# Cubic trihedral corner entanglement for a free scalar

## TL;DR

This paper computes the universal logarithmic contribution to the Rènyi entropy from a cubic trihedral corner in a 3+1D free scalar field, revealing unique magnitude and sign properties compared to spherical boundaries.

## Contribution

It provides the first numerical calculation of the universal coefficient for a trihedral corner in 3+1D free scalar fields, highlighting its distinct magnitude and sign from smooth boundaries.

## Key findings

- Universal coefficient has larger magnitude and opposite sign compared to spherical boundary.
- The functional dependence on Rènyi index α matches that of a sphere.
- Results suggest a universal structure in the α-dependence of corner contributions.

## Abstract

We calculate the universal contribution to the $\alpha$-Renyi 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, $v_{\alpha}$, 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 $v_{\alpha}$ on the R\'enyi index $\alpha$ is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this $\alpha$-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 R\'enyi entropy.