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 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.
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.
Entanglement in J 1 – J 2 , S = 1/2 quantum spin chains with an impurity is studied using analytic methods as well as large scale numerical density matrix renormalization group methods. The entanglement is investigated in terms of the von Neumann entropy, S = −Trρ A logρ A , for a subsystem A of size r of the chain. The impurity contribution to the uniform part of the entanglement entropy, S imp , is defined and analysed in detail in both the gapless, J 2 ≤ J 2 c , as well as the dimerized phase, J 2 > J 2 c , of the model. This quantum impurity model is in the universality class of the single channel Kondo model and it is shown that in a quite universal way the presence of the impurity in the gapless phase, J 2 ≤ J 2 c , gives rise to a large length scale, ξ K , associated with the screening of the impurity, the size of the Kondo screening cloud. The universality of Kondo physics then implies scaling of the form S imp ( r /ξ K , r / R ) for a system of size R . Numerical results are presented clearly demonstrating this scaling. At the critical point, J 2 c , an analytic approach based on a Fermi liquid picture, valid at distances and energy scales , is developed and analytic results at T = 0 are obtained showing S imp = πξ K [1+π(1− r / R )cot(π r / R )]/(12 R ) for finite R . For T >0, in the thermodynamic limit, we find S imp = [π 2 ξ K T /(6 v )]coth(2π rT / v ), with v the spin-wave velocity. In the dimerized phase an appealing picture of the entanglement is developed in terms of a thin soliton (TS) ansatz and the notions of impurity valence bonds (IVB) and single particle entanglement (SPE) are introduced. The TS-ansatz permits a variational calculation of the complete entanglement in the dimerized phase that appears to be exact in the thermodynamic limit at the Majumdar–Ghosh point, J 2 = J 1 /2, and surprisingly precise even close to the critical point J 2 c . In the appendices the TS-ansatz is further used to calculate and with high precision at the Majumdar–Ghosh point and the relation between the finite temperature entanglement entropy, S ( T ), and the thermal entropy, S th ( T ), is discussed. Finally, the alternating part of S imp is discussed, together with its relation to the boundary induced dimerization.
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 yr 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.
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.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
Impurities are ubiquitous in condensed matter. Boundary Conformal Field Theory (BCFT) provides a powerful method to study a localized quantum impurity interacting with a gapless continuum of excitations. The results can also be implied to nanoscopic devices like quantum dots. In these lecture notes, I review this field, including the following topics: I. General Renormalization Group (RG) framework for quantum impurity problems: example of simplest Kondo model II. Multi-channel Kondo model III. Quantum Dots: experimental realizations of one and two channel Kondo models IV. Impurities in Luttinger liquids: point contact in a quantum wire V. Quantum impurity entanglement entropy VI. Y-junctions of Luttinger liquids VII. Boundary condition changing operators and the X-ray edge problem.