Cubic trihedral corner entanglement for a free scalar

Abstract

We calculate the universal contribution to the α-Rényi entropy from a cubic trihedral corner in the boundary of the entangling region in 3 + 1 dimensions for a massless free scalar. The universal number, vα, is manifest as the coefficient of a scaling term that is logarithmic in the size of the entangling region. Our numerical calculations find that this universal coefficient has both larger magnitude and the opposite sign to that induced by a smooth spherical entangling boundary in 3 + 1 dimensions, for which there is a well-known subleading logarithmic scaling. Despite these differences, up to the uncertainty of our finite-size lattice calculations, the functional dependence of the trihedral coefficient vα on the Rényi index α is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this α-dependence arising from the general structure of (3 + 1)-dimensional conformal field theories, and suggest calculations past the free scalar which could further illuminate the general structure of the trihedral divergence in the Rényi entropy.

Generalizing the entanglement entropy of singular regions in conformal field theories

“…1-for which k = 0. This explains why, as previously observed [58][59][60][61][62], trihedral corners have a universal log δ divergence, instead of a log 2 δ one. The above analysis also reveals that, in the case of polyhedral corners, the corresponding logarithmic contribution will not come from some simple local integral along any curve on S 2 .…”

“…Let us close the paper with some final words regarding a few possible directions. One of our main motivations was trying to gain a better understanding on the nature of the trihedral universal coefficient v n (θ 1 , θ 2 , θ 3 ), previously studied using lattice techniques in [58][59][60][61], and analytically in the nearly smooth limit [62]. Our results indicate that an analytic computation of this coefficient at general angles for free fields would be equivalent to evaluating partition functions on a S 3 with multiplicative boundary conditions on a two-dimensional spherical triangle.…”

“…In particular, this geometry is natural in lattice simulations. In that case, it has been observed that the universal contribution is not quadratically logarithmic, but just logarithmic, like in the case of smooth entangling regions, namely [58][59][60][61][62] S (4) polyhedral…”

“…will generally be a highly non-local term, and there is no reason to expect it to be controlled by a linear combination of trace-anomaly coefficients (or their Rényi entropy generalizations, f a (n) and f b (n)) as suggested in [60,61]. This observation, while relying on a free-field calculation, should be valid for general CFTs.…”

“…In [60,61], the idea that v n (θ 1 , θ 2 , · · · , θ j ) might be controlled by a simple linear combination of the functions f a (n) and f b (n) was put forward, although a definite conclusion was not reached. Here, we will use free-field calculations to understand the true origin of v n (θ 1 , θ 2 , · · · , θ j ), and rule out this possibility.…”