Competition between the Kondo effect and Ruderman-Kittel-Kasuya-Yosida interaction in the two-impurity Kondo problem can be phenomenologically described by the Rasul-Schlottmann spin model. We revisit this model from the quantum entanglement perspective by calculating both the inter-impurity entanglement and the local Kondo entanglement, the latter being the entanglement between a local magnetic impurity and its spatially nearby conduction electron. A groundstate phase diagram is derived and a discontinuous breakdown of the local Kondo entanglement is found at the singular point, associated concomitantly with a jump in the inter-impurity entanglement. An entanglement monogamy holds in the whole phase diagram. Our results identify the important role of the frustrated cross-coupling and demonstrate the local characteristic of the quantum phase transition in the two-impurity Kondo problem. The implications of these results for Kondo lattices and quantum information processing are also briefly discussed.
Entanglement is a defining feature of quantum mechanics, and its relation to quantum criticality is of considerable current interest. Here we show that measurable spin and charge fluctuations provide an entanglement witness of quantum criticality in the two-impurity Kondo model, which has experimental realizations in terms of coupled quantum dots and magnetic impurities added to surfaces. We use density-matrix renormalization group and numerical renormalization group calculations to resolve the non-Fermi-liquid critical point separating two independently Kondo-screened impurities from an inter-impurity singlet and interpret it as a change in the dominant entanglement partner of each local moment: from entanglement of the local moment with an extended set of conduction-electron degrees of freedom to entanglement with the other impurity. This reorganization is accompanied by a singular response of the impurity-bath entanglement and the inter-impurity susceptibility. We show how the same structure is encoded in the quantum Fisher information of collective spin and charge operators at zero and finite temperatures, connecting the entanglement picture to experimentally accessible dynamical response functions. Our results establish impurity systems as controlled settings in which quantum-critical entanglement can be detected through measurable correlations, and provide further insight into the possibility of understanding heavy-fermion physics in terms of entanglement.
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.
The problem of two magnetic impurities in a normal metal exposes the two opposite tendencies in the formation of a singlet ground state, driven respectively by the single-ion Kondo effect with conduction electrons to screen impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction between the two impurities to directly form impurity spin singlet. However, whether the competition between these two tendencies can lead to a quantum critical point has been debated over more than two decades. Here, we study this problem by applying the newly proposed natural orbitals renormalization group method to a lattice version of the two-impurity Kondo model with a direct exchange
K between the two impurity spins. The method allows for unbiased access to the ground state wave functions and low-lying excitations for sufficiently large system sizes. We demonstrate the existence of a quantum critical point, characterized by the power-law divergence of impurity staggered susceptibility with critical exponent $\ensuremath{\gamma}=0.60(1)$, on the antiferromagnetic side of
K when the interimpurity distance
R is even lattice spacing, while a crossover behavior is recovered when
R is odd lattice spacing. These results have ultimately resolved the long-standing discrepancy between the numerical renormalization group and quantum Monte Carlo studies, confirming a link of this two-impurity Kondo critical point to a hidden particle-hole symmetry predicted by the local Fermi liquid theory.
We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on a square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling
JK, longitudinal anisotropy
D, and transverse anisotropy
E. In the isotropic case, two impurities on opposite (the same) sublattices have a singlet (triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively fourfold-degenerate ground state, i.e., two decoupled impurities. For large enough
JK the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear nonperturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persist and the two effects coexist: the impurities are underscreened, and the dangling spin-
1/2 degrees of freedom are responsible for the interimpurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasiclassical correlations.
We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.