Entanglement scaling in two-dimensional gapless systems

We numerically determine subleading scaling terms in the ground-state entanglement entropy of several two-dimensional (2D) gapless systems, including a Heisenberg model with Neel order, a free Dirac fermion in the pi-flux phase, and the nearest-neighbor resonating-valence-bond wave function. For these models, we show that the entanglement entropy between cylindrical regions of length x and L - x, extending around a torus of length L, depends on the dimensionless ratio x/L. This can be well approximated on finite-size lattices by a function ln (sin(pi x/L)), akin to the familiar chord-length dependence in one dimension. We provide evidence, however, that the precise form of this bulk-dependent contribution is a more general function in the 2D thermodynamic limit.