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arXiv:cond-mat/0703037 · Journal of Statistical Mechanics Theory and Experiment

Quantum impurity entanglement

Erik Schwartz Sørensen, Ming-Shyang Chang, Nicolas Laflorencie, Ian K. Affleck

Published Aug 1, 2007• 75 citations
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Abstract

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.

DOI
10.1088/1742-5468/2007/08/p08003

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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.
The screening of an impurity spin by conduction electrons is associated with the formation of a large Kondo screening cloud, of size ξ K . We study the quantum entanglement between a region of size r surrounding the impurity and the rest of the sample (of total size R ) using density matrix renormalization group and analytic methods. The corresponding ‘impurity entanglement entropy’, S imp , is shown to be a universal scaling function of r /ξ K and r / R . We calculate this universal function using Fermi liquid theory in the regime ξ K ≪ r .
The contribution to the entanglement of an impurity attached to one end of a J – J 2 –δ quantum spin chain ( S = 1/2) is studied. Two different measures of the impurity contribution to the entanglement have been proposed: the impurity entanglement entropy S imp and the negativity . The first, S imp , is based on a subtractive procedure where the entanglement entropy in the absence of the impurity is subtracted from results with the impurity present. The other, , is the negativity of a part of the system separated from the impurity and the impurity itself. In this paper we compare the two measures and discuss their similarities and the differences between them. In the J – J 2 –δ model it is possible to perform very precise variational calculations close to the Majumdar–Ghosh point ( J 2 = J /2 and δ = 0) where the system is gapped with a dimerized ground state. We describe in detail how such calculations are done and how they can be used to calculate as well as S imp for any impurity coupling J K . We then study the complete crossover in the impurity entanglement as J K is varied between 0 and 1 close to the Majumdar–Ghosh point. In particular, we study the impurity entanglement when a staggered nearest neighbour interaction proportional to δ is introduced. In this case we observe a very rapid reduction in the impurity entanglement as δ is increased.
Abstract The screening of an impurity spin in the Kondo model occurs over a characteristic length scale ξ K , that defines the size of the Kondo screening cloud or ‘mist’. The presence of such a length scale in experimental and numerical results is rather subtle. A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations in terms of the single variable, r / ξ K instead of the correlations depending on r and ξ K separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength J K ′ . Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, g. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling J K ′ is varied, a precise determination of ξ K can then be obtained. The length scale, ξ K , is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain.
The numerical renormalization group is used to study quantum entanglement in the Kondo impurity model with a density of states ρ(ϵ)∝|ϵ|^{r} (0<r<1/2) that vanishes at the Fermi energy ϵ=0. This nonintegrable model features a Kondo-destruction quantum critical point (QCP) separating a partially screened phase from a local-moment phase. The impurity contribution S_{e}^{imp} to the entanglement entropy between a region of radius R around the magnetic impurity and the rest of the system reveals a length scale R^{*} that distinguishes a region R≪R^{*} of strong critical entanglement from one R≫R^{*} of weak entanglement. Within each phase, S_{e}^{imp} is a universal function of R/R^{*} with a power-law decay for R/R^{*}≫1. The entanglement length R^{*} diverges on approach to the interacting QCP, showing that the critical Kondo screening cloud subsumes the entire system as the impurity becomes maximally entangled with the conduction band. This work has implications for entanglement calculations in other models and for the nature of heavy-fermion quantum criticality.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with a nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels, we confirm field-theory predictions for the boundary entropy difference ln(gUV/gIR)=ln(2)/2ln({g}_{\text{UV}}/{g}_{\text{IR}})=ln(2)/2ln(gUV​/gIR​)=ln(2)/2 between the ultraviolet and infrared limits and the leading ln(x)/xln(x)/xln(x)/x impurity correction to the block entropy. The impurity entanglement Simp{S}_{\text{imp}}Simp​ is shown to scale with the characteristic length ${\ensuremath{\xi}}_{2\text{CK}}$. We show that both the Schmidt gap and the entanglement of the impurity with one of the channels---as measured by the negativity---faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, $\ensuremath{\nu}\ensuremath{\approx}2$ and $\ensuremath{\beta}\ensuremath{\approx}0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.