Abstract:
In order to quantify quantum entanglement in two impurity Kondo systems, we calculate the concurrence, negativity, and von Neumann entropy. The entanglement of the two Kondo impurities is shown to be determined by two competing many-body effects, the Kondo effect and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, $I$. Due to the spin-rotational invariance of the ground state, the concurrence and negativity are uniquely determined by the spin-spin correlation between the impurities. It is found that there exists a critical minimum value of the antiferromagnetic correlation between the impurity spins which is necessary for entanglement of the two impurity spins. The critical value is discussed in relation with the unstable fixed point in the two impurity Kondo problem. Specifically, at the fixed point there is no entanglement between the impurity spins. Entanglement will only be created (and quantum information processing (QIP) be possible) if the RKKY interaction exchange energy, $I$, is at least several times larger than the Kondo temperature, $T_K$. Quantitative criteria for QIP are given in terms of the impurity spin-spin correlation.

Abstract:
Based on Yosida's ground state of the single-impurity Kondo Hamiltonian, we study three kinds of entanglement between an impurity and conduction electron spins. First, it is shown that the impurity spin is maximally entangled with all the conduction electrons. Second, a two-spin density matrix of the impurity spin and one conduction electron spin is given by a Werner state. We find that the impurity spin is not entangled with one conduction electron spin even within the Kondo screening length $\xi_K$, although there is the spin-spin correlation between them. Third, we show the density matrix of two conduction electron spins is nearly same to that of a free electron gas. The single impurity does not change the entanglement structure of the conduction electrons in contrast to the dramatic change in electrical resistance.

Abstract:
The screening of an impurity spin by conduction electrons is associated with the formation of a large Kondo screening cloud, of size xi_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/xi_K and r/R. We calculate this universal function using Fermi liquid theory in the regime xi_K << r.

Abstract:
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 RKKY interaction between the two impurity spins. We then use a DMRG 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 non-perturbative approach enables us to uncover the precise dependence of the spatial extent $\xi_K$ of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature $T_K \sim 1/\xi_K$ 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.

Abstract:
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 large-distance screening cloud profile and thermodynamic observables such as the specific heat.

Abstract:
Motivated by proposals to employ RKKY-coupled spins as building blocks in a solid-state quantum computer, we analyze how the RKKY interaction in a 2D electron gas is influenced by spin-orbit interactions. Using a two-impurity Kondo model with added Dresselhaus and Rashba spin-orbit interactions we find that spin-rotational invariance of the RKKY interaction - essential for having a well-controllable two-qubit gate - is restored when tuning the Rashba coupling to have the same strength as the Dresselhaus coupling. We also discuss the critical properties of the two-impurity Kondo model in the presence of spin-orbit interactions, and extract the leading correction to the block entanglement scaling due to these interactions.

Abstract:
We investigate the entanglement properties of the Kondo spin chain when it is prepared in its ground state as well as its dynamics following a single bond quench. We show that a true measure of entanglement such as negativity enables to characterize the unique features of the gapless Kondo regime. We determine the spatial extent of the Kondo screening cloud and propose an ansatz for the ground state in the Kondo regime accessible to this spin chain; we also demonstrate that the impurity spin is indeed maximally entangled with the Kondo cloud. We exploit these features of the entanglement in the gapless Kondo regime to show that a single local quench at one end of a Kondo spin chain may always induce a fast and long lived oscillatory dynamics, which establishes a high quality entanglement between the individual spins at the opposite ends of the chain. This entanglement is a footprint of the presence of the Kondo cloud and may be engineered so as to attain - even for very large chains- a constant high value independent of the length; in addition, it is thermally robust. To better evidence the remarkable peculiarities of the Kondo regime, we carry a parallel analysis of the entanglement properties of the Kondo spin chain model in the gapped dimerised regime where these remarkable features are absent.

Abstract:
In the two-impurity Anderson model, the inter-impurity spin exchange interaction favors a spin singlet state between two impurities leading to the breakdown of the Kondo effect. We show that a local uniform magnetic field can delocalize the quasiparticles to restore the Kondo resonance. This transition is found to be continuous, accompanied by not only the divergence of the staggered (antiferromagnetic) susceptibility, but also the divergence of the uniform spin susceptibility. This may imply that the magnetic field induced quantum phase transitions in Kondo systems are in favor of the local critical type.

Abstract:
The method of continuous unitary transformations (CUTs) is applied to the Anderson impurity and the Kondo model aiming at the systematic derivation of convergent effective models. If CUTs are applied in a conventional way, diverging differential equations occur. Similar to poor man's scaling the energy scale, below which the couplings diverge, corresponds to the Kondo temperature $T_K$. We present a way to apply CUTs to the Kondo and to the Anderson impurity model so that no divergences occur but a converged effective low-energy model is derived with small fnite parameters at arbitrarily small energies. The ground state corresponds to a bound singlet with a binding energy given by the Kondo temperature $T_K$.

Abstract:
We study the impurity entanglement entropy $S_e$ in quantum impurity models that feature a Kondo-destruction quantum critical point (QCP) arising from a pseudogap in the conduction-band density of states or from coupling to a bosonic bath. On the local-moment (Kondo-destroyed) side of the QCP, the entanglement entropy contains a critical component that can be related to the order parameter characterizing the quantum phase transition. In Kondo models describing a spin-$\Simp$, $S_e$ assumes its maximal value of $\ln(2\Simp+1)$ at the QCP and throughout the Kondo phase, independent of features such as particle-hole symmetry and under- or over-screening. In Anderson models, $S_e$ is nonuniversal at the QCP, and at particle-hole symmetry, rises monotonically on passage from the local-moment phase to the Kondo phase; breaking this symmetry can lead to a cusp peak in $S_e$ due to a divergent charge susceptibility at the QCP. Implications of these results for quantum critical systems and quantum dots are discussed.