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arXiv:1412.6529 · Physical Review B

Particle partition entanglement of bosonic Luttinger liquids

C. M. Herdman, Adrian Del Maestro

Published May 11, 2015• 17 citations
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Abstract

We consider the R\'enyi entanglement entropy of bosonic Tomonaga-Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher-order corrections involve a microscopic length scale, the leading-order scaling depends only on this sole dimensionless parameter which characterizes the low-energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Tomonaga-Luttinger liquid theory for the Lieb-Liniger model of $\ensuremath{\delta}$-function interacting bosons in the one-dimensional spatial continuum.

DOI
10.1103/physrevb.91.184507

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We consider the R'enyi entanglement entropy of bosonic Tomonaga-Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher-order corrections involve a microscopic length scale, the leading-order scaling depends only on this sole dimensionless parameter which characterizes the low-energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Tomonaga-Luttinger liquid theory for the Lieb-Liniger model of \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interacting bosons in the one-dimensional spatial continuum.
Abstract We investigate the scaling of the Rényi entanglement entropies for a particle bipartition of interacting spinless fermions in one spatial dimension. In the Tomonaga–Luttinger liquid regime, we calculate the second Rényi entanglement entropy and show that the leading order finite-size scaling is equal to a universal logarithm of the system size plus a non-universal constant. Higher-order corrections decay as power-laws in the system size with exponents that depend only on the Luttinger parameter. We confirm the universality of our results by investigating the one dimensional t − V model of interacting spinless fermions via exact-diagonalization techniques. The resulting sensitivity of the particle partition entanglement to boundary conditions and statistics supports its utility as a probe of quantum liquids.
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.
Entanglement of spatial bipartitions, used to explore lattice models in condensed matter physics, may be insufficient to fully describe itinerant quantum many-body systems in the continuum. We introduce a procedure to measure the R\'enyi entanglement entropies on a particle bipartition, with general applicability to continuum Hamiltonians via path integral Monte Carlo methods. Via direct simulations of interacting bosons in one spatial dimension, we confirm a logarithmic scaling of the single-particle entanglement entropy with the number of particles in the system. The coefficient of this logarithmic scaling increases with interaction strength, saturating to unity in the strongly interacting limit. Additionally, we show that the single-particle entanglement entropy is bounded by the condensate fraction, suggesting a practical route towards its measurement in future experiments.
We consider the entanglement between two spatial subregions in the Lieb-Liniger model of bosons in one spatial dimension interacting via a contact interaction. Using ground-state path integral quantum Monte Carlo we numerically compute the R\'enyi entropy of the reduced density matrix of the subsystem as a measure of entanglement. Our numerical algorithm is based on a replica method previously introduced by the authors, which we extend to efficiently study the entanglement of spatial subsystems of itinerant bosons. We confirm a logarithmic scaling of the R\'enyi entropy with subsystem size that is expected from conformal field theory, and compute the nonuniversal subleading constant for interaction strengths ranging over two orders of magnitude. In the strongly interacting limit, we find agreement with the known free fermion result.
Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.
For the one-dimensional Holstein model, we show that the relations among the scaling exponents of various correlation functions of the Tomonaga Luttinger liquid (LL), while valid in the thermodynamic limit, are significantly modified by finite-size corrections. We obtain analytical expressions for these corrections and find that they decrease very slowly with increasing system size. The interpretation of numerical data on finite-size lattices in terms of LL theory must therefore take these corrections into account. As an important example, we re-examine the proposed metallic phase of the zero-temperature, half-filled one-dimensional Holstein model without employing the LL relations. In particular, using quantum Monte Carlo calculations, we study the competition between the singlet pairing and charge ordering. Our results do not support the existence of a dominant singlet pairing state.