#### Prof. Dr. Frithjof B. Anders

Theoretical Physics

TU Dortmund University

##### Contact

- frithjof[dot]anders[at]tu-dortmund[dot]de
- +49 231 755 7958
- personal website

##### Hub

**Cross-correlation spectra in interacting quantum dot systems**

Fischer, A. and Kleinjohann, I. and Sinitsyn, N.A. and Anders, F.B.*Physical Review B*105 (2022)Two-color spin-noise spectroscopy of interacting electron spins in singly charged semiconductor quantum dots provides information on the interquantum dot interactions. We investigate the spin cross-correlation function in a quantum dot ensemble employing a modified semiclassical approach. Spin-correlation functions are calculated using a Hamilton quaternion approach that maintains local quantum mechanical properties of the spins. This method takes into account the effects of the nuclear-electric quadrupolar interactions, the randomness of the coupling constants, and the variation of the electron g factor on the spin-noise power spectra. We demonstrate that the quantum dot ensemble can be mapped on an effective two-quantum dot problem and discuss how the characteristic length scale of the interdot interaction modifies the low-frequency cross-correlation spectrum. We argue that details on the interaction strength distribution can be extracted from the cross-correlation spectrum when applying a longitudinal or a transversal external magnetic field. ©2022 American Physical Societyview abstract 10.1103/PhysRevB.105.035303 **Microscopic Origin of the Effective Spin-Spin Interaction in a Semiconductor Quantum Dot Ensemble**

Vonhoff, F. and Fischer, A. and Deltenre, K. and Anders, F.B.*Physical Review Letters*129 (2022)view abstract 10.1103/PhysRevLett.129.167701 **Nuclear-spin polaron formation: Anisotropy effects and quantum phase transition**

Kleinjohann, I. and Fischer, A. and Glazov, M.M. and Anders, F.B.*Physical Review B*105 (2022)view abstract 10.1103/PhysRevB.105.195309 **Open Wilson chain numerical renormalization group approach to Green's functions**

Böker, J. and Anders, F.B.*Physical Review B*105 (2022)view abstract 10.1103/PhysRevB.105.235127 **Femtosecond phononic coupling to both spins and charges in a room-temperature antiferromagnetic semiconductor**

Bossini, D. and Dal Conte, S. and Terschanski, M. and Springholz, G. and Bonanni, A. and Deltenre, K. and Anders, F. and Uhrig, G.S. and Cerullo, G. and Cinchetti, M.*Physical Review B*104 (2021)Spintronics is postulated on the possibility to employ the magnetic degree of freedom of electrons for computation and couple it to charges. In this view the combination of the high frequency of spin manipulations offered by antiferromagnets, with the wide tunability of the electronic properties peculiar of semiconductors, provides a promising and intriguing platform. Here we explore this scenario in α-MnTe, which is a semiconductor antiferromagnetically ordered at room temperature. Relying on a Raman mechanism and femtosecond laser pulses, we drive degenerate modes of coherent optical phonons, which modulate the chemical bonds involved in the superexchange interaction. The spectrally resolved measurements of the transient reflectivity reveal a coherent modulation of the band gap at the frequency of 5.3 THz. The detection of the rotation of the polarization, typically associated with magneto-optical effects, shows coherent and incoherent contributions. Modeling how the ionic motion induced by the phonons affects the exchange interaction in the material, we calculate the photoinduced THz spin dynamics: the results predict both a coherent and incoherent response, the latter of which is consistent with the experimental observation. Our work demonstrates that the same phonon modes modulate both the charge and magnetic degree of freedom, suggesting the resonant pumping of phonons as a viable way to link spin and charge dynamics even in nonlinear regimes. ©2021 American Physical Societyview abstract 10.1103/PhysRevB.104.224424 **Kondo holes in strongly correlated impurity arrays: RKKY-driven Kondo screening and hole-hole interactions**

Eickhoff, F. and Anders, F.B.*Physical Review B*104 (2021)The emerging and screening of local magnetic moments in solids have been investigated for more than 60 years. Local vacancies as in graphene or in heavy fermions can induce decoupled bound states that lead to the formation of local moments. In this paper, we address the puzzling question how these local moments can be screened and what determines the additionally emerging low-temperature scale. We review the initial problem for half-filled conduction bands from two complementary perspectives: By a single-particle supercell analysis in the uncorrelated limit and by the Lieb-Mathis theorem for systems with a large Coulomb interaction U. Applying Wilson's numerical renormalization group approach to a recently developed mapping of the problem onto an effective low-energy description of a Kondo hole with up to Nf=7 correlated impurities as background, we proof that the stable local moments are subject to screening by three different mechanisms. Firstly the local moments are delocalized by a finite U beyond the single-particle bound state. We find a Kosterlitz-Thouless type transition governed by an exponentially suppressed low-energy scale of a counterintuitive Kondo form with Jeff∝Un for small U, where n>1 depends on the precise model. Secondly, we show that away from half-filling the local moment phase becomes unstable and is replaced by two types of singlet phases that are adiabatically connected. At a critical value for the band center, the physics is governed by an exponentially suppressed Kondo scale approaching the strong coupling phase that is replaced by a singlet formation via antiferromagnetic RKKY interaction for large deviation from the critical values. Thirdly, we show that the local magnetic moment can be screened by a Kondo hole orbital at finite energy, even though the orbital occupation is negligible: An additional low-energy scale emerges below which the localized moment is quenched. Similarities to the experimental findings in Ce1-xLaxPd3 are pointed out. © 2021 American Physical Society.view abstract 10.1103/PhysRevB.104.045115 **Lattice-driven femtosecond magnon dynamics in α-MnTe**

Deltenre, K. and Bossini, D. and Anders, F.B. and Uhrig, G.S.*Physical Review B*104 (2021)The light-induced femtosecond dynamics of the sublattice magnetizations in the antiferromagnetically ordered phase of the semiconductor α-MnTe is investigated theoretically as a function of an external driving field. The electromagnetic field is coupled to optical modes and the concomitant atomic displacements modulate the Heisenberg exchange couplings. We derive the equations of motion for the time-dependent sublattice magnetization in spin wave theory and analyze the contributions from the driven magnon modes. The antiferromagnetic order parameter exhibits coherent longitudinal oscillations determined by the external driving frequency which decay due to dephasing. Including a phenomenological dissipative term to mimic spin-lattice relaxation processes leads to relaxation back to thermal equilibrium. We provide approximate analytic solutions of the resulting differential equations which allow us to understand the effect of the driving light pulse on the amplitude, frequency, and lifetime of the coherent spin dynamics. © 2021 American Physical Society.view abstract 10.1103/PhysRevB.104.184419 **Magnetic blue shift of Mott gaps enhanced by double exchange**

