Universal entanglement crossover of coupled quantum wires*

We consider the entanglement between two one-dimensional quantum wires (Luttinger Liquids) coupled by tunneling through a quantum impurity. The physics of the system involves a crossover between weak and strong coupling regimes characterized by an energy scale $T_B$, and methods of conformal field theory therefore cannot be applied. The evolution of the entanglement in this crossover has led to many numerical studies, but has remained little understood, analytically or even qualitatively. This is, in part, due to the fact that the entanglement in this case is non-perturbative in the tunneling amplitude. We argue that the correct universal scaling form of the entanglement entropy $S$ (for an arbitrary interval containing the impurity) is $\partial S/\partial \ln L = f(L T_B)$. In the special case where the coupling to the impurity can be refermionized, we show how the universal function $f(L T_B)$ can be obtained analytically using recent results on form factors of twist fields and a defect massless-scattering formalism. Our results are carefully checked against numerical simulations.