Scaling of Entanglement Entropy in Point Contact Free Fermion Systems

The scaling of entanglement entropy is computationally studied in several $1\le d\le 2$ dimensional free fermion systems that are connected by one or more point contacts (PC). 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 {\sl single} PC is found to be generically $S\sim L$. 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 $S\sim a{L}^{d-1}+bm\log L$ and is thus consistent with the anomalous area law for free fermions ($S \sim L \log{L}$)as$m \rightarrow L$. For the Bethe lattice, the relevance of this result to Density Matrix Renormalization Group (DMRG) schemes for interacting fermions is discussed.