Hafez-Torbati, M. and Bossini, D. and Anders, F.B. and Uhrig, G.S.*Physical Review Research*3 (2021)A substantial energy gap of charge excitations induced by strong correlations is the characteristic feature of Mott insulators. We study how the Mott gap is affected by long-range antiferromagnetic order. Our key finding is that the Mott gap is increased by the magnetic ordering: A magnetic blue shift (MBS) occurs. Thus the effect is proportional to the exchange coupling in the leading order in the Hubbard model. In systems with additional localized spins the double-exchange mechanism induces an additional contribution to the MBS which is proportional to the hopping in the leading order. The coupling between spin and charge degrees of freedom bears the potential to enable spin-to-charge conversion in Mott systems on extreme time scales determined by hopping and exchange only, since a spin-orbit-mediated transfer of angular momentum is not involved in the process. In view of spintronic and magnonic applications, it is highly promising to observe that several entire classes of compounds show exchange and double-exchange effects. Exemplarily, we show that the magnetic contribution to the band-gap blue shift observed in the optical conductivity of α-MnTe is correctly interpreted as the MBS of a Mott gap. © 2021 authors. Published by the American Physical Society.view abstract 10.1103/PhysRevResearch.3.043232 **Spectral properties of strongly correlated multi-impurity models in the Kondo insulator regime: Emergent coherence, metallic surface states, and quantum phase transitions**

Eickhoff, F. and Anders, F.B.*Physical Review B*104 (2021)We investigate the real-space spectral properties of strongly correlated multi-impurity arrays in the Kondo insulator regime. Employing a recently developed mapping onto an effective correlated cluster problem makes the problem accessible to the numerical renormalization group. The evolution of the spectrum as a function of cluster size and cluster site is studied. We applied the extended Lieb-Mattis theorem to predict whether the spectral function must vanish at the Fermi energy developing a true pseudogap or whether the spectral function remains finite at . Our numerical renormalization group spectra confirm the predictions of the theorem and shows a metallic behavior at the surface of a cluster prevailing in arbitrary spatial dimensions. We present a conventional minimal extension of a particle-hole symmetric Anderson lattice model at that leads to a gapped bulk band but a surface band with mainly -orbital character for weak and moderate hybridization strength. The change in the site-dependent spectra upon introducing a Kondo hole in the center of the cluster is presented as a function of the hole-orbital energy. In particular, the spectral signatures across the Kosterlitz-Thouless-type quantum phase transition from a singlet to a local moment fixed point are discussed. © 2021 American Physical Societyview abstract 10.1103/PhysRevB.104.165105 **Analytical and numerical study of the out-of-equilibrium current through a helical edge coupled to a magnetic impurity**

Vinkler-Aviv, Y. and May, D. and Anders, F.B.*Physical Review B*101 (2020)We study the conductance of a time-reversal-symmetric helical electronic edge coupled antiferromagnetically to a magnetic impurity, employing analytical and numerical approaches. The impurity can reduce the perfect conductance G0 of a noninteracting helical edge by generating a backscattered current. The backscattered steady-state current tends to vanish below the Kondo temperature TK for time-reversal-symmetric setups. We show that the central role in maintaining the perfect conductance is played by a global U(1) symmetry. This symmetry can be broken by an anisotropic exchange coupling of the helical modes to the local impurity. Such anisotropy, in general, dynamically vanishes during the renormalization group (RG) flow to the strong-coupling limit at low temperatures. The role of the anisotropic exchange coupling is further studied using the time-dependent numerical renormalization group method, uniquely suitable for calculating out-of-equilibrium observables of strongly correlated setups. We investigate the role of finite-bias voltage and temperature in cutting the RG flow before the isotropic strong-coupling fixed point is reached, and extract the relevant energy scales and the manner in which the crossover from the weakly interacting regime to the strong-coupling backscattering-free screened regime is manifested. Most notably, we find that at low temperatures the conductance of the backscattering current follows a power-law behavior G∼(T/TK)2, which we understand as a strong nonlinear effect due to time-reversal-symmetry breaking by the finite bias. © 2020 American Physical Society. ©2020 American Physical Society.view abstract 10.1103/PhysRevB.101.165112 **Inelastic electron tunneling spectroscopy for probing strongly correlated many-body systems by scanning tunneling microscopy**

Eickhoff, F. and Kolodzeiski, E. and Esat, T. and Fournier, N. and Wagner, C. and Deilmann, T. and Temirov, R. and Rohlfing, M. and Tautz, F.S. and Anders, F.B.*Physical Review B*101 (2020)We present an extension of the tunneling theory for scanning tunneling microscopy (STM) to include different types of electron-vibrational couplings responsible for inelastic contributions to the tunnel current in the strong-coupling limit. It allows for a better understanding of more complex scanning tunneling spectra of molecules on a metallic substrate in separating elastic and inelastic contributions. The starting point is the exact solution of the spectral functions for the electronically active local orbitals in the absence of the STM tip. This includes electron-phonon coupling in the coupled system comprising the molecule and the substrate to arbitrary order including the antiadiabatic strong-coupling regime as well as the Kondo effect on a free-electron spin of the molecule. The tunneling current is derived in second order of the tunneling matrix element which is expanded in powers of the relevant vibrational displacements. We use the results of an ab initio calculation for the single-particle electronic properties as an adapted material-specific input for a numerical renormalization group approach for accurately determining the electronic properties of a 1,4,5,8-naphthalene-Tetracarboxylic acid dianhydride molecule on Ag(111) as a challenging sample system for our theory. Our analysis shows that the mismatch between the ab initio many-body calculation of the tunnel current in the absence of any electron-phonon coupling to the experimental scanning tunneling spectra can be resolved by including two mechanisms: (i) a strong unconventional Holstein term on the local substrate orbital leads to the reduction of the Kondo temperature and (ii) a further electron-vibrational coupling to the tunneling matrix element is responsible for inelastic steps in the dI/dV curve at finite frequencies. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.101.125405 **Kinetic approach to nuclear-spin polaron formation**

