ − ¼ 2. We show the corresponding geometry in Fig. 4 for the α0 μ μ 0 where is the Regge slope, and ; > . case in which δ ¼ 0. The linear dilaton regime controls the UV, while vacuum AdS3 controls the IR. They are 5 μ − μ 1=2 Our is equal to 4π2 from Ref. [11]. smoothly connected at a scale determined by k where k

045009-6 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020) is the level of the world-sheet SL 2; R algebra related In the rest of this subsection we consider the case in pffiffiffiffiffiffiffið Þ to the AdS radius as l ¼ kα0. Crucially, there is a which δ ¼ 0, λ ≠ 0. In this case, the entropy and the AdS 7 radial (renormalization) scale where local physics breaks interval length are given by [46] down, U . This is determined by the deformation max c 1 d 1 parameter. The length scale in the spacetime CFT where α 2α−α2 Π φ φ Sð Þ¼3α 1 ξ ξ 1 ð ;n;kÞ þFð ;kÞ ; this breaks down is6 þ d þ ξ¼0 rffiffiffiffiffiffiffiffi ð2:13Þ π cπμ β rffiffiffiffiffiffiffiffiffiffiffi l ¼ ¼ H : ð2:9Þ π 1 α min 2 6 4 l þ φ ¼ 4 α Eð ;kÞ; ð2:14Þ lmin β H is the Hagedorn temperature governing the asymptotic where (E → ∞) density of states sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α þ 1 3ϵ2 α S ¼ β E: ð2:10Þ φ ¼ arcsin 1 þ UV ; H 2α þ 1 2cπμ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Any observables in the field theory probing shorter dis- 2α þ 1 1 1 2α þ 1 n ¼ · ;k¼ · : ð2:15Þ tances than lmin cease to make sense. We will see that this α þ 1 ξ þ 1 α þ 1 α þ 1 nonlocality scale plays an important role for the mutual information and reflected entropy. We stress that all results Fðφ;kÞ, Πðφ;n;kÞ, and Eðφ;kÞ are the incomplete elliptic in this section and Sec. III are computed from the gravity integrals of the first, second, and third kinds respectively. In side and are thus contingent on the validity of the holo- our conventions, they are defined as graphic dualities, the RT formula, and its generalizations. Z φ dθ Πðφ;n;kÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 ð1 − nsin2θÞ 1 − k2sin2θ A. Vacuum Z φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eðφ;kÞ¼ dθ 1 − k2sin2θ; ð2:16Þ 1. Mutual Information 0 In this subsection, we study the mutual information for and Fðφ;kÞ¼Πðφ; 0;kÞ. two disjoint intervals of lengths lA and lB separated by a We have introduced the UV cutoff, ϵ , since the δ 0 λ 0 UV distance d. We begin with the case in which ¼ , ¼ entropy S is UV divergent. The mutual information corresponding to the vacuum of a conformal field theory. In this case, the mutual information takes the well-known IðA; BÞ¼SðlA; λÞþSðlB; λÞ − min½SAðlA; λÞ form [82] þ SðlB; λÞ;SðlA þ lB þ d; λÞþSðd; λÞ ð2:17Þ c lAlB is independent of the UV cutoff ϵ . However, we find that I ¼ max log ; 0 ; ð2:11Þ UV 3 dðlA þ lB þ dÞ a divergence emerges for the mutual information even when the intervals are separated by a finite distance. At where c is the Brown-Henneaux central charge [83]. The short distances, for intervals of equal length l, it diverges ∝ − −1 mutual information is independent of the ultraviolet cutoff. as I ðd lminÞ (Fig. 5). The same divergence was The critical distance where the phase transition between also noted for the entropic c-function in [20,46] at short connected and disconnected entanglement wedges occurs distances. This provides a fascinating breakdown of the is given by split property of quantum field theory and the mutual qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi information regulator of Ref. [56]. 1 d ¼ l2 þ l2 þ 6l l − l − l : ð2:12Þ 2 A B A B A B 2. Reflected entropy

Under the assumption that the minimal surfaces do not In this subsection, we consider the entanglement wedge break the symmetries of the spacetime metric, the von cross section for the intervals. This has been computed, for Neumann entropy of single intervals was evaluated using example, in the vacuum state of an undeformed conformal the Ryu-Takayanagi formula in Ref. [46]. In this section, field theory [50] we make the trivial generalization of these results to the holographic mutual information of disjoint intervals, 7We can invert the equation for the interval length to write α as though we find intriguing new physics. a function of l, c, μ, and we can use this in the entropy to get an μ ϵ expression that only depends on l, c, and the UV cutoff UV. Once we get the entropy, we can use it to compute the mutual 6 μ μ μ For the special case that ¼ þ þ −, lmin is doubled. information.

045009-7 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

1.2 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 pFIG.ffiffiffiffiffiffi 5. Left: the mutual information per central charge for disjoint intervals in the vacuum. Each subregion is of length 5 and we plot πcμ 0 1 1 3 6 ¼f ; 4 ; 2 ; 4g (blue to red respectively). We mark, with vertical dashed lines, the corresponding values of lmin where the mutual information intriguingly diverges. Right: the area of the entanglement wedge cross section for the same parameters. We find the same divergences at d ¼ lmin. pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 þ x l l π 1 α E ¼ log pffiffiffi ;x¼ A B : ð2:18Þ d þ þ W 6 1 − x ðl þ dÞðl þ dÞ ¼ 4 α A B lmin þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! This is a UV finite quantity and, as previously discussed, 1 χα 1 2χα þ þ þ þ has been proposed as a natural regulator for the von × E arcsin 1 2χα ; 1 χα 1 α : Neumann entropy which is universally UV divergent in þ þ ð þ þÞð þ þÞ 8 local quantum field theory [54]. One advantage of using ð2:21Þ the reflected entropy as a regulator instead of the mutual information is that it is actually an entropy. However, like These equations can be inverted to obtain equations for α α μ μ the mutual information regulator, we will see that this also − and þ in terms of l; d; c; ; . Once we have these breaks down when we turn on the deformations (Fig. 5). equations, we can use them to compute the wedge cross For simplicity, we will consider the case where lA ¼ section. Note that when δ ¼ 0, λ ¼ 0, this reprodu- ≡ lB l so that the minimal entanglement wedge cross ces (2.18). section is purely radial in the bulk metric. Taking into The coupling δ appears as δ2 and therefore, in perturba- account the factor of the dilaton in (1.3), tion theory, the OðδÞ term is zero. We will briefly discuss pffiffiffiffiffiffiffi Z this case in perturbation theory later tn the paper from α0 ϕ 2ϕ α k þ e ϕ c þ χ α − α the boundary field theory side. In contrast, the OðλÞ EW ¼ 3 d ¼ log þ ð þ −Þ ; 4 ð Þ ϕ h 12 α− contribution is nonzero. This comes from the second, GN − δ2 nonlogarithmic term in (2.19) giving χ 1 − ¼ λ ; ð2:19Þ 2 4α0λ c l þ d clðl þ dÞ O λ2 α α EW ¼ log þ 2 2 þ ð Þ: ð2:22Þ where − and þ are related to the turning points of the two 6 d 3d ðd þ 2lÞ minimal surfaces corresponding to the intervals of lengths 2 (along the noncompact direction, x) l þ d and d respec- Note that the nonlogarithmic term exists because the tively. They are related by [46] dilaton is not a constant. Using (2.3), we find sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 π 1 α l þ d þ − 2 2π 2μ ¼ c l þ d c lðl þ dÞ 2 4 α E ¼ log þ þ Oðμ Þ: ð2:23Þ lmin − W 6 9 2 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d d ðd þ lÞ 1 þ χα− 1 þ 2χα− × E arcsin ; ; Because μ > 0, this signals an increase in the correlations 1 þ 2χα− ð1 þ χα−Þð1 þ α−Þ between the subregions. The nonperturbative result (2.19) ð2:20Þ is depicted in (Fig. 5) with δ ¼ 0. It is nondecreasing along the renormalization group (RG) towards the UV and it diverges at short distances as the separation distance d 8 – See related work in Refs. [77,84 90]. approaches the nonlocality scale lmin.

