We investigate the equilibrium and real-time properties of the spin correlation function $\langle \vec{S}_1\vec{S}_2 \rangle$ in the two-impurity Kondo model for different distances $R$ between the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of $\langle \vec{S}_1\vec{S}_2 \rangle$. For distances $R$ exceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings $J$ between the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain length scale introduced by finite temperature. The real-time dynamics after a sudden quench of the system reveals that correlations propagate through the conduction band with Fermi velocity. We identify two distinct timescales for the long time behavior which reflects that for small $J$ the system is driven by the RKKY interaction while for large $J$ the Kondo effect dominates. Interestingly, we find that at certain distances a one-dimensional dispersion obeying $\epsilon(k)=\epsilon(-k)$ may lead to a local parity conservation of the impurities such that $\langle \vec{S}_1\vec{S}_2 \rangle$ becomes a conserved quantity for long times and does not decay to its equilibrium value.

We show that the RKKY interaction in the two-impurity Anderson model comprise two contributions: a ferromagnetic part stemming from the symmetrized hybridization functions and an antiferromagnetic part. We demonstrate that this antiferromagnetic contribution can also be generated by an effective local tunneling term between the two impurities. This tunneling can be analytically calculated for particle-hole symmetric impurities. Replacing the full hybridization functions by the symmetric part and this tunneling term leads to the identical low-temperature fixed point spectrum in the numerical renormalization group. Compensating this tunneling term allows us to restore the Varma-Jones quantum critical point between a strong-coupling phase and a local singlet phase even in the absence of particle-hole symmetry in the hybridization functions. We analytically investigate the spatial frequencies of the effective tunneling term based on the combination of the band dispersion and the shape of the Fermi surface. Numerical renormalization group calculations provide a comparison of the distance-dependent tunneling term and the local spin-spin correlation function. Deviations between the spatial dependency of the full spin-spin correlation function and the textbook RKKY interaction are reported.

We investigate the competition between superconductivity, charge-ordering, magnetic-ordering, and the Kondo effect in a heavy fermion s-wave superconductor described by a Kondo lattice model with an attractive on-site Hubbard interaction. The model is solved using the real-space dynamical mean field theory. For this purpose, we develop a numerical renormalization group (NRG) framework in Nambu space, which is used to solve the superconducting impurity problem. This extended NRG scheme also allows for SU(2) spin symmetry broken solutions, enabling us to examine the competition or cooperation between s-wave superconductivity and incommensurate spin-density waves (SDWs). At half filling, we find an intriguing phase where the magnetic ordering of the f -electrons lifts the degeneracy between the charge density wave (CDW) state and the superconducting state, leading to a strong suppression of superconductivity. In addition, the system may also become a half metal in this parameter regime. Away from half filling, the CDWs vanish and are replaced by superconductivity combined with incommensurate SDWs up to moderate Kondo couplings to the f -electrons. We find that both CDWs as well as superconductivity enhance magnetic ordering due to the suppression of Kondo screening.

We show that the two-impurity Anderson model exhibits an additional quantum critical point at infinitely many specific distances between both impurities for an inversion symmetric one-dimensional dispersion. Unlike the quantum critical point previously established, it is robust against particle-hole or parity symmetry breaking. The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, protected. A finite single-particle tunneling $t$ or an applied uniform gate voltage will drive the system across the quantum critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functions at low temperature, which might be useful for future spintronics devices. A local parity conservation will prevent the spin-spin correlation function from decaying to its equilibrium value after spin manipulations.

We show that the two impurity Anderson model exhibit an additional quantum critical point at infinitely many specific distances between both impurities for an inversion symmetric 1D dispersion. Unlike the quantum critical point previously established by Jones and Varma, it is robust against particle-hole or parity symmetry breaking. The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, topological protected. A finite single particle tunneling $t$ or an applied uniform gate voltages will drive the system across the quantum critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functions at low temperature which might be useful for future spintronics devices. A local parity conservation will prevent the spin-spin correlation function to decay to its equilibrium value after spin-manipulations.

