Entanglement entropy between two coupled Tomonaga-Luttinger liquids

TL;DR

This paper investigates the entanglement entropy between two coupled Tomonaga-Luttinger liquids, deriving universal formulas for the subleading constant term and confirming results through numerical analysis, providing insights into anisotropic quantum phases.

Contribution

The paper derives universal formulas for the subleading constant in the entanglement entropy of coupled TLLs and confirms their validity through numerical diagonalization, extending understanding of entanglement in anisotropic systems.

Findings

Entanglement entropy follows a boundary law with a universal subleading constant.

Derived formulas for the subleading constant using conformal field theory and wave functionals.

Numerical analysis supports the universality of the subleading constant in the von Neumann entropy.

Abstract

We consider a system of two coupled Tomonaga-Luttinger liquids (TLL) on parallel chains and study the Renyi entanglement entropy $S_n$ between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction as used in previous studies of one-dimensional systems. 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 as far as the coupled system remains in a gapless phase. We argue that in this system, $S_n$ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant $\gamma_n$ determined by the ratio of the two TLL parameters. The formulae of $\gamma_n$ for integer $n\ge 2$ are derived using (a) ground-state wave functionals of TLLs 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 $\gamma_n$ to $n\to 1$ 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.