Rényi entropy and conformal defects

Bianchi, Lorenzo; Meineri, Marco; Myers, Robert C.; Smolkin, Michael

doi:10.1007/jhep07(2016)076

Cited by 100 publications

(207 citation statements)

References 69 publications

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“…The key ingredient in the study is the OPE of spherical twist operator [58][59][60] in terms of the primary operators in the replicated theory. This allows the systematic study of the mutual information.…”

Section: Jhep06(2017)096mentioning

confidence: 99%

On the mutual information in conformal field theory

Chen

Hao

et al. 2017

J. High Energ. Phys.

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Abstract:In this work, we study the universal behaviors in the mutual information of two disjoint spheres in a conformal field theory (CFT). By using the operator product expansion of the spherical twist operator in terms of the conformal family, we show that the large distance expansion of the mutual information can be cast in terms of the conformal blocks. We develop the 1/n prescription to compute the coefficients before the conformal blocks. For a single conformal family, the leading nonvanishing contribution to the mutual information comes from the bilinear operators. We show that the coefficients of these operators take universal forms and such universal behavior persists in the bilinear operators with derivatives as well. Consequently the first few leading order contributions to the mutual information in CFT take universal forms. To illustrate our framework, we discuss the free scalars and free fermions in various dimensions. For the free scalars, we compute the mutual information to the next-to-leading order and find good agreement with the improved numerical lattice result. For the free fermion, we compute the leading order result, which is of universal form, and find the good match with the numerical study. Our formalism could be applied to any CFT potentially.

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Section: Jhep06(2017)096mentioning

confidence: 99%

On the mutual information in conformal field theory

Chen

Hao

et al. 2017

J. High Energ. Phys.

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“…This conjecture has passed numerical test for free scalar and free fermion [17]. According to [12], it seems that the relation (1.8) holds only for free CFTs. Evidence includes an analytic proof for free scalar.…”

Section: Jhep12(2016)036mentioning

confidence: 92%

“…It is found that entanglement plays an important role in the emergence of space-time and gravitational dynamics [1][2][3][4][5]. In addition to entanglement entropy, Rényi entropy has drawn much attention recently, including the holographic formula of Rényi entropy [6,7], the shape dependence of Rényi entropy [8][9][10], the holographic dual of boundary cones [11] and Rényi twist displacement operator [12,13].…”

Section: Jhep12(2016)036mentioning

confidence: 99%

“…The coefficient f a can be obtained by studying the thermal free energy of CFTs on a hyperboloid [6]. The coefficients f c and f b are determined by the stress tensor onepoint function and two-point function on the hyperboloid background [12,16]. Remarkably, it is found in [16] that f c is completely determined by f a :…”

Section: Jhep12(2016)036mentioning

confidence: 99%

“…In the notation of [12], our conjecture (1.12) for 4d CFTs can be written in the form 14) where C T = 40 π 4 c, h n (n) and C D (n) are CFT data associated with the presence of the entangling surface. In general, for a d-dimensional CFT and an entangling surface Σ 1 We have assume f ′′ b (1) = f ′′ c (1) for free fermions and vectors.…”

Section: Jhep12(2016)036mentioning

confidence: 99%

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Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity

Chu

Miao

2016

J. High Energ. Phys.

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We consider higher derivative gravity and obtain universal relations for the shape coefficients (f a , f b , f c ) of the shape dependent universal part of the Rényi entropy for four dimensional CFTs in terms of the parameters (c, t 2 , t 4 ) of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients f a and f c satisfy the differential relation as derived previously for the Rényi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of f a for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we find a combination of our holographic relations which are satisfied by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we find there are similar universal laws for holographic Rényi entropy in general dimensions.

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Exact results in defect conformal field theories**

Lauria

2016

Fortschritte der Physik

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Defect Conformal Field Theories describe critical points of Quantum Field Theories excited or modified in the neighbourhood of a large p-dimensional submanifold. We elucidate the constraints of conformal invariance on correlation functions and we provide both recurrence relations, and in some cases exact results, for the conformal blocks with scalar external operators. MotivationsConformal defects appear in a variety of situations of theoretical and phenomenological interest. Boundaries and interfaces [14,15] provide a prototypical example of codimension one conformal defects, while Wilson and 't Hooft operators [12], as well as surface operators [5,11] are examples of higher codimension defects, which play a role in gauge theories. Low energy example of defects include vortices [6], magnetic-like impurities in spin systems [3], localized particles acting as sources for the order parameter of some bosonic system [1], higher dimensional descriptions of theories with long range interactions [16] etc. Moreover, conformal defects provide a useful tool to study the quantum Entanglement [2]. Given the importance of conformal defect in physics, our main goal [4] is to provide a toolbox to compute correlation functions and conformal blocks for CFT with conformal defects. The latter are important to generalize the conformal bootstrap program, recently extended to codimension one defects [8,10,13], to higher codimension defects. Correlation functions and CFT dataWe consider correlation functions of local operators constrained by SO(p + 1, 1) × SO(q) symmetry. This is the symmetry preserved by a flat p−dimensional extended operator inserted in R d , which is usually denoted as a conformal defect (of codimension q ≡ d − p).

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Rényi entropy and conformal defects

Cited by 100 publications

References 69 publications

On the mutual information in conformal field theory

On the mutual information in conformal field theory

Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity

Exact results in defect conformal field theories**

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