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arXiv:2307.05164 · Physical Review Letters

Conformal Bounds in Three Dimensions from Entanglement Entropy

Pablo Bueno, Horacio Casini, Óscar Lasso Andino, Javier Moreno

Published Oct 25, 2023• 11 citations
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Abstract

The entanglement entropy of an arbitrary spacetime region A in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, F(A). For general theories, the value of F(A) is minimized when A is a round disk, F_{0}, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity F(A)/F_{0} is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies C_{T}/F_{0}≤3/(4π^{2}log2-6ζ[3])≃0.14887 for general CFTs. We verify the validity of this bound for free scalars and fermions, general O(N) and Gross-Neveu models, holographic theories, N=2 Wess-Zumino models and general ABJM theories.

DOI
10.1103/physrevlett.131.171601

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The entanglement entropy of an arbitrary spacetime region A in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, F(A). For general theories, the value of F(A) is minimized when A is a round disk, F_{0}, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity F(A)/F_{0} is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies C_{T}/F_{0}≤3/(4π^{2}log2-6ζ[3])≃0.14887 for general CFTs. We verify the validity of this bound for free scalars and fermions, general O(N) and Gross-Neveu models, holographic theories, N=2 Wess-Zumino models and general ABJM theories.
A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, (−1)d−12 F(A) {\left(-1\right)}^{\frac{d-1}{2}}\ F(A) (−1)2d−1​ F(A) − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.
Abstract The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, −F ⊂ SEE. For a disk region, F|disk ≡ F0 coincides with the free energy on 𝕊3 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 nB boundaries, the general bound can be improved to F ≥ nBF0. 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.
We propose bounds relating thermal entropy to entanglement in unitary conformal field theories: f≤4π2af\le4π^2af≤4π2a in four dimensions and f≤2.99830949… Ff\le2.99830949\ldots\,Ff≤2.99830949…F in three dimensions. Here fff is the dimensionless coefficient of the cosmological constant term in the thermal effective action, aaa is the Euler anomaly coefficient, and FFF is the sphere free energy. Spherical Rényi entropy relates the four-dimensional bound to the conformal collider bounds. We formulate a stronger conjectural inequality involving the stress-tensor three-point function, saturated by every free theory, and establish it in controlled regimes of weakly coupled gauge theories, large-NNN vector models, and holographic CFTs. Finite-NNN numerical results provide additional tests. The bounds constrain the Cardy growth of the density of states at high energy.
The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function a(θ) when the entangling surface contains a sharp corner with opening angle θ. In the limit of a smooth surface (θ → π), this corner contribution vanishes as a(θ) = σ(θ − π)2. In arXiv:1505.04804 , we provided evidence for the conjecture that for any d = 3 CFT, this corner coefficient σ is determined by C T , the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is an instance of a much more general relation connecting the analogous corner coefficient σ n appearing in the nth Rényi entropy and the scaling dimension h n of the corresponding twist operator. In particular, we find the simple relation h n /σ n = (n − 1)π. We show how it reduces to our previous result as n → 1, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as n → 0, σ n yields the coefficient of the thermal entropy, c S . We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict σ n for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval Rényi entropies of d = 2 CFTs.
We study the contribution to the entanglement entropy of (2+1)-dimensional conformal field theories (CFTs) coming from a sharp corner in the entangling surface. This contribution is encoded in a function a(θ) of the corner opening angle, and was recently proposed as a measure of the degrees of freedom in the underlying CFT. We show that the ratio a(θ)/C(T), where C(T) is the central charge in the stress tensor correlator, is an almost universal quantity for a broad class of theories including various higher-curvature holographic models, free scalars, and fermions, and Wilson-Fisher fixed points of the O(N) models with N=1,2,3. Strikingly, the agreement between these different theories becomes exact in the limit θ→π, where the entangling surface approaches a smooth curve. We thus conjecture that the corresponding ratio is universal for general CFTs in three dimensions.
The entanglement entropy of a generic d-dimensional conformal field theory receives a regulator independent contribution when the entangling surface contains a (hyper)conical singularity of opening angle Ω, codified in a function a (d)(Ω). In arXiv:1505.04804 , we proposed that for three-dimensional conformal field theories, the coefficient σ (3) characterizing the limit where the surface becomes smooth is proportional to the central charge C T appearing in the two-point function of the stress tensor. In this paper, we prove this relation for general three-dimensional holographic theories, and extend the result to general dimensions. In particular, we define a generalized coefficient σ (d) to characterize the almost smooth limit of a (hyper)conical singularity in entangling surfaces in higher dimensions. We show then that this coefficient is universally related to C T for general holographic theories and provide a general formula for the ratio σ (d) /C T in arbitrary dimensions. We conjecture that the latter ratio is universal for general CFTs. Further, based on our recent results in arXiv:1507.06997 , we propose an extension of this relation to general Rényi entropies, which we show passes several consistency checks in d = 4 and 6.