We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy ${S}_{n}$ 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 $n\ensuremath{\rightarrow}1$ 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, ${S}_{n}$ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ 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 ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ 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.We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy ${S}_{n}$ 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 $n\ensuremath{\rightarrow}1$ 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, ${S}_{n}$ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ 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 ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ 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.

arXiv:1009.3016Physical Review BFeb 28, 201143 citations

Junction of Tomonaga–Luttinger liquids

Masaki Oshikawa

Physica E Low-dimensional Systems and NanostructuresJul 29, 20050 citations

Entanglement spectra between coupled Tomonaga-Luttinger liquids: Applications to ladder systems and topological phases

Rex Lundgren, Yohei Fuji, Shunsuke Furukawa +1 more

We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.

arXiv:1310.0829Physical Review BDec 30, 201354 citations

We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy

{S}_{n}

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 $n\ensuremath{\rightarrow}1$ 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,

{S}_{n}

is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ 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 ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ 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.