045009-8 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

B. Finite temperature We first compute the von Neumann entropy for an It is interesting to consider the generalization of the interval of length l. In Ref. [42], two distinct surfaces “ ” results obtained at zero temperature to finite temperature. were considered as saddle points, the connected surface, Thermal states are holographically dual to black holes. which is the standard surface shown in Fig. 4, and the “ ” Roughly, increasing the temperature decreases the “quan- disconnected surface which consists of two disconnected tumness” of the state and quantum correlations are radial geodesics that terminate on the horizon of the black destroyed at length scales larger than the inverse temper- hole. The Ryu-Takayanagi prescription tells us to take the ature β. We are able to verify this explicitly by writing the minimum of these two saddles. While this disconnected generalization of the asymptotically linear dilaton metric surface is natural to consider for confining geometries that have compact dimensions “capping off” at the horizon (with δ ¼ 0) to include the black hole solution. The spatial because a zero area tube may connect them (e.g., metric is Refs. [91,92]), this surface does not obey the homology 2 2 2 constraint for the theories we study, so it should not be ds ¼ α0fdϕ þ hdx ; ð2:24Þ considered. In particular, the λ, δ → 0 limit should repro- −1 2 ϕ −ϕ −1 −2ϕ duce the universal CFT result f ¼ 1 − e ð H Þ;hðϕÞ¼λ þ e ; ð2:25Þ c β πl ϕ S ; : where H is determined by the temperature as ¼ 3 log πϵ sinh β ð2 27Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UV 0 −1 β ¼ 2π α h ðϕHÞ: ð2:26Þ and including the disconnected surface in this limit causes the bulk and boundary computations to disagree. For these This metric describes a BTZ black hole deep in the IR and reasons, we only consider the connected regime for the von linear dilaton asymptotics in the UVas was shown in Fig. 4. Neumann entropy. In what follows, we generalize the computations of The connected entanglement entropy is given by the Ref. [46] and Sec. II A of this paper for finite temperature integral δ 0 δ ≠ 0 with ¼ . We leave the case in which for future pffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi work. In this paper, we work on the black hole back- kα0 x∞ αx þ 1 SC ¼ 3 dx ground (2.24). 4 ð Þ 1 ðx − x Þðx − 1Þðαx þ α þ 1Þ GN H αx 1 ; : 1. Mutual information × ð þ Þ ð2 28Þ In this subsection, we study the mutual information for which is UV divergent. We solve for the entanglement intervals of lengths lA and lB separated by a distance d. entropy in closed form:

1 γ γ α c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2α − α2 d φ þ Π φ φ SC ¼ 3 ξ ξγ α Fð ;kÞþ ξγ α ξ 1 ð ;n;kÞ þ Fð ;kÞ ; ðα þ 1Þðγ þ α þ 1Þ d þ ð þ Þð þ Þ ξ¼0 ð2:29Þ where [93] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3L2 α uγ α 1 1 þ Λ 2α 1 ξγ α γ 1 2α 1 φ t þ þ 2cπμ þ þ þ þ ¼ arcsin · 3 2 γ;n¼ · ;k¼ · ; ð2:30Þ 2α þ 1 1 LΛ γ þ α þ 1 ξα þ α γ þ α þ 1 α þ 1 þ 2cπμ and rffiffiffiffiffiffiffiffi β2 −1 cπμ γ − 1 ; β 2π : : ¼ β2 H ¼ 6 ð2 31Þ H The interval length l in terms of the turning point of the minimal curve is given by the integral sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi Z α0 ∞ dx 1 α 1 αx ð þ Þð þ Þ 2 ϕ0 l ¼ ; α ¼ λU0;U0 ¼ e ;xH ≤ 1; ð2:32Þ U0 1 x ðx − xHÞðx − 1Þðαx þ α þ 1Þ

045009-9 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

1.2 1.0

1.0 0.8 0.8 0.6 0.6 0.4 0.4

0.2 0.2

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 pFIG.ffiffiffiffiffiffi 6. Left: the mutual information per central charge for disjoint intervals at finite temperature. Each subregion is of length 5 and πcμ 1 β 6 is set to 10. The temperature is varied as 2π ¼f100; 3; 2; 1g from blue to red respectively. Right: the entanglement wedge cross section per central charge for the same parameters. We observe the divergences at d ¼ lmin for both quantities. which solves to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π α 1 1 1 γ 1 l 1 ν − Π ν 2 ¼ 4 γ α 1 þ γ Fð ;qÞþ α γ ; γ α 1 þ α ;q ; ð2:33Þ lmin þ þ þ þ where While we do not consider the disconnected regions to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi contribute to the von Neumann entropy, it does contribute to γ þ α þ 1 ðγ þ 1Þð2α þ 1Þ the entanglement wedge cross section of a single interval. sin ν ¼ ;q¼ : 2α þ 1 ðα þ 1Þðγ þ α þ 1Þ The contributions from the disconnected regions are pffiffiffiffiffiffiffi Z H ð2:34Þ α0 x∞ α 1 2 k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHx þ SD ¼ · 3 dx ; 4 ð Þ 1 − 1 We can invert the equation for the interval length to write GN xðx Þ α in terms of l, μ, c, β. We then can use this to write the 2 2 H U∞ α ¼ λU ;x∞ ¼ ; ð2:37Þ entropy in terms of only CFT data. The interval length at H H U2 leading order in the coupling is H rffiffiffi giving l 1 γ pffiffiffiffiffiffi ffiffiffi 2arctanh O α2 : 2:35 cπμ ¼ pγ α þ ð Þ ð Þ 6 c 4cπμ 8cπμ S ¼ þð2 þ γÞ log : ð2:38Þ D 6 3L2 3γL2 To leading order in the coupling, the entropy then becomes Λ Λ pffiffiffi The entanglement wedge cross section at finite temperature 2 πμ 2β ffiffiffi πl γ c c 2 γ H pγ for an interval of length l is then given by SC ¼ 6 3ϵ2 þð þ Þ log πϵ sinh β UV UV H 2 þ Oðμ Þ: ð2:36Þ EWðμ;l;γÞ¼min ðSDðμ; γÞ;SCðμ;l;γÞÞ: ð2:39Þ

At μ ¼ 0, this reduces upon using (2.35) to (2.27). 2. Reflected entropy The mutual information at finite temperature is obtained using the entropy (2.29) in (2.17). We find that the mutual In this subsection, we compute the entanglement wedge information is UV cutoff independent. We plot the mutual cross section for intervals of equal length l and separation information for disjoint intervals in Fig. 6. Again, there is a distance d.Itisgivenby(2.19) divergence in the mutual information at short distances c α when d ¼ lmin, the same place where the zero temperature þ χ α − α EW ¼ log þ ð þ −Þ : ð2:40Þ mutual information result diverges. Heating up the system 12 α− causes the information shared between the disjoint regions α α to monotonically decrease. Here, þ and − are related to l and d via (2.33)

045009-10 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ d π α− 1 1 1 γ 1 ¼ 1 þ Fðν;qÞþ − Π ν; 2 þ ;q ; l 4 γ þ α− þ 1 γ α− γ γ þ α− þ 1 α− min rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π α 1 1 1 γ 1 d þ 1 ν − Π ν 2 ¼ 4 γ α 1 þ γ Fð ;qÞþ α γ ; γ α 1 þ α ;q : ð2:41Þ lmin þ þ þ þ þ þ þ þ

In what follows, we study the first few leading terms in (2.40) in perturbation theory. At λ ¼ 0, we find using (2.35) " # πd c coth β E ¼ log : ð2:42Þ W 6 πð2lþdÞ coth β The OðλÞ correction comes from the nonlogarithmic term in (2.40), and including this we get [upon using (2.35)]

πd c coth β c λ d 2l þ d E ¼ log þ · · U2 coth2 pffiffiffiffi U − coth2 pffiffiffiffi U þ Oðλ2Þ W 6 πð2lþdÞ 6 2 H 2 α0 H 2 α0 H coth β πd 2 c coth β cπ α0λ πd πð2l þ dÞ ¼ log þ coth2 − coth2 þ Oðλ2Þ: ð2:43Þ 6 πð2lþdÞ 48β2 4β 4β coth β

Expanding the OðλÞ correction in powers of the inverse temperature β gives ∂E 4clðd þ lÞα0λ 4clðd þ lÞπ4α0λ 16clðd þ lÞðd2 þ 2dl þ 2l2Þπ6α0λ 1 λ W ¼ − þ þ O ∂λ 3d2ðd þ 2lÞ2 45β4 567β6 β8 2πc2lðd þ lÞμ 2c2lðd þ lÞπ5μ 8c2lðd þ lÞðd2 þ 2dl þ 2l2Þπ7μ 1 ¼ − þ þ O : ð2:44Þ 9d2ðd þ 2lÞ2 135β4 1701β6 β8

