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arXiv:1310.0829 · Physical Review B

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

Rex Lundgren, Yohei Fuji, Shunsuke Furukawa, Masaki Oshikawa

Published Dec 30, 2013• 54 citations
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Abstract

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.

DOI
10.1103/physrevb.88.245137

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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 consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy Sn{S}_{n}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 $n\ensuremath{\rightarrow}1$ 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, Sn{S}_{n}Sn​ 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 study momentum-space entanglement in quantum spin-half ladders consisting of two coupled critical XXZXXZXXZ spin-half chains, using field theoretical methods developed by Lundgren et al. [Phys. Rev. B 88, 245137 (2013)] and exact diagonalization. When the system is gapped, we analytically find that the momentum-space entanglement Hamiltonian is described by a chiral conformal field theory with a central charge of 2. This is in contrast to entanglement Hamiltonians of various real-space partitions of gapped-spin ladders that have a central charge of 1. When the system is gapless, the entanglement Hamiltonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. In the gapless region, we extend the work of Lundgren et al. to include the effect of a certain irrelevant term on the entanglement spectrum. In both the gapless and gapped phases, the momentum-space entanglement entropy obeys a volume law. In the gapless region, we find a subleading constant term in the entanglement entropy which contains information about the underlying field theory of the system. The analytical predictions for the entanglement spectrum are consistent with results from exact diagonalization.
The concept of Tomonaga--Luttinger liquids (TLL) on the basis of the free-boson models is ubiquitous in theoretical descriptions of low-energy properties in one-dimensional quantum systems. In this work, we develop a squeezed-field path-integral description for gapless one-dimensional systems beyond the free-boson picture of the TLL paradigm. In the squeezed-field description, the parameter of the Bogoliubov transformation for the TL Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise to corrections to the free-boson results. We derive an effective nonlinear Lagrangian describing the dispersion relation of the squeezing field, and interactions between the excitations of the TLL and the squeezing modes. Using the effective Lagrangian, we analyze the imaginary-time correlation function of a vertex operator in the non-interacting limit. We show that a side-band branch emerges due to the fluctuation of the squeezing field, in addition to the standard branch of the free-boson model of the TLL paradigm. Furthermore, we perturbatively analyze the spectral function of the density fluctuations for an ultracold Bose gas in one dimension. We evaluate the renormalized values of the phase velocities and spectral weights of the TLL and side-band branches due to the interaction between the TLL and the squeezing modes. At zero temperature, the renormalized dispersion relations are linear in the momentum, but at nonzero temperatures, these acquire a nonlinear dependence on the momentum due to the thermal population of the excitation branches.
Topological quantum matter is typically associated with gapped phases and edge modes protected by the bulk gap. In contrast, recent work (Phys. Rev. B 104, 075132) proposed intrinsically gapless topological phases that, in one dimension, carry protected edge modes only when the bulk is a gapless Luttinger liquid. The edge modes of such a topological Luttinger liquid (TLL) descend from a nonlocal string order that is forbidden in gapped phases and whose precise form depends on the symmetry class of the system. In this work, we propose a powerful and unbiased entanglement-based smoking gun signature of the TLL. In particular, we show that the entanglement entropy profile of a TLL lacks Friedel oscillations that are invariably present in other gapless one dimensional phases such as ordinary Luttinger liquids, and argue that their absence is closely related to a long-ranged string order which is an intrinsic property of the TLL. Crucially, such a diagnostic is more robust against numerical errors and relatively easier to measure in experiments as it relies on the entanglement entropy rather than entanglement spectrum, unlike the entanglement-based diagnostics of gapped topological phases in one dimension.
Abstract Quantum entanglement exists in nature but is absent in classical physics, hence it fundamentally distinguishes quantum from classical theories. While entanglement is routinely observed for few-body systems, it is significantly more challenging to witness in quantum many-body systems. Here, we theoretically study entanglement between two parallel and spatially separated Tomonaga–Luttinger liquids (TLLs) partitioned along the longitudinal axis. In particular, we focus on 1D Bose gases as a realization of TLLs and investigate two experimentally relevant situations: tunnel-coupled gases at finite temperatures and after coherent splitting. In both scenarios, we analytically calculate the logarithmic negativity and identify a threshold temperature below which the system is entangled. Notably, this threshold temperature is accessible in the current and near-term 1D Bose gas experiments. Furthermore, we investigate the crossover between quantum and classical correlations in the vicinity of the threshold temperature by comparing logarithmic negativity with mutual information. We argue that the initial mutual information established by the coherent splitting is conserved in TLL dynamics, thus preventing certain generalized Gibbs ensembles from being reached during prethermalization. Moreover, both logarithmic negativity and mutual information are found to scale extensively with the subsystem’s length. Although the ground-state entanglement between coupled TLLs has been predicted to be extensive, this setting is largely overlooked compared to other partitions. Our work extends the study of entanglement between coupled TLLs to finite temperatures and out-of-equilibrium regimes, and provides a strategy towards experimental detection of extensive entanglement in quantum many-body systems at finite temperatures.
The von Neumann entanglement entropy is studied with the density-matrix renormalization-group technique. We propose a simple approach to calculate the central charge using the entanglement entropy for a one-dimensional (1D) quantum system. This approach is applied to a couple of quantum systems: (i) a 1D frustrated spin model and (ii) a 1D half-filled spinless fermions with nearest-neighbor repulsion; it is confirmed that the central charge is estimated very accurately for the both systems. Also, a new method to determine the critical point between Tomonaga-Luttinger-liquid and gapped (or ordered) phases from the proposed approach is suggested. Furthermore, we mention that the Tomonaga-Luttinger parameter can be obtained in a like manner as the central charge, using the charge-density fluctuation of a part of the 1D system.