Fischer, A. and Kleinjohann, I. and Anders, F.B. and Glazov, M.M.*Physical Review B*102 (2020)Under optical cooling of nuclei, a strongly correlated nuclear-spin polaron state can form in semiconductor nanostructures with localized charge carriers due to the strong hyperfine interaction of the localized electron spin with the surrounding nuclear spins. Here we develop a kinetic-equation formalism describing the nuclear-spin polaron formation. We present a derivation of the kinetic equations for an electron-nuclear spin system coupled to reservoirs of different electron and nuclear spin temperatures which generate the exact thermodynamic steady state for equal temperatures independent of the system size. We illustrate our approach using the analytical solution of the central spin model in the limit of an Ising form of the hyperfine coupling. For homogeneous hyperfine coupling constants, i.e., the box model, the model is reduced to an analytically solvable form. Based on the analysis of the nuclear-spin distribution function and the electron-nuclear spin correlators, we derive a relation between the electron and nuclear spin temperatures, where the correlated nuclear-spin polaron state is formed. In the limit of large nuclear baths, this temperature line coincides with the critical temperature of the mean-field theory for polaron formation. The criteria of the polaron formation in a finite-size system are discussed. We demonstrate that the system's behavior at the transition temperature does not depend on details of the hyperfine-coupling distribution function but only on the effective number of coupled bath spins. In addition, the kinetic equations enable the analysis of the temporal formation of the nuclear-polaron state, where we find the build-up process predominated by the nuclear spin-flip dynamics. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.102.165309 **Restoring the continuum limit in the time-dependent numerical renormalization group approach**

Böker, J. and Anders, F.B.*Physical Review B*102 (2020)The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite-size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the nonequilibrium dynamics on the finite-size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter Λ>1. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation, a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic QISs. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interacting resonant-level model and the single-impurity Anderson model. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.102.075149 **Strongly correlated multi-impurity models: The crossover from a single-impurity problem to lattice models**

Eickhoff, F. and Anders, F.B.*Physical Review B*102 (2020)We present a mapping of various correlated multi-impurity Anderson models to a cluster model coupled to a number of effective conduction bands capturing its essential low-energy physics. The major ingredient is the complex single-particle self-energy matrix of the uncorrelated problem that encodes the influence to the host conduction band onto the dynamics of a set of correlated orbitals in a given geometry. While the real part of the self-energy matrix generates an effective hopping between the cluster orbitals, the imaginary part, or hybridization matrix, determines the coupling to the effective conduction electron bands in the mapped model. The rank of the hybridization matrix determines the number of independent screening channels of the problem, and allows the replacement of the phenomenological exhaustion criterion by a rigorous mathematical statement. This rank provides a distinction between multi-impurity models of the first kind and of the second kind. For the latter, there are insufficient screening channels available, so that a singlet ground state must be driven by the intercluster spin correlations. This classification provides a fundamental answer to the question of why ferromagnetic exchange interactions between local moments are irrelevant for the spin-compensated ground state in dilute multi-impurity models, whereas the formation of large spins competes with the Kondo scale in dense impurity arrays, without evoking a spin density wave. The low-temperature physics of three examples taken from the literature are deduced from the analytic structure of the mapped model, demonstrating the potential power of this approach. Numerical renormalization group calculations are presented for up to five-site clusters. We investigate the appearance of frustration-induced non-Fermi-liquid fixed points in the trimer, and demonstrate the existence of several critical points of Kosterlitz-Thouless type at which ferromagnetic correlations suppress the screening of an additional effective spin-1/2 degree of freedom. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.102.205132 **Symmetric single-impurity Kondo model on a tight-binding chain: Comparison of analytical and numerical ground-state approaches**

Barcza, G. and Bauerbach, K. and Eickhoff, F. and Anders, F.B. and Gebhard, F. and Legeza, Ö.*Physical Review B*101 (2020)We analyze the ground-state energy, local spin correlation, impurity spin polarization, impurity-induced magnetization, and corresponding zero-field susceptibilities of the symmetric single-impurity Kondo model on a tight-binding chain with bandwidth W=2D where a spin-12 impurity at the chain center interacts with coupling strength JK with the local spin of the bath electrons. We compare perturbative results and variational upper bounds from Yosida, Gutzwiller, and first-order Lanczos wave functions to the numerically exact extrapolations obtained from the density-matrix renormalization group (DMRG) method and from the numerical renormalization group (NRG) method performed with respect to the inverse system size and Wilson parameter, respectively. In contrast to the Lanczos and Yosida wave functions, the Gutzwiller variational approach becomes exact in the strong-coupling limit JK≫W, and reproduces the ground-state properties from DMRG and NRG for large couplings JKâ‰W, with a high accuracy. For weak coupling, the Gutzwiller wave function describes a symmetry-broken state with an oriented local moment, in contrast to the exact solution. We calculate the impurity spin polarization and its susceptibility in the presence of magnetic fields that are applied globally or only locally to the impurity spin. The Yosida wave function provides qualitatively correct results in the weak-coupling limit. In DMRG, chains with about 103 sites are large enough to describe the susceptibilities down to JK/D≈0.6. For smaller Kondo couplings, only the NRG provides reliable results for a general host-electron density of states ρ0(ϵ). To compare with results from Bethe ansatz that become exact in the wide-band limit, we study the impurity-induced magnetization and zero-field susceptibility. For small Kondo couplings, the zero-field susceptibilities at zero temperature approach χ0(JKâ‰ªD)/(gμB)2≈exp[1/(ρ0(0)JK)]/[2CDπeρ0(0)JK], where ln(C) is the regularized first inverse moment of the density of states. Using NRG, we determine the universal subleading corrections up to second order in ρ0(0)JK. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.101.075132 **Fourth-order spin correlation function in the extended central spin model**

Fröhling, N. and Jäschke, N. and Anders, F.B.*Physical Review B*99 (2019)Spin-noise spectroscopy has developed into a very powerful tool to access the electron spin dynamics. While the spin-noise power spectrum in an ensemble of quantum dots in a magnetic field is essentially understood, we argue that the investigation of the higher-order cumulants promises to provide additional information not accessible by the conventional power-noise spectrum. We present a quantum-mechanical approach to the correlation function of the spin-noise power operators at two different frequencies for small spin bath sizes and compare the results with a simulation obtained from the classical spin dynamics for large number of nuclear spins. This bispectrum is defined as a two-dimensional frequency cut in the parameter space of the fourth-order spin correlation function. It reveals information on the influence of the nuclear-electric quadrupolar interactions on the long-time electron spin dynamics dominated by a magnetic field. For large bath sizes and spin lengths the quantum-mechanical spectra converge to those of the classical simulations. The broadening of the bispectrum across the diagonal in the frequency space is a direct measure of the quadrupolar interaction strength. A narrowing is found with increasing magnetic field indicating a suppression of the influence of quadrupolar interactions in favor of the nuclear Zeeman effect. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.99.155305 **Effective low-energy description of the two-impurity Anderson model: RKKY interaction and quantum criticality**

