Disks globally maximize the entanglement entropy in $2+1$ dimensions

Abstract

The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, $-F\subset S_{\text{EE}}$. For a disk region, $F{|}_{\mathrm{disk}}\equiv F_0$ coincides with the free energy on ${\mathbb{S}}^3$ and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of $F$ generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that $F$ is precisely minimized by disks. Here, we argue that $F$ is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with $n_B$ boundaries, the general bound can be improved to $F\ge n_B{F}_0$. In addition, we provide accurate approximations to $F$ valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate $F$ numerically for more general shapes in the so-called "Extensive Mutual Information model", verifying the general bound.

arXiv comment38 pages, 10 figures. v2: some new references, comments and improved discussions added; matches version to appear in JHEP

References in corpus (51)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Aspects of Holographic Entanglement Entropy
• A Covariant Holographic Entanglement Entropy Proposal
• Towards a derivation of holographic entanglement entropy
• Generalized gravitational entropy
• Calculation of reduced density matrices from correlation functions
• Quantum corrections to holographic entanglement entropy
• Implications of Conformal Invariance in Field Theories for General Dimensions
• Entanglement entropy in free quantum field theory
• On the RG running of the entanglement entropy of a circle
• Entanglement entropy, conformal invariance and extrinsic geometry
• Entanglement and alpha entropies for a massive Dirac field in two dimensions
• D=4 Super Yang Mills, D=5 gauged supergravity and D=4 conformal supergravity
• A refinement of entanglement entropy and the number of degrees of freedom
• F-Theorem without Supersymmetry
• Universal terms for the entanglement entropy in 2+1 dimensions
• Remarks on the entanglement entropy for disconnected regions
• Universality of corner entanglement in conformal field theories
• AdS/CFT and Strong Subadditivity of Entanglement Entropy
• Markov property of the CFT vacuum and the a-theorem
• Modular Hamiltonians on the null plane and the Markov property of the vacuum state
• Entanglement and alpha entropies for a massive scalar field in two dimensions
• Entanglement entropy across a deformed sphere
• Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories
• Some results on the shape dependence of entanglement and Rényi entropies
• Universal corner entanglement from twist operators
• Exact and Numerical Results on Entanglement Entropy in (5+1)-Dimensional CFT
• Entanglement Renyi Entropies in Conformal Field Theories and Holography
• On shape dependence of holographic mutual information in AdS4
• Area terms in entanglement entropy
• Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice
• Positivity, negativity, and entanglement
• Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?
• Comments on Squashed-sphere Partition Functions
• Entanglement entropy for odd spheres
• What surface maximizes entanglement entropy?
• On shape dependence of holographic entanglement entropy in AdS4/CFT3
• Einstein-AdS action, renormalized volume/area and holographic Renyi entropies
• Entropy localization and extensivity in the semiclassical black hole evaporation
• Generalizing the entanglement entropy of singular regions in conformal field theories
• Entanglement Entropy of Annulus in Three Dimensions
• Cornering gapless quantum states via their torus entanglement
• Shape dependence of renormalized holographic entanglement entropy
• On Universality of Holographic Results for (2+1)-Dimensional CFTs on Curved Spacetimes
• Renormalized holographic entanglement entropy for Quadratic Curvature Gravity
• Renormalized entanglement entropy and curvature invariants
• What spatial geometry does the (2+1)-dimensional QFT vacuum prefer?
• Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds
• Does the Round Sphere Maximize the Free Energy of (2+1)-Dimensional QFTs?
• Geometric Flow Description of Minimal Surfaces
• Holographic entanglement entropy for hollow cones and banana shaped regions

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