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arXiv:2406.05029 · Physical review. B./Physical review. B

Kondo screening and indirect magnetic exchange through a conventional superconductor studied by the density matrix renormalization group

Cassian Plorin, Michael Potthoff

Published Aug 9, 2024• 1 citation
arXivHTMLPDF

Abstract

The competition between the Kondo screening and indirect magnetic exchange in systems with two magnetic impurities coupled to a conventional $s$-wave superconductor gives rise to a nontrivial ground-state phase diagram. Here, we utilize the density matrix renormalization group (DMRG) method and exploit the non-Abelian spin-SU(2) symmetry to study the phase diagram for two quantum spin-$\frac{1}{2}$ impurities locally exchange coupled to large one-dimensional chains. The nonlocal interimpurity exchange is treated as an emergent Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We find qualitatively different phase diagrams for impurity spins coupled to sites with odd or even distances $d$ on the chain and a partial Kondo-screened spin-doublet phase that extends over the whole range of local exchange couplings $J$ in the limit of weak superconducting pairing strength $\mathrm{\ensuremath{\Delta}}$. Our numerical DMRG studies are complemented by exact diagonalization of small (quantum-box) systems and by perturbative-in-$J$ computations of the $d$- and $\mathrm{\ensuremath{\Delta}}$-dependent RKKY interaction. It is thereby demonstrated that the specific system geometry is essential for our understanding of magnetic impurity interactions in superconducting hosts, and thus for insights into the control of quantum-state properties in nanoparticle systems and topological superconductivity.

DOI
10.1103/physrevb.110.085119

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The goal of this book is the investigation of physical effects at low energies caused by magnetic impurity adatoms, coupled to a small non-magnetic metallic substrate surface. Such systems can be understood with help of multi-impurity Kondo models, describing the substrate as a tight-binding model of finite-size and the impurities as quantum spins, which are antiferromagnetically coupled locally to their respective substrate sites. Focusing on one-dimensional finite-size systems with open boundary conditions at half-filling numerical exact diagonalisation and the variation of matrix product states (VMPS) method are used for the investigation. It is the aim to study finite-size effects in different regimes of the magnetic exchange interaction. For example due to the finite size of the system, the interplay of the Kondo effect and the indirect magnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between impurity spins is site dependent and non-trivial for at least three impurity Kondo systems. Furthermore evidence for an unconventional and novel magnetic coupling mechanism, causing magnetic order in the actually non-magnetic substrate, due to inverse indirect magnetic exchange, will be given.
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 KKK 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 KKK when the interimpurity distance RRR is even lattice spacing, while a crossover behavior is recovered when RRR 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 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.
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.
To analyze the physical properties arising from indirect magnetic exchange between several magnetic adatoms and between complex magnetic nanostructures on metallic surfaces, the real-space extension of dynamical mean-field theory (R-DMFT) appears attractive as it can be applied to systems of almost arbitrary geometry and complexity. While R-DMFT describes the Kondo effect of a single adatom exactly, indirect magnetic (RKKY) exchange is taken into account on an approximate level only. Here, we consider a simplified model system consisting of two magnetic Hubbard sites (``adatoms'') hybridizing with a noninteracting tight-binding chain (``substrate surface''). This two-impurity Anderson model incorporates the competition between the Kondo effect and indirect exchange but is amenable to an exact numerical solution via the density-matrix renormalization group (DMRG). The particle-hole symmetric model at half filling and zero temperature is used to benchmark R-DMFT results for the magnetic coupling between the two adatoms and for the magnetic properties induced in the substrate. In particular, the dependence of the local adatom and the nonlocal adatom-adatom static susceptibilities as well as the magnetic response of the substrate on the distance between the adatoms and on the strength of their coupling with the substrate is studied. We find excellent agreement with the DMRG data even on subtle details of the competition between RKKY exchange and the Kondo effect but also complete failure of the R-DMFT, depending on the parameter regime considered. R-DMFT calculations are performed using the Lanczos method as impurity solver. With the real-space extension of the two-site DMFT, we also benchmark a simplified R-DMFT variant.
Abstract In a superconductor embedded with a quantum magnetic impurity, the Kondo effect is involved, leading to the competition between the Kondo singlet phase and the superconductivity phase. By means of the natural orbitals renormalization group (NORG) method, we revisit the problem of a quantum magnetic impurity coupled with a conventional s-wave superconductor. Here we present a detailed study focusing on the impurity spin polarization and susceptibility, the Kondo screening cloud, as well as the number and structures of the active natural orbitals (ANOs). In the superconducting phase, the impurity spin is partially polarized, indicating that the impurity remains partially screened by the quantum fluctuations. Furthermore, the impurity spin susceptibility becomes divergent, resulting from the presence of residual local moment formed at the impurity site. Correspondingly, a non-integral (incomplete) Kondo cloud is formed, although the ground state is a spin doublet in this phase. In comparison, the Kondo cloud is complete in the Kondo singlet phase as expected. We also quantify the critical point, where the quantum phase transition from a Kondo singlet phase to a superconducting phase occurs, which is consistent with that in previous works. On the other hand, it is illustrated that only one ANO emerges in both quantum phases. The structures of the ANO, projected into both the real space and momentum space, are distinct in the Kondo singlet phase from that in the superconducting phase. More specifically, in the Kondo singlet phase, the ANO keeps fully active with half-occupied, and the superconducting gap has negligible influence on its structure. On the contrary, in the superconducting phase, the ANO tends to be inactive and its structure changes significantly as the superconducting gap increases. Additionally, our investigation demonstrates that the NORG method is reliable and convenient to solve the quantum impurity problems in superconductors as well, which will promote further theoretical studies on the Kondo problems in such systems using numerical methods.