Eickhoff, F. and Lechtenberg, B. and Anders, F.B.*Physical Review B*98 (2018)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. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.115103 **Electron spin noise under the conditions of nuclei-induced frequency focusing**

Jäschke, N. and Anders, F.B. and Glazov, M.M.*Physical Review B*98 (2018)We study theoretically the electron spin noise in quantum dots under nonequilibrium conditions caused by the pumping by a train of circularly polarized optical pulses. In such a situation, the nuclear spins are known to adjust in such a way that the electron spin precession frequencies become multiples of the pump pulse repetition frequency. This so-called phase synchronization effect was uncovered in A. Greilich et al. [Science 317, 1896 (2007)SCIEAS0036-807510.1126/science.1146850] and termed nuclei-induced frequency focusing of electron spin coherence. Using the classical approach to the central spin model, we evaluate the nuclear spin distribution function and the electron spin noise spectrum. We show that the electron spin noise spectrum consists of sharp peaks corresponding to the phase synchronization conditions and directly reveal the distribution of the nuclear spins. We discuss the effects of nuclear spin relaxation after the pumping is over and analyze the corresponding evolution of nuclear spin distributions and electron spin noise spectra. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.045307 **Equilibrium and real-time properties of the spin correlation function in the two-impurity Kondo model**

Lechtenberg, B. and Anders, F.B.*Physical Review B*98 (2018)We investigate the equilibrium and real-time properties of the spin-correlation function (S - 1S - 2) 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 (S - 1S - 2). 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 ϵ(k)=ϵ(-k) may lead to a local parity conservation of the impurities such that (S - 1S - 2) becomes a conserved quantity for long times and does not decay to its equilibrium value. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.035109 **Inducing Kondo screening of vacancy magnetic moments in graphene with gating and local curvature**

Jiang, Y. and Lo, P.-W. and May, D. and Li, G. and Guo, G.-Y. and Anders, F.B. and Taniguchi, T. and Watanabe, K. and Mao, J. and Andrei, E.Y.*Nature Communications*9 (2018)In normal metals the magnetic moment of impurity-spins disappears below a characteristic Kondo temperature which marks the formation of a cloud of conduction-band electrons that screen the local-moment. In contrast, moments embedded in insulators remain unscreened at all temperatures. What then is the fate of magnetic-moments in intermediate, pseudogap systems, such as graphene? Theory predicts that coupling to the conduction-band electrons will drive a quantum phase transition between a local-moment phase and a Kondo-screened phase. However, attempts to experimentally confirm this prediction and its intriguing consequences, such as electrostatically tunable magnetic-moments, have been elusive. Here we report the observation of Kondo-screening and the quantum phase-transition between screened and unscreened phases of vacancy magnetic moments in graphene. Using scanning tunneling spectroscopy and numerical renormalization-group calculations we show that this transition enables to control the screening of local moments by tuning the gate voltage and the local curvature of the graphene membrane. © 2018 The Author(s).view abstract 10.1038/s41467-018-04812-6 **Magnetic field dependence of the electron spin revival amplitude in periodically pulsed quantum dots**

Kleinjohann, I. and Evers, E. and Schering, P. and Greilich, A. and Uhrig, G.S. and Bayer, M. and Anders, F.B.*Physical Review B*98 (2018)Periodic laser pulsing of singly charged semiconductor quantum dots in an external magnetic field leads to a synchronization of the spin dynamics with the optical excitation. The pumped electron spins partially rephase prior to each laser pulse, causing a revival of electron spin polarization with its maximum at the incidence time of a laser pulse. The amplitude of this revival is amplified by the frequency focusing of the surrounding nuclear spins. Two complementary theoretical approaches for simulating up to 20 million laser pulses are developed and employed that are able to bridge between 11 orders of magnitude in time: a fully quantum mechanical description limited to small nuclear bath sizes and a technique based on the classical equations of motion applicable for a large number of nuclear spins. We present experimental data of the nonmonotonic revival amplitude as function of the magnetic field applied perpendicular to the optical axis. The dependence of the revival amplitude on the external field with a profound minimum at 4T is reproduced by both of our theoretical approaches and is ascribed to the nuclear Zeeman effect. Since the nuclear Larmor precession determines the electronic resonance condition, it also defines the number of electron spin revolutions between pump pulses, the orientation of the electron spin at the incidence time of a pump pulse, and the resulting revival amplitude. The magnetic field of 4T, for example, corresponds to half a revolution of nuclear spins between two laser pulses. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.155318 **Modeling of the gate-controlled Kondo effect at carbon point defects in graphene**

May, D. and Lo, P.-W. and Deltenre, K. and Henke, A. and Mao, J. and Jiang, Y. and Li, G. and Andrei, E.Y. and Guo, G.-Y. and Anders, F.B.*Physical Review B*97 (2018)We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound σ orbital and the vacancy-induced bound π state in an effective two-orbital single-impurity model. The local magnetic moments are stabilized by the Coulomb interaction as well as a significant ferromagnetic Hund's rule coupling between the orbitals predicted by a density functional theory calculation. A hybridization between the orbitals and the Dirac fermions is generated by the curvature of the graphene sheet in the vicinity of the vacancy. We present results for the local spectral function calculated using Wilson's numerical renormalization group approach for a realistic graphene band structure and find three different regimes depending on the filling, the controlling chemical potential, and the hybridization strength. These different regions are characterized by different magnetic properties. The calculated spectral functions qualitatively agree with recent scanning tunneling spectra on graphene vacancies. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.97.155419 **Nuclear spin noise in the central spin model**

Fröhling, N. and Anders, F.B. and Glazov, M.*Physical Review B*97 (2018)We study theoretically the fluctuations of the nuclear spins in quantum dots employing the central spin model which accounts for the hyperfine interaction of the nuclei with the electron spin. These fluctuations are calculated both with an analytical approach using homogeneous hyperfine couplings (box model) and with a numerical simulation using a distribution of hyperfine coupling constants. The approaches are in good agreement. The box model serves as a benchmark with low computational cost that explains the basic features of the nuclear spin noise well. We also demonstrate that the nuclear spin noise spectra comprise a two-peak structure centered at the nuclear Zeeman frequency in high magnetic fields with the shape of the spectrum controlled by the distribution of the hyperfine constants. This allows for direct access to this distribution function through nuclear spin noise spectroscopy. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.97.195311 **Signatures of long-range spin-spin interactions in an (In,Ga)As quantum dot ensemble**

Fischer, A. and Evers, E. and Varwig, S. and Greilich, A. and Bayer, M. and Anders, F.B.*Physical Review B*98 (2018)We present an investigation of the electron spin dynamics in an ensemble of singly charged semiconductor quantum dots subject to an external magnetic field and laser pumping with circularly polarized light. The spectral laser width is tailored such that ensembles with an increasing number of quantum dots are coherently pumped. Surprisingly, the dephasing time T∗ of the electron spin polarization depends only weakly on the laser spectral width. These findings can be consistently explained by a cluster theory of coupled quantum dots with a long-range electronic spin-spin interaction. We present a numerical simulation of the spin dynamics based on the central spin model that includes a quantum mechanical description of the laser pulses as well as a time-independent Heisenberg interaction between each pair of electron spins. We discuss the individual dephasing contributions stemming from the Overhauser field, the distribution of the electron g factors, and the electronic spin-spin interaction as well as the spectral width of the laser pulse. This analysis reveals counterbalancing effects on the total dephasing time when increasing the spectral laser width. On one hand, the increasing deviations of the electron g factors reduce the dephasing time. On the other hand, more electron spins are coherently pumped and synchronize due to the electronic spin-spin interaction which extends the dephasing time. We find an excellent agreement between the experimental data and the dephasing time in the simulation using an exponential distribution of Heisenberg couplings with a mean value J≈0.26μeV. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.205308 **Exchange couplings for Mn ions in CdTe: Validity of spin models for dilute magnetic II-VI semiconductors**

Linneweber, T. and Bünemann, J. and Löw, U. and Gebhard, F. and Anders, F.*Physical Review B - Condensed Matter and Materials Physics*95 (2017)We employ density-functional theory (DFT) in the generalized gradient approximation (GGA) and its extensions GGA+U and GGA+Gutzwiller to calculate the magnetic exchange couplings between pairs of Mn ions substituting Cd in a CdTe crystal at very small doping. DFT(GGA) overestimates the exchange couplings by a factor of 3 because it underestimates the charge-transfer gap in Mn-doped II-VI semiconductors. Fixing the nearest-neighbor coupling J1 to its experimental value in GGA+U, in GGA+Gutzwiller, or by a simple scaling of the DFT(GGA) results provides acceptable values for the exchange couplings at second-, third-, and fourth-neighbor distances in Cd(Mn)Te, Zn(Mn)Te, Zn(Mn)Se, and Zn(Mn)S. In particular, we recover the experimentally observed relation J4>J2,J3. The filling of the Mn 3d shell is not integer, which puts the underlying Heisenberg description into question. However, using a few-ion toy model the picture of a slightly extended local moment emerges so that an integer 3d-shell filling is not a prerequisite for equidistant magnetization plateaus, as seen in experiment. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.95.045134 **Influence of the nuclear Zeeman effect on mode locking in pulsed semiconductor quantum dots**

Beugeling, W. and Uhrig, G.S. and Anders, F.B.*Physical Review B*96 (2017)The coherence of the electron spin in a semiconductor quantum dot is strongly enhanced by mode locking through nuclear focusing, where the synchronization of the electron spin to periodic pulsing is slowly transferred to the nuclear spins of the semiconductor material, mediated by the hyperfine interaction between these. The external magnetic field that drives the Larmor oscillations of the electron spin also subjects the nuclear spins to a Zeeman-like coupling, albeit a much weaker one. For typical magnetic fields used in experiments, the energy scale of the nuclear Zeeman effect is comparable to that of the hyperfine interaction, so that it is not negligible. In this work, we analyze the influence of the nuclear Zeeman effect on mode locking quantitatively. Within a perturbative framework, we calculate the Overhauser-field distribution after a prolonged period of pulsing. We find that the nuclear Zeeman effect can exchange resonant and nonresonant frequencies. We distinguish between models with a single type and with multiple types of nuclei. For the latter case, the positions of the resonances depend on the individual g factors, rather than on the average value. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.96.115303 **Long-time coherence in fourth-order spin correlation functions**

Fröhling, N. and Anders, F.B.*Physical Review B*96 (2017)We study the long-time decay of fourth-order electron spin correlation functions for an isolated singly charged semiconductor quantum dot. The electron spin dynamics is governed by the applied external magnetic field as well as the hyperfine interaction. While the long-time coherent oscillations in the correlation functions can be understood within a semiclassical approach treating the Overhauser field as frozen, the field dependent decay of its amplitude reported in different experiments cannot be explained by the central-spin model indicating the insufficiency of such a description. By incorporating the nuclear Zeeman splitting and the strain induced nuclear-electric quadrupolar interaction, we find the correct crossover from a fast decay in small magnetic fields to a slow exponential asymptotic in large magnetic fields. It originates from a competition between the quadrupolar interaction inducing an enhanced spin decay and the nuclear Zeeman term that suppressed the spin-flip processes. We are able to explain the magnetic field dependency of the characteristic long-time decay time T2 depending on the experimental setups. The calculated asymptotic values of T2=3-4μs agree qualitatively well with the experimental data. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.96.045441 **Nonequilibrium nuclear spin distribution function in quantum dots subject to periodic pulses**

Jäschke, N. and Fischer, A. and Evers, E. and Belykh, V.V. and Greilich, A. and Bayer, M. and Anders, F.B.*Physical Review B*96 (2017)Electron spin dephasing in a singly charged semiconductor quantum dot can partially be suppressed by periodic laser pulsing. We propose a semiclassical approach describing the decoherence of the electron spin polarization governed by the hyperfine interaction with the nuclear spins as well as the probabilistic nature of the photon absorption. We use the steady-state Floquet condition to analytically derive two subclasses of resonance conditions excellently predicting the peak locations in the part of the Overhauser field distribution which is projected in the direction of the external magnetic field. As a consequence of the periodic pulsing, a nonequilibrium distribution develops as a function of time. The numerical simulation of the coupled dynamics reveals the influence of the hyperfine coupling constant distribution onto the evolution of the electron spin polarization before the next laser pulse. Experimental indications are provided for both subclasses of resonance conditions. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.96.205419 **Open Wilson chains for quantum impurity models: Keeping track of all bath modes**

Bruognolo, B. and Linden, N.-O. and Schwarz, F. and Lee, S.-S.B. and Stadler, K. and Weichselbaum, A. and Vojta, M. and Anders, F.B. and Von Delft, J.*Physical Review B*95 (2017)view abstract 10.1103/PhysRevB.95.121115 **Realistic quantum critical point in one-dimensional two-impurity models**