We note that the order μ small temperature leading nontrivial. We note that the torus partition function itself correction is negative. We plot the entanglement wedge diverges precisely at β ¼ βH. cross section in Fig. 6. Both the finite distance divergence as d approaches the nonlocality scale l and the mono- min III. DOUBLE-TRACE TT¯ AND CUTOFF AdS tonic decrease with temperature is clear. We now compute extremal surfaces in the cutoff C. Comments on the split property AdS geometry proposed to be holographically dual to the double-trace TT¯ deformation [11]. Importantly, this We briefly comment on some interesting consequences of prescription uses the opposite sign of the deformation the computations in this section. We have found the mutual parameter9 such that the spectrum always contains complex information and reflected entropy to generically diverge energies. β 4 when the distance between intervals approaches H= .This Starting with the metric for the BTZ black hole, means that the split property must have failed; i.e., there does A A ¯ not exist a type I factor that splits ðAÞ and ðBÞ.Inthe r2 − r2 l2 ds2 H dt2 AdS dr2 r2dx2; : geometric regularization scheme, we are then unable to take ¼ 2 þ 2 − 2 þ ð3 1Þ lAdS r rH the LA=ϵ → 0 limit. This limits our ability to extract the relevant physical constants that serve as c-functions. It is an → ∞ important question how to define c-functions for theories where the asymptotic boundary is at r and the like these that are nonlocal at short distances along the horizon is at rH, the deformation corresponds to a finite renormalization group flow towards the UV. radial cutoff with Dirichlet boundary conditions with An additional curiosity is the factor of 4 in lmin.In 6l4 theories with Hagedorn divergences, the minimum distance 2 − AdS rc ¼ μπ : ð3:2Þ for a split is βH [76], but we do not see the divergence in the c ¯ reflected entropy until d ¼ βH=4.ForTT-deformed theories, the Hagedorn density of states only starts to dominate The horizon radius is related to the temperature as at some energy scale (which depends quadratically on βH), β 9 thus H only acts as an upper bound for the minimum In this section, we will alwaysR take μ < 0 so the deformation distance needed for a split. This factor of 4 is therefore quite of the Lagrangian is L → L − μ TT¯.

045009-11 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

1 1 3 FIG. 7. The mutual information (left) and entanglement wedge cross section (right) for zc ¼f0; 4 ; 2 ; 4g in descending order. When the deformation parameter is finite (solid lines), EW and I are manifestly finite even as the intervals become adjacent. We use symmetric intervals of length l ¼ 5. pffiffiffiffiffiffiffiffiffiffi 2π μπ [23] if one identifies − c with the UV cutoff. Thus, the rH ¼ β : ð3:3Þ 6 deformation provides a natural cutoff for this sign of the The metric on the finite cutoff boundary is deformation. We stress that this resultp isffiffiffiffiffiffi extremely different from that 2 − 2 μπc ≡ 2 rc rH 2 2 2 of Sec. II where the value 6 lmin determined the finite ds ¼ 2 dt þ rcdx : ð3:4Þ lAdS distance between the intervals where the mutual information diverged. Clearly, the correlation structure is drasti- Thus, we must rescale the metric such that t is a physical cally changed in opposite ways given the sign of the time on this surface: deformation. The results are plotted in Fig. 7. l2 r2 2 → 2 AdS c 2 ds dt þ 2 − 2 dx : ð3:5Þ 2. Reflected entropy rc rH Note that this subtlety is absent at zero temperature We now investigate how this UV regulation emerges for (r → 0). the reflected entropy by returning to the entanglement H wedge cross section. As previously noted, at zero temperature and μ ¼ 0, the area of the entanglement wedge cross A. Vacuum section is a simple function of the conformally invariant 1. Mutual information cross ratio [50] ffiffiffi At zero temperature, the von Neumann entropy of an 1 p ˜ ˜ c þ ffiffiffix x21x43 interval of length l can straightforwardly be computed: EW ¼ 6 log 1 − p ;x¼ ˜ ˜ ; ð3:9Þ x x42x31 c 3l2 Sðl; μÞ¼ cosh−1 1 − : ð3:6Þ 6 μπc where the coordinates with tildes correspond to those on the asymptotic boundary. The mutual information between disjoint intervals is simply In the majority of parameter space, the entanglement wedge cross section has the same area as the undeformed μ μ − μ I ¼ max½SðlA; ÞþSðlB; Þ SðlA þ lB þ d; Þ theory because it lies in the IR part of the geometry which is − Sðd; μÞ; 0: ð3:7Þ entirely unchanged in the cutoff AdS story. Though the area of the geometric object remains unchanged, the dependence It is interesting to consider the adjacent intervals limit of the cross ratio on the boundary positions will flow. In (d → 0) of the mutual information. Interestingly, the mutual particular, the cross ratio will no longer be conformally information in this limit is UV finite for any value of μ.At invariant. We can map the points from the asymptotic leading order in the deformation parameter, we have boundary to the finite cutoff boundary along the geodesics (see Fig. 8), which are semicircles at zero temperature when c lAlBpffiffiffiffiffiffiffiffiffiffi 2 I ¼ log μπ : ð3:8Þ we work in Poincaré coordinates (r → l =z) 3 ðl þ l Þ − c AdS A B 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 − 4x2 This is identical to the universal value of the mutual r ¼ : ð3:10Þ information for adjacent intervals in a conformal field theory 2

045009-12 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi This leads to the following mapping of boundary coor- 1 ˜ − 2 4 2 dinates: x3 ¼ 2 x2 þ x3 þ ðx2 x3Þ þ zc ; ð3:13Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ˜ − − 2 4 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 ¼ x1 þ x4 ðx1 x4Þ þ zc ; ð3:11Þ 1 2 2 2 x˜4 ¼ x1 þ x4 þ ðx1 − x4Þ þ 4z : ð3:14Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c 1 2 2 x˜2 ¼ x2 þ x3 − ðx2 − x3Þ þ 4z ; ð3:12Þ 2 c Thus, in terms of the cutoff coordinates, the cross ratio is

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − − 2 4 2 − 2 4 2 − 2 − 2 4 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 x4Þ þ zcp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 x3Þ þ zc þ x1ðx2 þ x3 x4Þþx4ðx2 þ x3Þ x2x3 þ zc x ¼ 2 2 2 2 2 : ð3:15Þ ðx1 − x4Þ þ 4zc ðx2 − x3Þ þ 4zc þ x1ðx2 þ x3 − 2x4Þþx4ðx2 þ x3Þ − 2x2x3 þ 4zc

Inserting (3.15) into (3.9), we arrive at the full non- Let us now look at the adjacent intervals limit. We find perturbative area of the entanglement wedge cross section. EW to be UV finite as d → 0 for any finite value of μ. There is a correction at leading order in μ. For e.g., equal In particular, the leading order correction (in μ)is length intervals of length l and distance d, we find c 2l l E ¼ log A pB ffiffiffiffiffiffiffiffiffiffi : ð3:17Þ 2 W 6 − μπc c 2l 2c lðd þ lÞπμ ðlA þ lBÞ 6 E 1 O μ2 : : W ¼ 6 log þ þ 9 2 2 2 þ ð Þ ð3 16Þ d d ðd þ lÞ Analogous to the mutual information, this is equivalent to the adjacent intervals limit of (half of the) reflectedpffiffiffiffiffiffiffiffiffiffi entropy Interestingly, this leading order correction is identical to the − μπc one found for the linear dilaton geometry (2.23). However, in a conformal field theory once identifying 6 with we again stress that we take μ < 0 such that this decreases the UV cutoff. The results are plotted in Fig. 7. the correlations, the opposite effect found from the linear dilaton analysis. The sign of the change in correlations will B. Finite temperature “ ” crucially depend on how one flows up the RG. We progress to finite temperature where it will be easier This correction, of course, is only valid when we are to compare with boundary computations of the following within the connected regime of the entanglement wedge section. For finite temperature calculations, we simply set (I>0). There is a phase transition of the entanglement rH > 0 in (3.1). wedge to the disconnected regime which depends on the deformation parameter. The phase transition occurs when 1. Mutual information S½x1;x2 þ S½x3;x4 ¼ S½x1;x4 þ S½x2;x3. In the disconnected regime, EW ¼ 0,soΔEW ¼ 0. We plot the corresponding The von Neumann entropy for an interval can be found at EW and mutual information in Fig. 7. finite temperature from the Ryu-Takayanagi formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c r 2 r l r2 − r2 S ¼ cosh−1 1 þ 2 c sinh2 H c H : ð3:18Þ 6 rH 2rc This is UV finite and transitions from logarithmic growth for l smaller than thethermallength to lineargrowthfor largel.The linear growth signals that it operates as an extensive thermo- dynamic entropy.The mutual information is again determined by (3.7). To leading order in the deformation parameter,

