Entanglement Entropy in a Boundary Impurity Model

Boundary impurities are known to dramatically alter certain bulk properties of ($1+1$)-dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling or bosonization scheme. For a Luttinger liquid of length $2L$ and UV cutoff $ϵ$, the boundary impurity correction (δSᵢₘₚ) to the logarithmic entanglement entropy (Sₑₙₜ∝lnL/ϵ) scales as δSᵢₘₚ~yᵣlnL/ϵ, where yᵣ is the renormalized backscattering coupling constant. In this way, the entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case ($g<1$), δSᵢₘₚ diverges (negatively) suggesting that the entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow.