Scaling of entanglement entropy in point-contact, free-fermion systems

Abstract

The scaling of entanglement entropy is computationally studied in several ($1\ensuremath{\le}d\ensuremath{\le}2$)-dimensional free-fermion systems that are connected by one or more point contacts (PCs). For both the $k$-leg Bethe lattice $(d=1)$ and $d=2$ rectangular lattices with a subsystem of ${L}^{d}$ sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$ entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by $m$ PCs, the area law is found to be $S\ensuremath{\sim}a{L}^{d\ensuremath{-}1}+bm\phantom{\rule{0.16em}{0ex}}logL$ and is thus consistent with the anomalous area law for free fermions ($S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$) as $m\ensuremath{\rightarrow}L$. For the Bethe lattice, the relevance of this result to density-matrix renormalization-group schemes for interacting fermions is discussed.