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arXiv:1412.6246 · Physical Review B

Effect of single impurity on free fermion entanglement entropy

Mohammad Pouranvari, Kun Yang, Alexander Seidel

Published Feb 18, 2015• 5 citations
arXivHTMLPDF

Abstract

The one-dimensional free Fermi gas is a prototype conformally invariant system whose entanglement properties are well understood. In this paper, the effects of a single impurity on one-dimensional free fermion entanglement entropy are studied both analytically and numerically. Such an impurity represents an exactly marginal perturbation to the bulk conformally invariant fixed point. We find that the impurity leads to subleading contributions to the entanglement entropy that scale inversely with the subsystem size. The origins of such contributions are identified.

DOI
10.1103/physrevb.91.075115

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The one-dimensional free Fermi gas is a prototype conformally invariant system whose entanglement properties are well understood. In this paper, the effects of a single impurity on one-dimensional free fermion entanglement entropy are studied both analytically and numerically. Such an impurity represents an exactly marginal perturbation to the bulk conformally invariant fixed point. We find that the impurity leads to subleading contributions to the entanglement entropy that scale inversely with the subsystem size. The origins of such contributions are identified.
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.
We study the entanglement in the ground state of a chain of free spinless fermions with a single side-coupled impurity. We find a logarithmic scaling for the entanglement entropy of a segment neighboring the impurity. The prefactor of the logarithm varies continuously and contains an impurity contribution described by a one-parameter function while the contribution of the unmodified boundary enters additively. The coefficient is found explicitly by pointing out similarities with other models involving interface defects. The proposed formula gives excellent agreement with our numerical data. If the segment has an open boundary, one finds a rapidly oscillating subleading term in the entropy that persists in the limit of large block sizes. The particle-number fluctuation inside the subsystem is also reported. It is analogous with the expression for the entropy scaling, however, with a simpler functional form for the coefficient.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
Abstract We study the entanglement properties of non-Hermitian free fermionic models with translation symmetry using the correlation matrix technique. Our results show that the entanglement entropy has a logarithmic correction to the area law in both one-dimensional and two-dimensional systems. For any one-dimensional one-band system, we prove that each Fermi point of the system contributes exactly 1/2 to the coefficient c of the logarithmic correction. Moreover, this relation between c and Fermi point is verified for more general one-dimensional and two-dimensional cases by numerical calculations and finite-size scaling analysis. In addition, we also study the single-particle and density–density correlation functions.
The entanglement entropy in Kondo impurity systems is studied analytically using conformal field theory. From the impurity contribution to the scaling corrections of the entanglement entropy we extract information about the screening cloud profile for general non-Fermi-liquid fixed points. By also considering the finite-temperature corrections to scaling of the von Neumann entropy we point out a direct connection between the long-distance screening cloud profile and thermodynamic observables such as the specific heat.
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the \log{L} log L scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log L L log L scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.