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arXiv:cond-mat/0408366 · Physical Review Letters

Entanglement Entropy in a Boundary Impurity Model

Greg Levine

Published Dec 20, 2004• 65 citations
arXivHTMLPDF

Abstract

Boundary impurities are known to dramatically alter certain bulk properties of ($1+1$)-dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling or bosonization scheme. For a Luttinger liquid of length $2L$ and UV cutoff $ϵ$, the boundary impurity correction ($\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}$) to the logarithmic entanglement entropy (${S}_{\mathrm{e}\mathrm{n}\mathrm{t}}\ensuremath{\propto}\mathrm{ln}L/ϵ$) scales as $\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}\ensuremath{\sim}{y}_{r}\mathrm{ln}L/ϵ$, where ${y}_{r}$ is the renormalized backscattering coupling constant. In this way, the entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case ($g<1$), $\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}$ diverges (negatively) suggesting that the entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow.

DOI
10.1103/physrevlett.93.266402

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Using a fermionic renormalization group method we present a simple real space picture of the strong influence an impurity has on the electronic properties of a Luttinger liquid. We compute the flow of the renormalized impurity potential for a single impurity over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. We confirm that low energy properties close to the impurity are as if the chain is cut in two pieces with open boundary conditions at the end points, but show that this universal behavior is only reached for extremely large systems. The accuracy of the renormalization group scheme is demonstrated by a direct comparison with data obtained from the density-matrix renormalization group method.
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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 $\ensuremath{\delta}$-function interacting bosons in the one-dimensional spatial continuum.
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We review research on a number of situations where a quantum impurity or a physical boundary has an interesting effect on entanglement entropy. Our focus is mainly on impurity entanglement as it occurs in one dimensional systems with a single impurity or a boundary, in particular quantum spin models, but generalizations to higher dimensions are also reviewed. Recent advances in the understanding of impurity entanglement as it occurs in the spin-boson and Kondo impurity models are discussed along with the influence of boundaries. Particular attention is paid to 1+1 dimensional models where analytical results can be obtained for the case of conformally invariant boundary conditions and a connection to topological entanglement entropy is made. New results for the entanglement in systems with mixed boundary conditions are presented. Analytical results for the entanglement entropy obtained from Fermi liquid theory are also discussed as well as several different recent definitions of the impurity contribution to the entanglement entropy.
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