We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the Rényi entanglement entropy Sn between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction, as has been done in previous studies of one-dimensional systems. The limit n→1 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 and antisymmetric channels as far as the coupled system remains in a gapless phase. We argue that in this system, Sn is a linear function of the length of the chains (boundary law) followed by a universal subleading constant γn determined by the ratio of the two TLL parameters. The formulas of γn for integer n≥2 are derived using (a) ground-state wave functionals of TLL's and (b) boundary conformal field theory, 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 the analytic continuation of γn to n→1 turns out to be a difficult problem, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal. Our results may provide useful characterization of inherently anisotropic quantum phases such as the sliding Luttinger liquid phase via qualitatively different behaviors of the entanglement entropy with the entanglement partitions along different directions.
|Physical Review B - Condensed Matter and Materials Physics
|Published - 2011 Feb 28
ASJC Scopus subject areas
- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics