Entanglement entropy between two coupled Tomonaga-Luttinger liquids

We consider a system of two coupled Tomonaga-Luttinger liquids (TLL) on parallel chains and study the R\'enyi entanglement entropy $S_n$ between the two chains. The limit $n\to1$ corresponds to the von Neumann entanglement entropy. The system is effectively described by two-component bosonic field theory with different TLL parameters in the symmetric/antisymmetric channels. We argue that in this system, $S_n$ is a linear function of the length of the chains followed by a universal subleading constant $\gamma_n$ determined by the ratio of the two TLL parameters. We derive the formulae of $\gamma_n$ for integer $n\ge 2$ using (a) ground-state wave functionals of TLLs and (b) conformal boundary states, which lead to the same result. These predictions are checked in a numerical diagonalization analysis of a hard-core bosonic model on a ladder. Although our formulae of $\gamma_n$ are not analytic in the limit $n\to 1$, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal.