The (111) surface of noble metals is usually treated as an isolated two-dimensional (2D) triangular lattice completely decoupled from the bulk. However, unlike in topological insulators, bulk bands also cross the Fermi level. We here introduce an effective tight-binding model that accurately reproduces results from first-principles calculations, accounting for both surface and bulk states. We numerically solve the many-body problem of two quantum impurities sitting on the surface by means of the density matrix renormalization group. By performing simulations in a star geometry, we are able to study the nonperturbative problem in the thermodynamic limit with machine precision accuracy. We find that there is a nontrivial competition between Kondo and RKKY physics and as a consequence, ferromagnetism is never developed, except at short distances. The bulk introduces a variation in the period of the RKKY interactions, and therefore the problem departs considerably from the simpler 2D case. In addition, screening and the magnitude of the effective indirect exchange are enhanced by the contributions from the bulk states.
The (111) surface of noble metals is usually treated as an isolated two-dimensional (2D) triangular lattice completely decoupled from the bulk. However, unlike in topological insulators, bulk bands also cross the Fermi level. We here introduce an effective tight-binding model that accurately reproduces results from first-principles calculations, accounting for both surface and bulk states. We numerically solve the many-body problem of two quantum impurities sitting on the surface by means of the density matrix renormalization group. By performing simulations in a star geometry, we are able to study the nonperturbative problem in the thermodynamic limit with machine precision accuracy. We find that there is a nontrivial competition between Kondo and RKKY physics and as a consequence, ferromagnetism is never developed, except at short distances. The bulk introduces a variation in the period of the RKKY interactions, and therefore the problem departs considerably from the simpler 2D case. In addition, screening and the magnitude of the effective indirect exchange are enhanced by the contributions from the bulk states.
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-21 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.
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.
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.
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.
Historically, the RKKY or indirect exchange, interaction has been accepted as being able to be described by second order perturbation theory. A typical universal expression is usually given in this context. This approach, however, fails to incorporate many body ef- fects, quantum fluctuations, and other important details. In Chapter 2, a novel numerical approach is developed to tackle these problems in a quasi-exact, non-perturbative manner. Behind the method lies the main concept of being able to exactly map an n-dimensional lattice problem onto a 1-dimensional chain. The density matrix renormalization group algorithm is then employed to solve the newly cast Hamiltonian.
The Kondo effect, the screening of the spin of an impurity by the conduction electrons of a metallic host system, is a standard problem in solid state physics. At weak coupling strengths between the host system and a group of impurities, the Kondo competes with indirect (RKKY) exchange, mediated by the conduction electrons, which promotes magnetic order among the impurities.
This work deals with the physics emerging in impurity systems with spatially confined metallic host materials, relevant to the quantum dots and artificial nanostructures created by scanning tunnelling techniques. It is shown that (i) the finite system size leads to an unconventional reentrant competition, where the Kondo prevails at weak couplings in contradiction to the standard Doniach picture. (ii) A novel mechanism is proposed which ferromagnetically correlates the impurities' magnetic moments without any indirect exchange. (iii) In case of degenerate Fermi levels, a multi-channel Kondo effect is observed. (iv) In the regime of strong couplings magnetic ordering is shown to be driven by a novel inverse indirect magnetic exchange (IIME), which is mediated by magnetically inert Kondo singlets.
The evolution of this physics is studied from the single-impurity Kondo box limit via the Kondo-vs.-RKKY quantum box to the regime of the Kondo-lattice box. Key insights are gained analytically by deriving effective low-energy Hamiltonians in the respective regimes by perturbation theory, and numerically by means of density-matrix renormalisation group (DMRG), dynamical mean-field theory (DMFT), and full diagonalisation calculations.
The presented theoretical concept is robust against weak environmental influences. The predicted effects await an experimental confirmation and exploration.
Der Kondo-Effekt, die Abschirmung eines Storstellen-Spins durch Leitungselektronen eines metallischen Wirtssystems, ist ein Standardproblem in der Festkorperphysik. Bei schwachen Kopplungen zwischen dem Wirtssystem und einer Gruppe von Storstellen konkurriert er mit dem indirekten, durch die Leitungselektronen vermittelten (RKKY) Austausch, der magnetische Ordnungen der Storstellen begunstigt.
Die vorliegende Arbeit behandelt die Physik von Storstellensystemen mit raumlich eingeschrankten metallischen Wirtsmaterialien, die von groser Bedeutung ist fur Quantenpunkte und kunstliche Nanostrukturen, die mit Rastertunneltechniken erschaffen werden konnen. Es wird gezeigt, dass (i) die endliche Systemgrose zu einem unkonventionellen Wettstreit fuhrt, in dem der Kondo-Effekt sich entgegen den Standardvorstellungen nach Doniach bei kleinen Kopplungen durchsetzt. (ii) Ein neuartiger Mechanismus wird vorgeschlagen, der die magnetischen Momente von Storstellen ohne jeglichen indirekten Austausch ferromagnetisch korreliert. (iii) Im Fall von entarteten Fermikanten-Zustanden wird ein multi-channel finite-size Kondo-Effekt beobachtet. (iv) Magnetische Ordnung im Regime groser Kopplungen wird durch einen neuartigen inversen indirekten magnetischen Austausch (IIMA) verursacht, der durch magnetisch inerte Kondo-Singlets vermittelt wird.
Die Entwicklung dieser Physik wird untersucht von der Eine-Storstelle-Kondo-Box, uber die Kondo-vs.-RKKY-Quantum-Box, bis hin zum Regime der Kondo-Gitter-Box. Schlusselerkenntnisse werden analytisch gewonnen durch die storungstheoretische Ableitung von effektiven Niedrigenergie-Hamiltonoperatoren in den entsprechenden Grenzfallen und numerisch mithilfe von Dichtematrix-Renormierungsgruppe (DMRG), dynamischer Meanfield-Theorie (DMFT) und voller Diagonalisierung.
Das vorgestellte Konzept ist robust gegen schwache Umgebungseinflusse. Die vorhergesagten Effekte sehen einer experimentellen Bestatigung und Erforschung entgegen.