2 lπ c sinh ð β Þ I ¼ log 6 dπ ð2lþdÞπ sinhð β Þ sinhð β Þ π3 π π − μ c π d − 2 π l FIG. 8. In the cutoff AdS approach, the standard Dirichlet 18β3 d coth β l coth β boundary conditions of holography are moved from the con- ð2l þ dÞπ dπ formal boundary at r∞ to a hard Dirichlet boundary at rc. Because þðd þ 2lÞπ coth − β csch the geometry within the new boundary remains the same, so do β β the geodesics. In the figure, we show how the entangling surface ð2l þ dÞπ lπ points of the interval are mapped from the asymptotic boundary þ csch − 2csch : ð3:19Þ to the finite boundary. β β

045009-13 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

The length of the interval on the cutoff surface in terms of the length at the asymptotic boundary is ffiffiffiffiffiffiffiffiffi h ˜ p i cosh ðr lÞ r2−r2 2 −1 H2 c H rccosh r l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic ; ð3:22Þ 2 − 2 rH rc rH

or inverted ffiffiffiffiffiffiffiffiffi " p 2 2 # lr rc −r H H rc cosh 2r 2 cosh−1 pffiffiffiffiffiffiffiffiffic r2−r2 ˜ c H FIG. 9. The analog of the mapping in Fig. 8, this time with the l ¼ ; ð3:23Þ rH additional scale, rH, representing the black hole horizon radius.

where we have made sure to be careful in rescaling the 2. Reflected entropy physical length of the interval at the finite cutoff. To solve for the entanglement wedge cross section, we For simplicity, we consider disjoint intervals of equal again map the coordinates on the asymptotic boundary to length, so that the entanglement wedge cross section is the cutoff surface (Fig. 9). For this, we need to solve the purely radial: geodesic equation for a boundary anchored curve giving a Z parametrization of the Ryu-Takayanagi surface, r ðdÞ rðdÞ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffidr c −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffir EW ¼ 2 2 ¼ tanh 2 2 : 2 −1 2 6 2 − 6 − 2 = rð lþdÞ r rH r rH rð lþdÞ cosh ðrHxÞ rðxÞ¼rH 1 − : ð3:20Þ 2 rHl ð3:24Þ cosh ð 2 Þ

The turning point is at Evaluating this is straightforward but gives a complicated, r unenlightening expression for the nonperturbative correc- r ¼ H : ð3:21Þ tion to E . Instead, we consider the first order correction l˜ W tanh ðrH 2Þ (linear in μ)

" # πd c coth β E ¼ log W 6 πð2lþdÞ coth β

3 2 πd πd πðdþ2lÞ πðdþ2lÞ π c μððπd − β cothð β ÞÞcschð β Þþðβ cothð β Þþπð−d − 2lÞÞcschð β ÞÞ − O μ2 : : 18β3 þ ð Þ ð3 25Þ

In the low temperature expansion (taking μ → 0 before β → ∞), this gives " # πd c coth β E ¼ log W 6 πð2lþdÞ coth β 2πc2μlðl þ dÞ 13c2lðd þ lÞπ5μ 139c2μπ7lðd þ lÞðd2 þ 2dl þ 2l2Þ þ − þ þ Oðβ−8Þ: ð3:26Þ 9d2ðd þ 2lÞ2 540β4 34020β6

This has identical functional form to the linear dilaton results (2.44), though the numerical coefficients, after the zero temperature correction, disagree. We plot the nonperturbative results in Fig. 10 where it is clear that heating up the system destroys the quantum correlations between the subsystems. Again, EW is UV finite for all d.

045009-14 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

1.0 0.6

0.8 0.5

0.4 0.6 0.3 0.4 0.2 0.2 0.1

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

FIG. 10. Left: the mutual information per unit central charge for disjoint intervals of equal length (5). We set the radial cutoff to −2 rc ¼ 10 and vary the temperature rH ¼f10 ; 1; 2; 3g from blue to red respectively. Right: the same configurations, but for the entanglement wedge cross section.

IV. CONFORMAL PERTURBATION THEORY where we have used the definition R It is important to check whether the predictions from − Dϕ SCFT ¯ ¯ ≡ ½R ne ðTTÞ holography from the previous section agree with (per- hTTiM − : ð4:4Þ n Dϕ SCFT turbed) CFT computations. In order to nonperturbatively ½ ne understand how the entanglement structure changes under the flow induced by the irrelevant deformations from the To leading order in μ, the change in the logarithm of the field theory side, one must understand how the partition partition function due to the deformation is function on arbitrary genus Riemann surfaces flows. While Z there has been significant progress for sphere and torus ¯ δ log Z ¼ μ hTTiM : ð4:5Þ partition functions [3,5,20,37,94–100], little is known n n Mn about more generic higher genus partition functions. This is why, thus far, the only nonperturbative results for Generalizing to include the other irrelevant deformations, ¯ entanglement measures under TT flows are for the very we have special case of von Neumann entropy where the replica Z Z Riemann surface is genus zero [37,38]. The one exception δ Z μ ¯ μ ¯ log n ¼ hTTiM þ þ hJTiM to this is for 2D Yang-Mills theory because of its near M n M n topological behavior [99]. nZ n ¯ ¯ We consider the partition function of the TT deformed μ− TJ : 4:6 þ h iMn ð Þ theory from the path integral formalism, Mn Z R d ¯ −SCFTþμ d xTT Z ¼ ½Dϕ e Mn ; ð4:1Þ n n A. TT¯ at finite temperature We will focus on the reflected entropy at finite temper- M where n is the replica manifold of n connected copies of ature because there are ambiguous contact terms in the the theory which generically has highly nontrivial topology. correlation functions at zero temperature. Similar ambigu- We work perturbatively so that we can expand in the ities have previously been commented on in Ref. [42].To coupling as compute the reflected entropy directly in the field theory, Z Z we must compute the path integral on an mn-sheeted −S d ¯ 2 branched cover of the CFT: Zn ¼ ½Dϕne CFT 1 þ μ d xðTTÞþOðμ Þ ; M n 1 Z S ¼ lim log n;m ; ð4:7Þ ð4:2Þ R →1 n n;m 1 − n ðZ1;mÞ Z Z ∈ 2Z − 2 where m is the replica index for the given purification Dϕ SCFT 1 μ d ¯ O μ ¼ ½ ne þ d xhTTiM þ ð Þ ; ∈ Z M n and n is the standard replica index for the R´enyi n entropies. Recall that in holographic theories, the loga- ð4:3Þ rithmic negativity is equivalent to the following path

045009-15 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020) α −2 1 − 1 − hp hL n → 1=2 n → 1 h α−1 ð zÞ integral when taking instead of . The path V ¼ð1 − zÞ Lð Þ integrals may be computed by a correlation function of T α twist fields in the S orbifold theory. For example, when α mn × 2F1ðh ;h ; 2h ; 1 − ð1 − zÞ Þ; we consider the reflected entropy between two disjoint rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip p p intervals, we must compute [54] 24h α ≡ 1 − H; ð4:12Þ c Z ¼hσ ðz1Þσ −1 ðz2Þσ ðz3Þσ −1 ðz4Þi; ð4:8Þ n;m gA gA gB gB and VJ is the contribution only from the Uð1Þ descendent where the twist fields have conformal dimensions states: 2 q 2 − 1 − L qHqL cnðm Þ VJ ¼ z k ð1 − zÞ k : ð4:13Þ hg ¼ hg−1 ¼ hg ¼ hg−1 ¼ : ð4:9Þ B B A A 24m Here, k is the level of the Uð1Þ current algebra, σ −1 σ In the operator product expansion of g and g , the A B δ leading operator is another Virasoro primary field, σ −1 , ½Jn;Jm¼nk nþm;0; ð4:14Þ gA gB with conformal dimension and in the linear dilaton story is related to the AdS radius, 2cðn2 − 1Þ lAdS,as hg g−1 ¼ : ð4:10Þ B A 24 2 n lAdS k ¼ 2 : ð4:15Þ We will approximate these four-point functions by taking ls only the dominant conformal block in the conformal block Thus, in the semiclassical limit in the bulk, the level will decomposition. This is a valid approximation in the limit of be large. large central charge because contributions from subdomi- We now specify to two disjoint intervals where the nant conformal blocks are exponentially suppressed in c M ¯ ¯ correlation functions on n;m may be computed by [101,102]. When considering the JT and TJ deformations, correlation functions of twist fields on M: we, by definition, have a conserved Uð1Þ current. We are then required to take the dominant Vir × Uð1Þ block rather 1 ¯ ¯ ¯ σ σ −1 σ hTTiM ¼ hTðwÞTðwÞ g ðz1Þ g ðz2Þ g ðz3Þ than just the Virasoro conformal block. This extended n;m nm A A B conformal block was shown to factorize at large c into σ −1 × g ðz4ÞiM; ð4:16Þ Virasoro and Uð1Þ blocks as [103] B 2 where the stress tensors on the right-hand side live in the V V − 1 − qi orbifold theory, leading to the prefactor. A similar expres- TþJðc; hi;k;qi;hp;zÞ¼ T c ;hi ;hp;z 2k sion holds for the other two terms. The strategy is to apply V the Ward identities to evaluate these correlation functions × Jðk; qi;zÞ; ð4:11Þ and then integrate over the thermal cylinder. The calcu- 1 where qi are the Uð Þ charges and hp is the conformal lation is technical and involves long formulas and subtle- weight of the intermediate operator. VT is the Virasoro ties, so we relegate some details to Appendix C. block that may be evaluated in the heavy-heavy-light-light After evaluating, we find the change in the reflected limit as entropy to be

