Generalizing the entanglement entropy of singular regions in conformal field theories

Daniel Jaud

Pablo Bueno, Horacio Casini, William Witczak‐Krempa

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 $$ {S}_n^{\mathrm{univ}}=-\frac{1}{8\pi }{f}_b(n){\int}_{\gamma }{k}^2{\log}^2\left(R/\delta \right) $$ 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.

Mostafa Ghasemi, Shahrokh Parvizi

We study the entanglement entropy of theories that are derived from relevant perturbation of given CFTs for regions with a singular boundary by using the AdS/CFT correspondence. In the smooth case, it is well known that a relevant deformation of the boundary theory by the relevant operator with scaling dimension $Δ=\frac{d+2}{2}$ generates a logarithmic universal term to the entanglement entropy. As the smooth case, when the boundary CFT deformed by a relevant operator, we find that the entanglement entropy of singular surface also contains a new logarithmic term which is due to relevant perturbation of the conformal field theory, and depends on the scaling dimension of relevant operator. We also find for extended singular surfaces, $c_{n}\times R^{m}$, as well as logarithmic term, the new universal double logarithmic terms may appear depending on the scaling dimension of relevant operator and spacetime dimensions. These new terms are due to relevant perturbation of the boundary theory.

K. Sakai

Seth Whitsitt, William Witczak‐Krempa, Subir Sachdev A

We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with $\mathrm{O}(N)$ symmetry in the limit of large $N$ for general entanglement geometries. We show that the leading large-$N$ result can be obtained from the entanglement entropy of $N$ Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of $N$ massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=\ensuremath{\infty}$. Notably, for a semi-infinite cylindrical region it scales as ${N}^{0}$ in the Wilson-Fisher theory, in stark contrast to the $N$-linear result of the Gaussian fixed point.

Benjamin Hsu, Michael Mulligan, Eduardo Fradkin +1 more