The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function
a(θ) when the entangling surface contains a sharp corner with opening angle
θ. In the limit of a smooth surface (
θ→π), this corner contribution vanishes as
a(θ)=σ(θ−π)2. In arXiv:1505.04804, we provided evidence for the conjecture that for any
d=3 CFT, this corner coefficient
σ is determined by
CT, the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is a particular instance of a much more general relation connecting the analogous corner coefficient
σn appearing in the
nth Rényi entropy and the scaling dimension
hn of the corresponding twist operator. In particular, we find the simple relation
hn/σn=(n−1)π. We show how it reduces to our previous result as
n→1, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as
n→0,
σn yields the coefficient of the thermal entropy,
cS. We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict
σn for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval Rényi entropies of
d=2 CFTs.