We prove a conjectured lower bound on T −− (x) ψ in any state ψ of a relativistic QFT dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a "probe" operator O in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality in modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point. We also prove a higher spin version of the QNEC.ArXiv ePrint: 1706.09432 arXiv:1706.09432v2 [hep-th]

We give a recipe for computing correlation functions of the displacement operator localized on a spherical or planar higher dimensional twist defect using AdS/CFT. Such twist operators are typically used to construct the n'th Renyi entropies of spatial entanglement in CFTs and are holographically dual to black holes with hyperbolic horizons. The displacement operator then tells us how the Renyi entropies change under small shape deformations of the entangling surface. We explicitly construct the bulk to boundary propagator for the displacement operator insertion as a linearized metric fluctuation of the hyperbolic black hole and use this to extract the coefficient of the displacement operator two point function CD in any dimension. The n → 1 limit of the twist displacement operator gives the same bulk response as the insertion of a null energy operator in vacuum, which is consistent with recent results on the shape dependence of entanglement entropy and modular energy.Introduction. The study of entanglement in quantum field theories (QFT) has led to new perspectives on strongly correlated phenomena. Some recent applications include insight into the renormalization group flow in the space of QFTs [1][2][3], the dynamics of excited states [4][5][6], topological phases [7][8][9], emergent space-time and gravitational dynamics hidden behind holographic dualities [10][11][12] and proofs of energy conditions in QFTs [13][14][15]. Conformal field theories (CFTs) offer hope for studying entanglement in strongly correlated systems where the available symmetries fit nicely with entanglement computations [16][17][18]. CFTs live at quantum critical points where long range entanglement is a basic characteristic [19,20]. Indeed the spatial Renyi entropies, the entanglement measure we plan to study in this paper, can even be studied in the lab [21] and are often employed in numerical modeling [22], providing signatures of quantum phase transitions and topological phases. All of the above motivates development of theoretical tools for the study of Renyi entropies in CFTs and in this paper we will add to already available holographic results [23][24][25][26][27][28] to meet this goal.The Renyi entanglement entropies between a spatial region A and its complement can be constructed via the replica trick which computes Trρ n A for integer n by considering a product orbifold CFT n /Z n and introducing a co-dimension-2 cyclic twist defect on the boundary ∂A separating the spatial regions. An analytic continuation away from integer n for correlation functions involving twist defects is often assumed, although is sometimes subtle [29,30]. To exploit the maximal symmetries of the problem it has been recently suggested [31] that we should use the full technology of defect CFTs (dCFTs). Indeed the theoretical study of CFTs, including dCFTs [32][33][34], in higher than 2 dimensions has recently undergone a renaissance [35]. We might hope to exploit as many of these results as possible in the study of entanglement.In this paper we plan to s...

We conjecture that the asymptotic behavior of the numbers of solid (three-dimensional) partitions is identical to the asymptotics of the threedimensional MacMahon numbers. Evidence is provided by an exact enumeration of solid partitions of all integers ≤ 68 whose numbers are reproduced with surprising accuracy using the asymptotic formula (with one free parameter) and better accuracy on increasing the number of free parameters. We also conjecture that similar behavior holds for higherdimensional partitions and provide some preliminary evidence for four and five-dimensional partitions. * suresh@physics.iitm.ac.inThe purpose of computation is insight, not numbers. -Richard Hamming

We study half-space/Rindler modular Hamiltonians for excited states created by turning on sources for local operators in the Euclidean path integral in relativistic quantum field theories. We derive a simple, manifestly Lorentzian formula for the modular Hamiltonian to all orders in perturbation theory in the sources. We apply this formula to the case of shapedeformed half spaces in the vacuum state, and obtain the corresponding modular Hamiltonian to all orders in the shape deformation in terms of products of half-sided null energy operators, i.e., stress tensor components integrated along the future and past Rindler horizons. In the special case where the shape deformation is purely null, our perturbation series can be resummed, and agrees precisely with the known formula for vacuum modular Hamiltonians for null cuts of the Rindler horizon. Finally, we study some universal properties of modular flow (corresponding to Euclidean path integral states) of local operators inside correlation functions in conformal field theories. In particular, we show how the flow becomes the local boost in the limit where the operator being flowed approaches the entanglement cut.

We study the defect operator product expansion (OPE) of displacement operators in free and interacting conformal field theories using replica methods. We show that as n approaches 1 a contact term can emerge when the OPE contains defect operators of twist d − 2. For interacting theories and general states we give evidence that the only possibility is from the defect operator that becomes the stress tensor in the n → 1 limit. This implies that the quantum null energy condition (QNEC) is always saturated for CFTs with a twist gap.As a check, we show independently that in a large class of near vacuum states, the second variation of the entanglement entropy is given by a simple correlation function of averaged null energy operators as studied by Hofman and Maldacena. This suggests that sub-leading terms in the the defect OPE are controlled by a defect version of the spin-3 non-local light ray operator and we speculate about the possible origin of such a defect operator. For free theories this contribution condenses to a contact term that leads to violations of QNEC saturation.