Two-cylinder entanglement entropy under a twist

We study the von Neumann and Renyi entanglement entropy (EE) of the scale-invariant theories defined on the tori in 2 + 1 and 3 + 1 spacetime dimensions. We focus on the spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), the fermionic quadratic band touching and the boson with z = 2 Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find the non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk.