Entanglement scaling in two-dimensional gapless systems

Abstract

We numerically determine subleading scaling terms in the ground-state entanglement entropy of several two-dimensional (2D) gapless systems, including a Heisenberg model with N\'eel order, a free Dirac fermion in the $\ensuremath{\pi}$-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 $L\ensuremath{-}x$, 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 $\mathrm{ln}(\mathrm{sin}(\ensuremath{\pi}x/L)),$ 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.