Lechtenberg, B. and Eickhoff, F. and Anders, F.B.*Physical Review B*96 (2017)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. © 2017 American Physical Society.view abstract 10.1103/PhysRevB.96.041109 **A chemically driven quantum phase transition in a two-molecule Kondo system**

Esat, T. and Lechtenberg, B. and Deilmann, T. and Wagner, C. and Krüger, P. and Temirov, R. and Rohlfing, M. and Anders, F.B. and Tautz, F.S.*Nature Physics*12 (2016)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. © 2006 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.view abstract 10.1038/nphys3737 **Decoherence of a single spin coupled to an interacting spin bath**

Wu, N. and Fröhling, N. and Xing, X. and Hackmann, J. and Nanduri, A. and Anders, F.B. and Rabitz, H.*Physical Review B - Condensed Matter and Materials Physics*93 (2016)Decoherence of a central spin coupled to an interacting spin bath via inhomogeneous Heisenberg coupling is studied by two different approaches, namely an exact equations of motion (EOMs) method and a Chebyshev expansion technique (CET). By assuming a wheel topology of the bath spins with uniform nearest-neighbor XX-type intrabath coupling, we examine the central spin dynamics with the bath prepared in two different types of bath initial conditions. For fully polarized baths in strong magnetic fields, the polarization dynamics of the central spin exhibits a collapse-revival behavior in the intermediate-time regime. Under an antiferromagnetic bath initial condition, the two methods give excellently consistent central spin decoherence dynamics for finite-size baths of N ≤ 14 bath spins. The decoherence factor is found to drop off abruptly on a short time scale and approach a finite plateau value which depends on the intrabath coupling strength nonmonotonically. In the ultrastrong intrabath coupling regime, the plateau values show an oscillatory behavior depending on whether N/2 is even or odd. The observed results are interpreted qualitatively within the framework of the EOM and perturbation analysis. The effects of anisotropic spin-bath coupling and inhomogeneous intrabath bath couplings are briefly discussed. Possible experimental realization of the model in a modified quantum corral setup is suggested. © 2016 American Physical Society. ©2016 American Physical Society.view abstract 10.1103/PhysRevB.93.035430 **Interplay of Coulomb interaction and spin-orbit coupling**

Bünemann, J. and Linneweber, T. and Löw, U. and Anders, F.B. and Gebhard, F.*Physical Review B - Condensed Matter and Materials Physics*94 (2016)We employ the Gutzwiller variational approach to investigate the interplay of Coulomb interaction and spin-orbit coupling in a three-orbital Hubbard model. Already in the paramagnetic phase we find a substantial renormalization of the spin-orbit coupling that enters the effective single-particle Hamiltonian for the quasiparticles. Only close to half band-filling and for sizable Coulomb interaction do we observe clear signatures of Hund's atomic rules for spin, orbital, and total angular momentum. For a finite local Hund's rule exchange interaction we find a ferromagnetically ordered state. The spin-orbit coupling considerably reduces the size of the ordered moment, it generates a small ordered orbital moment, and it induces a magnetic anisotropy. To investigate the magnetic anisotropy energy, we use an external magnetic field that tilts the magnetic moment away from the easy axis (1,1,1). © 2016 American Physical Society.view abstract 10.1103/PhysRevB.94.035116 **Quantum criticality of the two-channel pseudogap Anderson model: Universal scaling in linear and non-linear conductance**

Wu, T.-P. and Wang, X.-Q. and Guo, G.-Y. and Anders, F. and Chung, C.-H.*Journal of Physics Condensed Matter*28 (2016)The quantum criticality of the two-lead two-channel pseudogap Anderson impurity model is studied. Based on the non-crossing approximation (NCA) and numerical renormalization group (NRG) approaches, we calculate both the linear and nonlinear conductance of the model at finite temperatures with a voltage bias and a power-law vanishing conduction electron density of states, ρ(c)(ω) α | ω-µ(F)|(r) (0 < r < 1) near the Fermi energy µ(F). At a fixed leadimpurity hybridization, a quantum phase transition from the two-channel Kondo (2CK) to the local moment (LM) phase is observed with increasing r from r = 0 to r = r(c) < 1. Surprisingly, in the 2CK phase, different power-law scalings from the well-known √T or √V form is found. Moreover, novel power-law scalings in conductances at the 2CK-LM quantum critical point are identified. Clear distinctions are found on the critical exponents between linear and non-linear conductance at criticality. The implications of these two distinct quantum critical properties for the non-equilibrium quantum criticality in general are discussed. © 2016 IOP Publishing Ltd.view abstract 10.1088/0953-8984/28/17/175003 **Quantum model for mode locking in pulsed semiconductor quantum dots**

Beugeling, W. and Uhrig, G.S. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*94 (2016)Quantum dots in GaAs/InGaAs structures have been proposed as a candidate system for realizing quantum computing. The short coherence time of the electronic quantum state that arises from coupling to the nuclei of the substrate is dramatically increased if the system is subjected to a magnetic field and to repeated optical pulsing. This enhancement is due to mode locking: oscillation frequencies resonant with the pulsing frequencies are enhanced, while off-resonant oscillations eventually die out. Because the resonant frequencies are determined by the pulsing frequency only, the system becomes immune to frequency shifts caused by the nuclear coupling and by slight variations between individual quantum dots. The effects remain even after the optical pulsing is terminated. In this work, we explore the phenomenon of mode locking from a quantum mechanical perspective. We treat the dynamics using the central-spin model, which includes coupling to 10-20 nuclei and incoherent decay of the excited electronic state, in a perturbative framework. Using scaling arguments, we extrapolate our results to realistic system parameters. We estimate that the synchronization to the pulsing frequency needs time scales in the order of 1s. © 2016 American Physical Society.view abstract 10.1103/PhysRevB.94.245308 **Spin noise of electrons and holes in (In,Ga)As quantum dots: Experiment and theory**

Glasenapp, P. and Smirnov, D.S. and Greilich, A. and Hackmann, J. and Glazov, M.M. and Anders, F.B. and Bayer, M.*Physical Review B - Condensed Matter and Materials Physics*93 (2016)The spin fluctuations of electron and hole doped self-assembled quantum dot ensembles are measured optically in the low-intensity limit of a probe laser for absence and presence of longitudinal or transverse magnetic fields. The experimental results are modeled by two complementary approaches based either on a semiclassical or quantum mechanical description. This allows us to characterize the hyperfine interaction of electron and hole spins with the surrounding bath of nuclei on time scales covering several orders of magnitude. Our results demonstrate (i) the intrinsic precession of the electron spin fluctuations around the effective Overhauser field caused by the host lattice nuclear spins, (ii) the comparably long time scales for electron and hole spin decoherence, as well as (iii) the dramatic enhancement of the spin lifetimes induced by a longitudinal magnetic field due to the decoupling of nuclear and charge carrier spins. © 2016 American Physical Society.view abstract 10.1103/PhysRevB.93.205429 **Influence of the Nuclear Electric Quadrupolar Interaction on the Coherence Time of Hole and Electron Spins Confined in Semiconductor Quantum Dots**