πd 2πd 2πl 2πl 2πðdþ1Þ π 2 4 2 β πd β β β β ðdþ lÞ π c μe ðcothð β Þ − 1Þð2lðe − 1Þe − dðe − 1Þðe þ 1ÞÞðcothð β Þ − 1Þ ΔS : : R ¼ 18β3 ð4 17Þ

The zero temperature (β → ∞) limit is

lim ΔS ¼ 0: ð4:18Þ β→∞ R

This, however, is not sensitive to the contact terms discussed earlier. Similarly, the β → ∞ limit of the von Neumann entropy is zero even though there are finite corrections at zero temperature due to contact terms. The leading low temperature correction comes at fourth order: " # πd 2 5 2 7 2 2 c coth β 2c lðd þ lÞπ μ 7c μπ lðd þ lÞðd þ 2dl þ 2l Þ S ¼ log − þ þ Oðβ−8Þ: ð4:19Þ R 3 πð2lþdÞ 27β4 405β6 coth β

045009-16 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

Besides the zero temperature term which may arise from a the mutual information [56] and reflected entropy [54] have contactterm,wenotethatitisinterestingthatthishasprecisely been proposed as valuable regularization scheme indepen- the same form as both holographic calculations (2.44) and dent “geometric regulators” that can be studied to extract (3.26). However, the numerical coefficients are different. This c-functions in general dimensions. These schemes rely on suggests that the correlation structures of the TT¯ deformed taking the ratio between the characteristic size of the CFTand the field theory dual to cutoff AdS geometry are very regions and the spatial distance between regions to zero. similar, but not quite identical. Thus, this suggests that Clearly, this breaks down when we hit the nonlocality scale modifications, perhaps similar to Refs. [14,104], must be lmin. It would be fascinating to consider more carefully how made to have a precise agreement for the two theories.10 We to incorporate nonlocal theories, such as the ones we have reiterate that the cutoff AdS results assumed the validity of studied, in renormalization group analysis. holographic duality between reflected entropy and the entan- For μ < 0 and cutoff AdS geometries, we have found the glement wedge cross section in generic spacetimes, an opposite effect of the theory becoming nonlocal. Rather assumption that is on solid footing but not rigorously derived than enhanced divergences, we have found all divergences [43,54]. In order to claim modifications must be made for the in the mutual information and reflected entropy to be tamed linear dilaton proposal, we would have had to compute the even when the distance between intervals goes to zero. The single-trace correction in the CFT. square root of the deformation parameter acts akin to a UV cutoff. This phenomenon is also never seen in local B. JT¯ at finite temperature quantum field theory and is more reminiscent of finite We can also apply the Ward identities for the currents for dimensional lattice models. We were able to compare the the JT¯ correction, bulk and boundary computations of the reflected entropy of disjoint intervals at finite temperature perturbatively. Z Z Xk 1 1 2πz qj While we found the corrections to be formally equivalent, hJTiMðn;mÞ ¼ the numerical coefficients were distinct. There are two Mðn;mÞ h iC M nm β ðz − z Þ j¼1 j possible explanations for this tension. Either nonper- 11 2 Xk turbative corrections save the day as in Ref. [38] or 2πz hj × β − 2 modifications must be made to have a consistent duality j 1 ðz zjÞ ¼ between the deformed field theory and cutoff AdS. Such 1 π2nmc modifications may be analogous to the ones shown to ∂ − : þ zj 2 h iC be necessary for the inclusion of matter fields in z − zj 6β Refs. [14,104]. We believe resolving this tension is an ð4:20Þ important direction for future work. It is also interesting to consider a recent holographic The twist fields are not charged under the Uð1Þ (see proposal that we have largely neglected in this work where Appendix B), so the leading term is trivial. The same can be the dual of the TT¯ deformation is proposed to be an said for the TJ¯ deformation. We found this to also be the ensemble of AdS spacetimes with randomly fluctuating case holographically in Sec. II. boundary diffeomorphisms [10]. Such a proposal is a natural combination of Cardy’s interpretation of the TT¯ V. DISCUSSION deformation as random geometry [6] and the standard In this work, we have studied the mutual information GKPW dictionary of AdS=CFT [12,13]. We see no reason and reflected entropy in holographic TT¯ deformed two- why bulk and boundary computations of mutual informa- dimensional quantum field theories. We also studied these tion and reflected entropy in this proposal should not match entanglement measures perturbatively in TJ¯ and JT¯ defor- exactly, though the computations may be quite technical. mations. The results of our calculations have led to several Confirming this would certainly be worthwhile. notable physical phenomena. There are also several inter- Lastly, we comment on a somewhat tangential future esting directions to take from here. direction. The twist fields needed for the computation For μ > 0 and asymptotically linear dilaton geometries, for R´enyi entropies and reflected entropy are uncharged we have found the mutual information and reflected under Uð1Þ, leading to somewhat mundane results in entropy of disjoint intervals to diverge when the intervals perturbation theory for the JT¯ and TJ¯ deformations. are a finite, regulator independent distance (lmin) away from However, there are finer grained probes of charged states, each other. Such a phenomenon never occurs in local such as the symmetry-resolved entanglement [105], whose quantum field theory and signals a breakdown of locality, corresponding twist fields are charged under Uð1Þ. The specifically a breakdown of the split property. In particular, symmetry-resolved entanglement explains the contribution

10Another logical possibility is that nonperturbative correc- 11In this analysis, taking the correct order of limits also played tions restore consistency. a central role.

