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.
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 extend previous work on the perturbative expansion of the Rényi entropy, S q , around q = 1 for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Rényi entropies. On the other hand, the perturbative results agree with known Rényi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Rényi entropy of N=4 super-Yang-Mills near q = 1. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.
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.
A bstract We study the structure of divergences and universal terms of the entanglement and Rényi entropies for singular regions. First, we show that for (3 + 1)-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution Snuniv=−8π1fb(n)∫γk2log2(R/δ) S n univ = − 1 8 π f b n ∫ γ k 2 log 2 R / δ , where γ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and k the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the (2 + 1)-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces.
A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFTs in even dimensions. In odd dimensions, the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a nontrivial integrated anomaly if an odd-dimensional spacetime has boundaries. We show that, similarly, there exists a logarithmic term in the entanglement entropy when the entangling surface crosses the boundary of spacetime. The relation of the entanglement entropy to the integrated conformal anomaly is elaborated for three-dimensional theories. Distributional properties of intrinsic and extrinsic geometries of the boundary in the presence of conical singularities in the bulk are established. This allows one to find contributions to the entropy that depend on the relative angle between the boundary and the entangling surface.