Hackmann, J. and Glasenapp, Ph. and Greilich, A. and Bayer, M. and Anders, F.B.*Physical Review Letters*115 (2015)The real-time spin dynamics and the spin noise spectra are calculated for p and n-charged quantum dots within an anisotropic central spin model extended by additional nuclear electric quadrupolar interactions and augmented by experimental data. Using realistic estimates for the distribution of coupling constants including an anisotropy parameter, we show that the characteristic long time scale is of the same order for electron and hole spins strongly determined by the quadrupolar interactions even though the analytical form of the spin decay differs significantly consistent with our measurements. The low frequency part of the electron spin noise spectrum is approximately 1/3 smaller than those for hole spins as a consequence of the spectral sum rule and the different spectral shapes. This is confirmed by our experimental spectra measured on both types of quantum dot ensembles in the low power limit of the probe laser. © 2015 American Physical Society.view abstract 10.1103/PhysRevLett.115.207401 **Quantum transport through a molecular level: A scattering states numerical renormalization group study**

Jovchev, A. and Anders, F.B.*Physica Scripta*2015 (2015)We use the scattering states numerical renormalization group (SNRG) approach to calculate the current I(V) through a single molecular level coupled to a local molecular phonon. The suppression of I for asymmetric junctions with increasing electron-phonon coupling, the hallmark of the Franck-Condon blockade, is discussed. We compare the SNRG currents with recently published data obtained by an iterative summation of path integrals approach (ISPI). Our results excellently agree with the ISPI currents for small and intermediate voltages. In the linear response regime I(V) approaches the current calculated from the equilibrium spectral function. We also present the temperature and voltage evolution of the non-equilibrium spectral functions for a particle-hole asymmetric junction with symmetric coupling to the lead. © 2015 The Royal Swedish Academy of Sciences.view abstract 10.1088/0031-8949/2015/T165/014007 **Transfering spin into an extended Ï orbital of a large molecule**

Esat, T. and Deilmann, T. and Lechtenberg, B. and Wagner, C. and Krüger, P. and Temirov, R. and Anders, F.B. and Rohlfing, M. and Tautz, F.S.*Physical Review B - Condensed Matter and Materials Physics*91 (2015)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 Ï 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 TK=(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 TK and by taking into account the full energy dependence of the coupling function between the molecule-metal complex and the metallic substrate. © 2015 American Physical Society.view abstract 10.1103/PhysRevB.91.144415 **Conservation laws protect dynamic spin correlations from decay: Limited role of integrability in the central spin model**

Uhrig, G.S. and Hackmann, J. and Stanek, D. and Stolze, J. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*90 (2014)Mazur's inequality renders statements about persistent correlations possible. We generalize it in a convenient form applicable to any set of linearly independent constants of motion. This approach is used to show rigorously that a fraction of the initial spin correlations persists indefinitely in the isotropic central spin model unless the average coupling vanishes. The central spin model describes a major mechanism of decoherence in a large class of potential realizations of quantum bits. Thus the derived results contribute significantly to the understanding of the preservation of coherence. We will show that persisting quantum correlations are not linked to the integrability of the model but are caused by a finite operator overlap with a finite set of constants of motion. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.90.060301 **From thermal equilibrium to nonequilibrium quench dynamics: A conserving approximation for the interacting resonant level**

Vinkler-Aviv, Y. and Schiller, A. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*90 (2014)We develop a low-order conserving approximation for the interacting resonant-level model (IRLM), and apply it to (i) thermal equilibrium, (ii) nonequilibrium steady state, and (iii) nonequilibrium quench dynamics. Thermal equilibrium is first used to carefully gauge the quality of the approximation by comparing the results with other well-studied methods, and finding good agreement for small values of the interaction. We analytically show that the power-law exponent of the renormalized level width usually derived using renormalization group approaches can also be correctly obtained in our approach in the weak interaction limit. A closed expression for the nonequilibrium steady-state current is derived and analytically and numerically evaluated. We find a negative differential conductance at large voltages, and the exponent of the power-law suppression of the steady-state current is calculated analytically at zero temperature. The response of the system to quenches is investigated for a single lead as well as for two-lead setup at finite voltage bias at particle-hole symmetry using a self-consistent two-times Keldysh Green function approach, and results are presented for the time-dependent current for different bias and contact interaction strength. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.90.155110 **Gate-tunable Kondo resistivity and dephasing rate in graphene studied by numerical renormalization group calculations**

Lo, P.-W. and Guo, G.-Y. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*89 (2014)Motivated by the recent observation of the Kondo effect in graphene in transport experiments, we investigate the resistivity and dephasing rate in the Kondo regime due to magnetic impurities in graphene with different chemical potentials (μ). The Kondo effect due to either carbon vacancies or magnetic adatoms in graphene is described by the single-orbital pseudogap asymmetric Anderson impurity model which is solved by the accurate numerical renormalization group method. We find that although the Anderson impurity model considered here is a mixed-valence system, it can be driven into either the Kondo [μ>μc (critical value) >0], mixed-valency (μ ≈ μc), or empty-orbital (μ<μc) regime by a gate voltage, giving rise to characteristic features in resistivity and dephasing rate in each regime. Specifically, in the case of μ<μc, the shapes of the resistivity (dephasing rate) curves for different μ are nearly identical. However, as temperature decreases, they start to increase to their maxima at a lower T/TK, but more rapidly [as (TK/T)3/2] than in normal metals [here, T (TK) denotes the (Kondo) temperature]. As T further decreases, after reaching the maximum, the dephasing rate drops more quickly than in normal metals, behaving as (T/TK)3 instead of (T/TK)2. Furthermore, the resistivity has a distinct peak above the saturation value near TK. In the case of μ>μc, in contrast, the resistivity curve has an additional broad shoulder above 10TK and the dephasing rate exhibits an interesting shoulder-peak shape. In the narrow boundary region (μ ≈ μc), both the resistivity and dephasing rate curves are similar to the corresponding ones in normal metals. This explains the conventional Kondo-like resistivity from recent experiments on graphene with defects, although the distinct features in the resistivity in the other cases (μ<μc or μ>μc) were not seen in the experiments. The interesting features in the resistivity and dephasing rate are analyzed in terms of the calculated T-dependent spectral function, correlation self-energy, and renormalized impurity level. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.89.195424 **Real-time dynamics induced by quenches across the quantum critical points in gapless Fermi systems with a magnetic impurity**