045009-17 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020) to the von Neumann entropy from each charge sector. APPENDIX B: U(1) CHARGES OF These will presumably have corrections at leading order in TWIST OPERATORS μ and it would be fascinating to understand how these In order to evaluate the conformal blocks, we must contributions are effected under the charged deformations. find the charges under the Uð1Þ symmetry for the twist operators. Let us warm up by computing these for the twist ACKNOWLEDGMENTS operators used for entanglement entropy, σn, labeled by a We thank Tom Faulkner, Christian Ferko, David Kutasov, single replica index. These have conformal dimensions Shinsei Ryu, Gautam Satishchandran, Sav Sethi, Ronak 1 Soni, and Tadashi Takayanagi for useful discussions and ¯ c hn ¼ hn ¼ n − : ðB1Þ comments. M. A. is supported in part by DOE Grant No. de- 24 n sc0009924. J. K. F. is supported through a Simons Inves- tigator Award to Shinsei Ryu from the Simons Foundation. These can be determined by considering the n-sheeted branched cover of the complex plane, Rn, used to compute the nth power of the reduced density matrix of a single APPENDIX A: DERIVATION OF HOLOGRAPHIC interval, ðu; vÞ. We know that on the complex plane, the ENTANGLEMENT ENTROPY IN one-point function of the Uð1Þ current is trivial by rota- STRING FRAME tional and translational invariance Under reasonable assumptions, the Ryu-Takayanagi 0 formula has been derived for theories with arbitrary matter hJðzÞiC ¼ : ðB2Þ content in Einstein frame12 Z qffiffiffiffiffiffiffiffi JðzÞ is a chiral primary field of dimension (1,0), so it 1 transforms covariantly under conformal maps, S ¼ dd−2x gðEÞ: ðA1Þ 4 ðdÞ γ GN A ∂z JðwÞ¼ JðzÞ: ðB3Þ We would like to know what the analog is in string ∂w frame. The relation between the string and Einstein frame In particular, we consider the conformal map that takes R metrics is n to C: ðsÞ Φ=2 ðEÞ gμν ¼ e gμν : ðA2Þ w − u 1=n z ¼ : ðB4Þ Thus, the determinants of the metrics are related as w − v ðsÞ ðd−2ÞΦ=2 ðEÞ This means that J w 0. We now compare this with g ¼ e g : ðA3Þ h ð ÞiRn ¼ the Ward identity for Kac-Moody symmetries, The reason why we have (d − 2) is because we are working with the induced metrics on codimension-two surfaces. hJðzÞσnðw1Þσ¯nðw2Þi Thus, the RT formula becomes Z qffiffiffiffiffiffiffi q1 q2 σ σ¯ 1 ¼ − þ − h nðw1Þ nðw2Þi: ðB5Þ S ¼ dd−2xe−ðd−2ÞΦ=4 gðsÞ: ðA4Þ z w1 z w2 4 ðdÞ γ GN A For this to equal zero for any value of w1 and w2, we must Now, let us specify to the d ¼ 10 supergravity theory that have q1 ¼ q2 ¼ 0, so the twist fields are neutral under we are generally concerned with: the Uð1Þ. Z qffiffiffiffiffiffiffi 1 Now let us proceed to the generalized twist operators 8 −2Φ ðsÞ (4.9). The simplest to study is the two point function S ¼ 10 d xe g : ðA5Þ ð Þ γ 4G A hσ −1 σ −1 i because the manifold this describes decou- N gBgA gAgB ples into two copies of R , so running through the same This is the formula posited in Ref. [30]. It is important to n argument, we find these operators are also uncharged. This note that the extremal surface in (A5) is not described by immediately tells us that the two operators that fuse to form the same coordinates as the one in (A1), though it is the them must have opposite charge. We can fix these by same surface. In the string frame coordinates, it is extremal σ σ −1 considering h g g i which is needed for computing the with respect to the integrand including the dilaton prefactor. A A reflected entropy between A and the empty set. This factorizes into n R ’s, so the same argument goes through 12This is because in Einstein frame, the contribution to the on- m shell bulk action for all matter fields is proportional to n (the and we conclude that all twist operators are uncharged replica number) when performing the gravitational replica trick under the Uð1Þ. This means that the Uð1Þ contribution to while the metric term is proportional to (n − 1). the conformal block, (4.13), is unity.

045009-18 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

APPENDIX C: DETAILS OF PERTURBATIVE CFT COMPUTATIONS In this Appendix, we present a few details of the calculation leading to IVA. The conformal Ward identity implies

Z Z 2 Xk 2 1 1 2πz hj 1 π nmc ðn;mÞ ∂ − hTTiM ¼ 2 þ zj 2 Mðn;mÞ M nm β − z − z 6β h iC j¼1 ðz zjÞ j 2 Xk 2 2πz hj 1 π nmc × þ ∂ − h iC: ðC1Þ β − 2 z − z zj 6β2 j¼1 ðz zjÞ j

The correlation function of four twist operators may be computed using (4.12). The total integral becomes Z Z ∞ β −π4 2μ − − τ c ðz1 z2Þðz3 z4Þ dx d 9β4 −∞ 2 0 4πðxþiτÞ 6 e β 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × 2πðxþiτÞ 2πðxþiτÞ 2πðxþiτÞ 2πðxþiτÞ ðz1−z2Þðz3−z4Þ −z1 e β −z2 e β −z3 e β −z4 e β ð þ Þð þ Þð þ Þð þ Þ ðz1−z3Þðz2−z4Þ 3 4πðx−iτÞ e β 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 þ 2πðx−iτÞ 2πðx−iτÞ 2πðx−iτÞ 2πðx−iτÞ : ðC2Þ ðz2−z1Þðz3−z4Þ −z1 e β −z2 e β −z3 e β −z4 e β − ð þ Þð þ Þð þ Þð þ Þ ðz1−z3Þðz2−z4Þ

We first do the τ integral to find 2 Z 2πðxþiτÞ ∞ π3 2μ − − 6 − β i c ðz1 z2Þðz3 z4Þ − z1 log ð z1 þ e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ dx 3 4 −∞ 18β ðz1−z2Þðz3−z4Þ ðz1 − z2Þðz1 − z3Þðz1 − z4Þ ðz1−z3Þðz2−z4Þ

2πðxþiτÞ 2πðxþiτÞ z2 log ð−z2 þ e β Þ z3 log ð−z3 þ e β Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz1−z2Þðz3−z4Þ ðz1−z2Þðz3−z4Þ z1 − z2 z2 − z3 z2 − z4 z1 − z3 z3 − z2 z3 − z4 ð Þð Þð Þ ðz1−z3Þðz2−z4Þ ð Þð Þð Þ ðz1−z3Þðz2−z4Þ

2πðxþiτÞ 2πðx−iτÞ z4 log ð−z4 þ e β Þ z1 log ð−z1 þ e β Þ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz1−z2Þðz3−z4Þ ðz2−z1Þðz3−z4Þ z1 − z4 z2 − z4 z4 − z3 z1 − z2 z1 − z3 z1 − z4 − ð Þð Þð Þ ðz1−z3Þðz2−z4Þ ð Þð Þð Þ ðz1−z3Þðz2−z4Þ

2πðx−iτÞ 2πðx−iτÞ z2 log ð−z2 þ e β Þ z3 log ð−z3 þ e β Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz2−z1Þðz3−z4Þ ðz2−z1Þðz3−z4Þ ðz1 − z2Þðz3 − z2Þðz2 − z4Þ − − − ðz1 − z3Þðz3 − z2Þðz3 − z4Þ − − − ðz1 z3Þðz2 z4Þ3 ðz1 z3Þðz2 z4Þ τ β 2πðx−iτÞ ¼ β z4 log ð−z4 þ e Þ 7 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 : ðC3Þ ðz2−z1Þðz3−z4Þ ðz4 − z1Þðz4 − z2Þðz4 − z3Þ − − − ðz1 z3Þðz2 z4Þ τ¼0

Every term that either does not depend on τ or depends on τ only exponentially and not within a logarithm is trivial when evaluating the difference of the indefinite integral at β and 0. The terms with the exponential within the logarithm must be treated with care. This has been discussed in e.g., Ref. [39]. Due to the branch cut, we have 2πðxþiτÞ 2πl τ β 0;x l:

Analogously, for the complex conjugate, we run around the branch cut the opposite direction, so

045009-19 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020) ˜ μ˜ 2πðx−iτÞ 2πl τ β 0;x l: 24π 4

After evaluating, we find IVA. However, in the symmetric orbifold theory, we have

XN APPENDIX D: NONPERTURBATIVE CFT T TðiÞ; D7 TT¯ ab ¼ ab ð Þ CALCULATION FOR SINGLE-TRACE i In this section, we repeat the analysis done in Ref. [37] ¯ for the entanglement entropy of a region with an entangling with each individual copy flowing under a TT deformation. surface of antipodal points on S2 except for the single-trace Thus, there is an extra factor of N: deformation dual to the asymptotically linear dilaton c μ a − − ab − a 2 geometry. Here, we assume that the spacetime conformal hTai¼ 24π R 4 ðhT ihTabi hTai Þ: ðD8Þ N field theory is a symmetric product orbifold M =SN. Each block has central charge c˜, so the total CFT has central We have absorbed the factors of N into c and μ due to our charge c ¼ Nc˜. We apply the replica trick by considering definitions of the central charge and deformation parameter. the n-sheeted cover of the sphere of radius r: By symmetry and (D8), the stress tensor must be proportional to the metric as ds2 ¼ r2ðdθ2 þ n2 sin θdϕ2Þ: ðD1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cμ The von Neumann entropy is then T ¼ 1 − 1 þ g ; ðD9Þ ab μ 24πr2 ab ∂ S ¼ 1 − n log Zj 1: ðD2Þ ∂n n¼ so we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The sphere partition function responds to a change in n as ∂ Z 16π μ Z log 2 c − ∂ Z 1 ffiffiffi ∂ ¼ μ r þ 24π r : ðD10Þ log − p r ∂ ¼ 2 gT; ðD3Þ n n 1 ¼ Imposing the boundary condition log Zðr ¼ 0Þ¼0,we where T is the trace of the stress tensor. Similarly, when have the entropy 1 n ¼ , the response of the sphere partition function to a sffiffiffiffiffiffiffiffi ! change in radius is c 24π Z S ¼ sinh−1 r : ðD11Þ d log Z 1 pffiffiffi 3 cμ ¼ − gT; ðD4Þ dr r This is identical to the double-trace formula from Ref. [37]. so the entropy may be rewritten as It would be interesting to check this holographically by r ∂ finding the Euclidean compactification of (2.4) to a sphere. S ¼ 1 − log Z: ðD5Þ 2 ∂r However, the boundary condition for the differential equation would need to be appropriately modified because, The trace of the stress tensor flows in a known way under a for the relevant sign of μ, we expect the partition function to TT¯ deformation: diverge at a finite value of r.