The magnetic properties of nanostructures that consist of a small number of atoms or molecules are typically determined by magnetic exchange interactions. Here, we show that non-magnetic, chemical interactions can have a similarly decisive effect if spin-moment-carrying orbitals extend in space and therefore allow the direct coupling of magnetic properties to wavefunction overlap and the formation of bonding and antibonding orbitals. We demonstrate this for a dimer of metal–molecule complexes on the Au(111) surface. A changing wavefunction overlap between the two monomers drives the surface-adsorbed dimer through a quantum phase transition from an underscreened triplet to a singlet ground state, with one configuration being located extremely close to a quantum critical point.

By means of low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS), we have investigated the adsorption of single Au atoms on a PTCDA monolayer physisorbed on the Au(111) surface. A chemical reaction between the Au atom and the PTCDA molecule leads to the formation of a radical that has an unpaired electron in its highest occupied orbital. This orbital is a pi orbital that extends over the whole Au-PTCDA complex. Because of the large Coulomb repulsion in this orbital, the unpaired electron generates a local moment when the molecule is adsorbed on the Au(111) surface. We demonstrate the formation of the radical and the existence of the local moment after adsorption by observing a zero-bias differential conductance peak that originates from the Kondo effect. By temperature dependent measurements of the zero-bias differential conductance, we determine the Kondo temperature to be T-K = (38 +/- 8) K. For the theoretical description of the properties of the Au-PTCDA complex we use a hierarchy of methods, ranging from density functional theory (DFT) including a van der Waals correction to many-body perturbation theory (MBPT) and the numerical renormalization group (NRG) approach. Regarding the high-energy orbital spectrum, we obtain an excellent agreement with experiments by both spin-polarized DFT/MBPT and NRG. Moreover, the NRG provides an accurate description of the low-energy excitation spectrum of the spin degree of freedom, predicting a Kondo temperature very close to the experimental value. This is achieved by a detailed analysis of the universality of various definitions of T-K and by taking into account the full energy dependence of the coupling function between the molecule-metal complex and the metallic substrate.

We address the fundamental question how the spatial Kondo correlations are building up in time assuming an initially decoupled impurity spin $\vec{S}_{\rm imp}$. We investigate the time-dependent spin-correlation function $\chi(\vec{r},t) = \langle \vec{S}_\mathrm{imp} \vec{s}(\vec{r}) \rangle (t)$ in the Kondo model with antiferromagnetic and ferromagnetic couplings where $ \vec{s}(\vec{r})$ denotes the spin density of the conduction electrons after switching on the Kondo coupling at time $t=0$. We present data obtained from a time-dependent numerical renormalisation group (TD-NRG) calculation. We gauge the accuracy of our two-band NRG by the spatial sum-rules of the equilibrium correlation functions and the reproduction of the analytically exactly known spin-correlation function of the decoupled Fermi sea. We find a remarkable building up of Kondo-correlation outside of the light cone defined by the Fermi velocity of the host metal. By employing a perturbative approach exact in second-order of the Kondo coupling, we connect these surprising correlations to the intrinsic spin-density entanglement of the Fermi sea. The thermal wave length supplies a cutoff scale at finite temperatures beyond which correlations are exponentially suppressed. We present data for the frequency dependent retarded spin-spin susceptibility and use the results to calculate the real-time response of a weak perturbation in linear response: within the spatial resolution no response outside of the light cone is found.

Before transport data can be understood quantitatively, a few prerequisites have to be fulfilled: the geometric and the electronic structures of the metal/molecule contacts have to be known, and electron correlation effects have to be taken into account. Here we discuss experimental and theoretical approaches to tackle these challenges. On the theoretical side, density‐functional theory (including van der Waals‐corrections for structural optimization) is combined with many‐body perturbation theory and numerical renormalization group theory in order to include all relevant correlation effects. We had already discussed such features in a previous study [Phys. Rev. B 84, 125413 (2011)], but some remaining differences between our calculated spectra and our experimental data from a scanning‐tunnelling microscope remained unexplained. Here we show that the explicit incorporation of van der Waals interaction in the calculations, that had been negleted before, yields improved geometric structure and leads to much better agreement with our measured spectra. This clearly demonstrates the significant sensitivity of electronic transport to structural details.PTCDA molecule in a junction between a silver surface and an STM tip.