Universal divergence of the Renyi entropy of a thinly sliced torus at the Ising fixed point

The entanglement entropy of a quantum critical system can provide new universal numbers that depend on the geometry of the entangling bipartition. We calculate a universal number called $\kappa$, which arises when a quantum critical system is embedded on a two-dimensional torus and bipartitioned into two cylinders. In the limit when one of the cylinders is a thin slice through the torus, $\kappa$ parameterizes a divergence that occurs in the entanglement entropy sub-leading to the area law. Using large-scale Monte Carlo simulations of an Ising model in 2+1 dimensions, we access the second Renyi entropy, and determine that, at the Wilson-Fisher (WF) fixed point, $\kappa_{2,\text{WF}} = 0.0174(5)$. This result is significantly different from its value for the Gaussian fixed point, known to be $\kappa_{2,\text{Gaussian}} \approx 0.0227998$.