[1] A. B. Zamolodchikov, Expectation value of composite [4] G. Bonelli, N. Doroud, and M. Zhu, TT¯ deformations in field TT¯ in two-dimensional quantum field theory, closed form, J. High Energy Phys. 06 (2018) 149. arXiv:hep-th/0401146. [5] S. Dubovsky, V. Gorbenko, and G. Hernández-Chifflet, TT¯ [2] F. A. Smirnov and A. B. Zamolodchikov, On space of partition function from topological gravity, J. High Energy integrable quantum field theories, Nucl. Phys. B915, 363 Phys. 09 (2018) 158. (2017). [6] J. Cardy, The TT¯ deformation of quantum field theory as [3] A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, TT¯ - random geometry, J. High Energy Phys. 10 (2018) 186. deformed 2D quantum field theories, J. High Energy Phys. [7] M. Guica, An integrable Lorentz-breaking deformation of 10 (2016) 112. two-dimensional CFTs, SciPost Phys. 5, 048 (2018).

045009-20 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

[8] J. Maldacena, The large-N limit of superconformal field [32] V. E. Hubeny, M. Rangamani, and T. Takayanagi, A theories and supergravity, Int. J. Theor. Phys. 38, 1113 covariant holographic entanglement entropy proposal, (1999). J. High Energy Phys. 07 (2007) 062. [9] P. Kraus, Lectures on black holes and the AdS3=CFT2 [33] H. Casini and M. Huerta, A finite entanglement entropy correspondence, Lect. Notes Phys. 755, 193 (2008). and the c-theorem, Phys. Lett. B 600, 142 (2004). [10] S. Hirano and M. Shigemori, Random boundary geometry [34] H. Casini and M. Huerta, Renormalization group running and gravity dual of TT¯ deformation, arXiv:2003.06300. of the entanglement entropy of a circle, Phys. Rev. D 85, [11] L. McGough, M. Mezei, and H. Verlinde, Moving the 125016 (2012). CFT into the bulk with TT¯, J. High Energy Phys. 04 (2018) [35] H. Liu and M. Mezei, A Refinement of entanglement 010. entropy and the number of degrees of freedom, J. High [12] E. Witten, Anti-de Sitter space and holography, Adv. Energy Phys. 04 (2013) 162. Theor. Math. Phys. 2, 253 (1998). [36] R. C. Myers and A. Sinha, Holographic c-theorems in [13] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge arbitrary dimensions, J. High Energy Phys. 01 (2011) 125. theory correlators from noncritical string theory, Phys. [37] W. Donnelly and V. Shyam, Entanglement Entropy and TT¯ Lett. B 428, 105 (1998). Deformation, Phys. Rev. Lett. 121, 131602 (2018). ¯ ¯ [14] P. Kraus, J. Liu, and D. Marolf, Cutoff AdS3 versus the TT [38] A. Lewkowycz, J. Liu, E. Silverstein, and G. Torroba, TT deformation, J. High Energy Phys. 07 (2018) 027. and EE, with implications for (A)dS subregion encodings, [15] A. Giveon, N. Itzhaki, and D. Kutasov, TT¯ and LST, J. High Energy Phys. 04 (2020) 152. J. High Energy Phys. 07 (2017) 122. [39] B. Chen, L. Chen, and P. x. Hao, Entanglement entropy in [16] A. Giveon, N. Itzhaki, and D. Kutasov, A solvable TT¯-deformed CFT, Phys. Rev. D 98, 086025 (2018). irrelevant deformation of AdS3=CFT2, J. High Energy [40] A. Banerjee, A. Bhattacharyya, and S. Chakraborty, Phys. 12 (2017) 155. Entanglement entropy for TT deformed CFT in general [17] M. Asrat, A. Giveon, N. Itzhaki, and D. Kutasov, Holog- dimensions, Nucl. Phys. B948, 114775 (2019). raphy beyond AdS, Nucl. Phys. B932, 241 (2018). [41] H. S. Jeong, K. Y. Kim, and M. Nishida, Entanglement and ¯ [18] J. Maldacena and H. Ooguri, Strings in AdS3 and the R´enyi entropy of multiple intervals in TT-deformed CFT SL(2,R) WZW model. I: The spectrum, J. Math. Phys. and holography, Phys. Rev. D 100, 106015 (2019). (N.Y.) 42, 2929 (2001). [42] S. Chakraborty, A. Giveon, N. Itzhaki, and D. Kutasov, [19] R. Argurio, A. Giveon, and A. Shomer, Superstrings on Entanglement beyond AdS, Nucl. Phys. B935, 290 ads3 and symmetric products, J. High Energy Phys. 12 (2018). (2000) 003. [43] C. Murdia, Y. Nomura, P. Rath, and N. Salzetta, Com- [20] S. Chakraborty, A. Giveon, and D. Kutasov, TT¯, JT¯, TJ¯ ments on holographic entanglement entropy in TT de- and string theory, J. Phys. A 52, 384003 (2019). formed conformal field theories, Phys. Rev. D 100, 026011 [21] L. Apolo, S. Detournay, and W. Song, TsT, TT¯ and black (2019). strings, J. High Energy Phys. 06 (2020) 109. [44] V. Gorbenko, E. Silverstein, and G. Torroba, dS/dS and [22] M. B. Hastings, An area law for one-dimensional quantum TT¯, J. High Energy Phys. 03 (2019) 085. systems, J. Stat. Mech. (2007) P08024. [45] C. Park, Holographic entanglement entropy in cutoff AdS, [23] C. Holzhey, F. Larsen, and F. Wilczek, Geometric and Int. J. Mod. Phys. A 33, 1850226 (2019). renormalized entropy in conformal field theory, Nucl. [46] M. Asrat, Entropic c-functions in TT;J¯ T;T¯ J¯ deforma- Phys. B424, 443 (1994). tions, arXiv:1911.04618. [24] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entangle- [47] S. He and H. Shu, Correlation functions, entanglement and ment in Quantum Critical Phenomena, Phys. Rev. Lett. 90, chaos in the TT=J¯ T¯-deformed CFTs, J. High Energy Phys. 227902 (2003). 02 (2020) 088. [25] P. Calabrese and J. Cardy, Entanglement entropy and [48] Y. Sun and J. R. Sun, Note on the R´enyi entropy of 2D quantum field theory, J. Stat. Mech. (2004) P06002. perturbed fermions, Phys. Rev. D 99, 106008 (2019). [26] P. Calabrese and J. Cardy, Entanglement entropy and [49] S. Grieninger, Entanglement entropy and TT¯ deformations conformal field theory, J. Phys. A 42, 504005 (2009). beyond antipodal points from holography, J. High Energy [27] M. Levin and X. G. Wen, Detecting Topological Order in a Phys. 09 (2019) 171. Ground State Wave Function, Phys. Rev. Lett. 96, 110405 [50] K. Umemoto and T. Takayanagi, Entanglement of purifi- (2006). cation through holographic duality, Nat. Phys. 14, 573 [28] A. Kitaev and J. Preskill, Topological Entanglement (2018). Entropy, Phys. Rev. Lett. 96, 110404 (2006). [51] P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel, and [29] S. Ryu and T. Takayanagi, Holographic Derivation of B. Swingle, Entanglement of purification: From spin Entanglement Entropy from the Anti-de Sitter Space/ chains to holography, J. High Energy Phys. 01 (2018) 098. Conformal Field Theory Correspondence, Phys. Rev. Lett. [52] J. Kudler-Flam and S. Ryu, Entanglement negativity and 96, 181602 (2006). minimal entanglement wedge cross sections in holographic [30] S. Ryu and T. Takayanagi, Aspects of holographic theories, Phys. Rev. D 99, 106014 (2019). entanglement entropy, J. High Energy Phys. 08 (2006) [53] K. Tamaoka, Entanglement Wedge Cross Section from the 045. Dual Density Matrix, Phys. Rev. Lett. 122, 141601 (2019). [31] M. van Raamsdonk, Building up spacetime with quantum [54] S. Dutta and T. Faulkner, A canonical purification for the entanglement, Gen. Relativ. Gravit. 42, 2323 (2010). entanglement wedge cross section, arXiv:1905.00577.

