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arXiv:2310.02096 · Journal of High Energy Physics

Rényi entropy with surface defects in six dimensions

Ma-Ke Yuan, Yang Zhou

Published Mar 6, 2024• 3 citations
arXivHTMLPDF

Abstract

A bstract We compute the surface defect contribution to Rényi entropy and supersymmetric Rényi entropy in six dimensions. We first compute the surface defect contribution to Rényi entropy for free fields, which verifies a previous formula about entanglement entropy with surface defect. Using conformal map to $$ {S}_{\beta}^1\times {H}^{d-1} $$ S β 1 × H d − 1 we develop a heat kernel approach to compute the defect contribution to Rényi entropy, which is applicable for p -dimensional defect in general d -dimensional free fields. Using the same geometry $$ {S}_{\beta}^1\times {H}^5 $$ S β 1 × H 5 with an additional background field, one can construct the supersymmetric refinement of the ordinary Rényi entropy for six-dimensional (2, 0) theories. We find that the surface defect contribution to supersymmetric Rényi entropy has a simple scaling as polynomial of Rényi index in the large N limit. We also discuss how to connect the free field results and large N results.

DOI
10.1007/jhep03(2024)031

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A bstract We compute the surface defect contribution to Rényi entropy and supersymmetric Rényi entropy in six dimensions. We first compute the surface defect contribution to Rényi entropy for free fields, which verifies a previous formula about entanglement entropy with surface defect. Using conformal map to Sβ1×Hd−1{S}_{\beta}^1\times {H}^{d-1}Sβ1​×Hd−1 S β 1 × H d − 1 we develop a heat kernel approach to compute the defect contribution to Rényi entropy, which is applicable for p -dimensional defect in general d -dimensional free fields. Using the same geometry Sβ1×H5{S}_{\beta}^1\times {H}^5Sβ1​×H5 S β 1 × H 5 with an additional background field, one can construct the supersymmetric refinement of the ordinary Rényi entropy for six-dimensional (2, 0) theories. We find that the surface defect contribution to supersymmetric Rényi entropy has a simple scaling as polynomial of Rényi index in the large N limit. We also discuss how to connect the free field results and large N results.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.
We propose a field theoretic framework for calculating the dependence of Rényi entropies on the shape of the entangling surface in a conformal field theory. Our approach rests on regarding the corresponding twist operator as a conformal defect and in particular, we define the displacement operator which implements small local deformations of the entangling surface. We identify a simple constraint between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which consolidates a number of distinct conjectures on the shape dependence of the Rényi entropy. As an example, using this approach, we examine a conjecture regarding the universal coefficient associated with a conical singularity in the entangling surface for CFTs in any number of spacetime dimensions. We also provide a general formula for the second order variation of the Rényi entropy arising from small deformations of a spherical entangling surface, extending Mezei’s results for the entanglement entropy.
We describe a holographic approach to explicitly computing the universal logarithmic contributions to entanglement and Rényi entropies for free conformal scalar and spinor fields on even-dimensional spheres. This holographic derivation proceeds in two steps: first, following Casini and Huerta, a conformal mapping to thermal entropy in a hyperbolic geometry; then identification of the hyperbolic geometry with the conformal boundary of a bulk hyperbolic space and use of an AdS/CFT holographic formula to compute the resultant functional determinant. We explicitly verify the connection with the type-A trace anomaly for the entanglement entropy, whereas the Rényi entropy is computed with the aid of the Sommerfeld formula in order to deal with a conical defect. We show that as a by-product, the log coefficient of the Rényi entropy for round spheres can be efficiently obtained as the q-analog of a procedure similar to the one found by Cappelli and D’Appollonio that rendered the type-A trace anomaly.
We study the defect contribution to the entanglement entropy of a spherical region centred on four-dimensional superconformal RG interfaces. These interfaces are supersymmetric codimension one defects separating the Leigh--Strassler SCFT vacuum and the N=4\mathcal{N}=4N=4 SYM theory deformed by spatially varying mass terms. Exploiting the holographic dual to these interfaces, we extract a cutoff-independent interface contribution to the entanglement entropy, CI\mathcal{C}^ICI. We quantify the non-trivial dependence of CI\mathcal{C}^ICI on the supersymmetric mass deformation, and observe a scaling regime for large deformations. We further relate CI\mathcal{C}^ICI to the renormalized on-shell action and the stress-tensor one-point function of the interface theory. Our results provide the first fully backreacted holographic computation of this entanglement quantity for superconformal RG interfaces in more than two dimensions.
We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D≥3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c theorem, as in 2D CFT. For a 2D defect, we show that another, d_{2}, interpreted as the defect's "conformal dimension," must be non-negative if the averaged null energy condition holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d_{2}. Using this and known holographic results, we compute b and d_{2} for 1/2-Bogomol'nyi-Prasad-Sommerfield surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.
We show that in any two dimensional conformal field theory with (2, 2) super-symmetry one can define a supersymmetric analog of the usual Renyi entropy of a spatial region A. It differs from the Renyi entropy by a universal function (which we compute) of the central charge, Renyi parameter n and the geometric parameters of A. In the limit n → 1 it coincides with the entanglement entropy. Thus, it contains the same information as the Renyi entropy but its computation only involves correlation functions of chiral and anti-chiral operators. We also show that this quantity appears naturally in string theory on AdS 3.