Exact results for corner contributions to the entanglement entropy and Renyi entropies of free bosons and fermions in 3d

TL;DR

This paper derives exact analytical results for corner contributions to entanglement and Renyi entropies in 3d free boson and fermion theories, confirming some universality conjectures and revealing new relationships.

Contribution

It provides the first exact integral evaluations for corner coefficients in 3d free theories for all Renyi indices, confirming a universal ratio for the entanglement entropy.

Findings

The corner coefficient for EE matches the conjectured universal ratio with the central charge.

Exact formulas for corner contributions to Renyi entropies are obtained for all n.

Asymptotic Renyi corner contributions relate to free energy on the n-covered sphere.

Abstract

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Renyi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient $\sigma$ for the EE and $\sigma_n$ for the Renyi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for $\sigma$ and $\sigma_n$ for all $n=2,3,\dots$. The results for $\sigma$ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that $\sigma/C_T = \pi^2/24$ in all 3d CFTs, where $C_T$ is the central charge. For the Renyi entropies, the ratios $\sigma_n/C_T$ do not indicate similar universality. However, in the limit $n \to \infty$, the asymptotic values satisfy a simple relationship and equal $1/(4\pi^2)$ times the asymptotic values of the free energy of free scalars/fermions on the $n$-covered 3-sphere.