045009-21 MESERET ASRAT and JONAH KUDLER-FLAM PHYS. REV. D 102, 045009 (2020)

[55] A. Lewkowycz and J. Maldacena, Generalized gravita- [77] P. Bueno and H. Casini, Reflected entropy, symmetries and tional entropy, J. High Energy Phys. 08 (2013) 090. free fermions, J. High Energy Phys. 05 (2020) 103. [56] H. Casini, M. Huerta, R. C. Myers, and A. Yale, Mutual [78] D. Buchholz, C. D’Antoni, and K. Fredenhagen, The information and the F-theorem, J. High Energy Phys. 10 universal structure of local algebras, Commun. Math. (2015) 003. Phys. 111, 123 (1987). [57] D. Buchholz, Product states for local algebras, Commun. [79] T. Grover, A. M. Turner, and A. Vishwanath, Entangle- Math. Phys. 36, 287 (1974). ment entropy of gapped phases and topological order in [58] S. Doplicher and R. Longo, Standard and split inclusions three dimensions, Phys. Rev. B 84, 195120 (2011). of von Neumann algebras, Inventiones Mathematicae 75, [80] T. Araujo, E. O. Colgáin, Y. Sakatani, M. M. Sheikh- 493 (1984). Jabbari, and H. Yavartanoo, Holographic integration of [59] T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum TT¯ and JT¯ via Oðd; dÞ, J. High Energy Phys. 03 (2019) corrections to holographic entanglement entropy, J. High 168. Energy Phys. 11 (2013) 074. [81] M. Banados, C. Teitelboim, and J. Zanelli, Black Hole in [60] N. Engelhardt and A. C. Wall, Quantum extremal surfaces: Three-Dimensional Spacetime, Phys. Rev. Lett. 69, 1849 Holographic entanglement entropy beyond the classical (1992). regime, J. High Energy Phys. 01 (2015) 073. [82] M. Headrick, Entanglement Rényi entropies in holo- [61] X. Dong, The gravity dual of Rényi entropy, Nat. Com- graphic theories, Phys. Rev. D 82, 126010 (2010). mun. 7, 12472 (2016). [83] J. D. Brown and M. Henneaux, Central charges in the [62] X. Dong, A. Lewkowycz, and M. Rangamani, Deriving canonical realization of asymptotic symmetries: An exam- covariant holographic entanglement, J. High Energy Phys. ple from three-dimensional gravity, Commun. Math. Phys. 11 (2016) 028. 104, 207 (1986). [63] X. Dong and A. Lewkowycz, Entropy, extremality, Euclid- [84] H. Narnhofer, Entropy density for relativistic quantum ean variations, and the equations of motion, J. High Energy field theory, Rev. Math. Phys. 06, 1127 (1994). Phys. 01 (2018) 081. [85] B. Schroer, Localization entropy from holography on null [64] B. M. Terhal, M. Horodecki, D. W. Leung, and D. P. surfaces and the split property, arXiv:0712.4403. DiVincenzo, The entanglement of purification, J. Math. [86] B. Schroer, Localization and the interface between Phys. (N.Y.) 43, 4286 (2002). quantum mechanics, quantum field theory and quantum [65] K. Umemoto, Quantum and classical correlations inside gravity I. The two antagonistic localizations and their the entanglement wedge, Phys. Rev. D 100, 126021 asymptotic compatibility, Stud. Hist. Phil. Mod. Phys. 41, (2019). 104 (2010). [66] J. Levin, O. DeWolfe, and G. Smith, Correlation measures [87] H. Narnhofer, Entanglement, split and nuclearity in quan- and distillable entanglement in AdS=CFT, Phys. Rev. D tum field theory, Rep. Math. Phys. 50, 111 (2002). 101, 046015 (2020). [88] H. Narnhofer, Entropy for quantum fields in curved [67] J. Eisert and M. B. Plenio, A comparison of entanglement space time, Classical Quantum Gravity 28, 145016 measures, J. Mod. Opt. 46, 145 (1999). (2011). [68] G. Vidal and R. F. Werner, Computable measure of [89] Y. Otani and Y. Tanimoto, Toward entanglement entropy entanglement, Phys. Rev. A 65, 032314 (2002). with UV-cutoff in conformal nets, Ann. Henri Poincaré 19, [69] M. B. Plenio, Logarithmic Negativity: A Full Entangle- 1817 (2018). ment Monotone That is Not Convex, Phys. Rev. Lett. 95, [90] R. Longo and F. Xu, Von Neumann entropy in QFT, 090503 (2005). Commun. Math. Phys. (2020). [70] Y. Kusuki, J. Kudler-Flam, and S. Ryu, Derivation of [91] T. Nishioka and T. Takayanagi, AdS bubbles, entropy and Holographic Negativity in AdS3=CFT2, Phys. Rev. Lett. closed string tachyons, J. High Energy Phys. 01 (2007) 123, 131603 (2019). 090. [71] J. Kudler-Flam, Y. Kusuki, and S. Ryu, Correlation [92] I. R. Klebanov, D. Kutasov, and A. Murugan, Entangle- measures and the entanglement wedge cross-section after ment as a probe of confinement, Nucl. Phys. B796, 274 quantum quenches in two-dimensional conformal field (2008). theories, J. High Energy Phys. 04 (2020) 074. [93] P. F. Byrd and M. D. Friedman, Handbook of Elliptic [72] Y. Kusuki and K. Tamaoka, Dynamics of entanglement Integrals for Engineers and Scientists (Springer-Verlag, wedge cross section from conformal field theories, arXiv: New York, 1971). 1907.06646. [94] S. Datta and Y. Jiang, TT¯ deformed partition functions, [73] Y. Kusuki and K. Tamaoka, Entanglement wedge cross J. High Energy Phys. 08 (2018) 106. section from CFT: Dynamics of local operator quench, [95] A. Hashimoto and D. Kutasov, Strings, symmetric prod- J. High Energy Phys. 02 (2020) 017. ucts, TT¯ deformations and Hecke operators, Phys. Lett. B [74] C. Akers and P. Rath, Entanglement wedge cross sections 806, 135479 (2020). require tripartite entanglement, J. High Energy Phys. 04 [96] A. Hashimoto and D. Kutasov, TT;J¯ T¯, TJ¯ partition (2020) 208. sums from string theory, J. High Energy Phys. 02 (2020) [75] R. Longo and F. Xu, Von Neumann entropy in QFT, 080. arXiv:1911.09390. [97] O. Aharony, S. Datta, A. Giveon, Y. Jiang, and D. Kutasov, [76] R. Haag, Local Quantum Physics: Fields, Particles, Modular covariance and uniqueness of JT¯ deformed CFTs, Algebras (Springer-Verlag, Berlin, 1992). J. High Energy Phys. 01 (2019) 085.

045009-22 TT¯, THE ENTANGLEMENT WEDGE CROSS … PHYS. REV. D 102, 045009 (2020)

[98] O. Aharony, S. Datta, A. Giveon, Y. Jiang, and D. Kutasov, [102] M. Beşken, S. Datta, and P. Kraus, Semiclassical Virasoro Modular invariance and uniqueness of TT¯ deformed CFT, blocks: Proof of exponentiation, J. High Energy Phys. 01 J. High Energy Phys. 01 (2019) 086. (2020) 109. ¯ [99] A. Ireland and V. Shyam, TT deformed YM2 on general [103] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters, Virasoro backgrounds from an integral transformation, J. High conformal blocks and thermality from classical back- Energy Phys. 07 (2020) 058. ground fields, J. High Energy Phys. 11 (2015) 200. [100] P. Caputa, S. Datta, and V. Shyam, Sphere partition fun- [104] M. Guica and R. Monten, TT¯ and the mirage of a bulk ctions and cutoff AdS, J. High Energy Phys. 05 (2019) 112. cutoff, arXiv:1906.11251. [101] A. B. Zamolodchikov, Conformal symmetry in two dimen- [105] M. Goldstein and E. Sela, Symmetry-Resolved Entangle- sions: An explicit recurrence formula for the conformal ment in Many-Body Systems, Phys. Rev. Lett. 120,200602 partial wave amplitude, Commun. Math. Phys. 96,419 (2018). (1984).

045009-23