Kleine, C. and Mußhoff, J. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*90 (2014)The energy-dependent scattering of fermions from a localized orbital at an energy-dependent rate Γ(ɛ) α ɛr gives rise to quantum critical points (QCPs) in the pseudogap single-impurity Anderson model separating a local moment phase with an unscreened spin moment from a strong-coupling phase which slightly deviates from the screened phase of standard Kondo problem. Using the time-dependent numerical renormalization group (TD-NRG) approach we show that local dynamic properties always equilibrate towards a steady-state value even for quenches across the QCP but with systematic deviations from the thermal equilibrium depending on the distance to the critical coupling. Local nonequilibrium properties are presented for interaction quenches and hybridization quenches. We augment our numerical data by an analytical calculation that becomes exact at short times and find excellent agreement between the numerics and the analytical theory. For interaction quenches within the screened phase we find a universal function for the time-dependent local double occupancy. We trace back the discrepancy between our results and the data obtained by a time-dependent Gutzwiller variational approach to restrictions of the wave-function ansatz in the Gutzwiller theory: while the NRG ground states properly account for the formation of an extended spin moment which decouples from the system in the unscreened phase, the Gutzwiller ansatz only allows the formation of the spin moment on the local impurity orbital. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.90.235145 **Spatial and temporal propagation of Kondo correlations**

Lechtenberg, B. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*90 (2014)We address the fundamental question how the spatial Kondo correlations are building up in time assuming an initially decoupled impurity spin S - imp. We investigate the time-dependent spin-correlation function Ï(r - ,t)=©S - imps - (r - )(t) in the Kondo model with antiferromagnetic and ferromagnetic couplings, where s - (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 wavelength 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. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.90.045117 **Spin noise in a quantum dot ensemble: From a quantum mechanical to a semi-classical description**

Hackmann, J. and Smirnov, D.S. and Glazov, M.M. and Anders, F.B.*Physica Status Solidi (B) Basic Research*251 (2014)Spin noise spectroscopy is a promising technique for revealing the microscopic nature of spin dephasing processes in quantum dots (QDs). We compare the spin-noise in an ensemble of singly charged QDs calculated by two complementary approaches. The Chebyshev polynomial expansion technique (CET) accounts for the full quantum mechanical fluctuation of the nuclear spin bath and a semi-classical approach (SCA) is based on the averaging the electron spin dynamics over all different static Overhauser field configurations. We observe a remarkable agreement between both methods in the high-frequency part of the spectra determined by static nuclear fields. The low-frequency part is determined by the long time fluctuations of the Overhauser field. We find small differences in the spectra depending on the distribution of hyperfine couplings. The spin-noise spectra in strong enough magnetic fields where the nuclear dynamics is quenched calculated by two complimentary approaches are in perfect agreement. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/pssb.201451103 **Spin noise in the anisotropic central spin model**

Hackmann, J. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*89 (2014)Spin-noise measurements can serve as a direct probe for the microscopic decoherence mechanism of an electronic spin in semiconductor quantum dots (QDs). We have calculated the spin-noise spectrum in the anisotropic central spin model using a Chebyshev expansion technique which exactly accounts for the dynamics up to an arbitrary long but fixed time in a finite-size system. In the isotropic case, describing QD charge with a single electron, the short-time dynamics is in good agreement with quasistatic approximations for the thermodynamic limit. The spin-noise spectrum, however, shows strong deviations at low frequencies with a power-law behavior of Ï-3/4 corresponding to a t-1/4 decay at intermediate and long times. In the Ising limit, applicable to QDs with heavy-hole spins, the spin-noise spectrum exhibits a threshold behavior of (Ï-ÏL)-1/2 above the Larmor frequency ÏL=gμBB. In the generic anisotropic central spin model we have found a crossover from a Gaussian type of spin-noise spectrum to a more Ising-type spectrum with increasing anisotropy in a finite magnetic field. In order to make contact with experiments, we present ensemble averaged spin-noise spectra for QD ensembles charged with single electrons or holes. The Gaussian-type noise spectrum evolves to a more Lorentzian shape spectrum with increasing spread of characteristic time scales and g factors of the individual QDs. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.89.045317 **Double occupancy and magnetic susceptibility of the Anderson impurity model out of equilibrium**

Dirks, A. and Schmitt, S. and Han, J.E. and Anders, F. and Werner, P. and Pruschke, T.*EPL*102 (2013)We use different numerical approaches to calculate the double occupancy and magnetic susceptibility as a function of a bias voltage in an Anderson impurity model. Specifically, we compare results from the Matsubara voltage quantum Monte Carlo approach (MV-QMC), the scattering states numerical renormalization group (SNRG), and real-time quantum Monte Carlo (RT-QMC), covering Coulomb repulsions ranging from the weak-coupling well into the strong-coupling regime. We observe a distinctly different behavior of the double occupancy and the magnetic response. The former measures charge fluctuations and thus only indirectly exhibits the Kondo scale, while the latter exhibits structures on the scale of the equilibrium Kondo temperature. The Matsubara voltage approach and the scattering states numerical renormalization group yield consistent values for the magnetic susceptibility in the Kondo limit. On the other hand, all three numerical methods produce different results for the behavior of charge fluctuations in strongly interacting dots out of equilibrium. © Copyright EPLA, 2013.view abstract 10.1209/0295-5075/102/37011 **Hybrid NRG-DMRG approach to real-time dynamics of quantum impurity systems**

Güttge, F. and Anders, F.B. and Schollwöck, U. and Eidelstein, E. and Schiller, A.*Physical Review B - Condensed Matter and Materials Physics*87 (2013)A hybrid approach to nonequilibrium dynamics of quantum impurity systems is presented. The numerical renormalization group serves as a means to generate a suitable low-energy Hamiltonian, allowing for an accurate evaluation of the real-time dynamics of the problem up to exponentially long times using primarily the time-adaptive density-matrix renormalization group. In particular, by constructing a suitable hybrid chain, discretization errors are essentially eliminated on all time scales of interest. We extract the decay time of the interaction-enhanced oscillations in the interacting resonant-level model and show their quadratic divergence with the interaction strength U. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in 1/U. © 2013 American Physical Society.view abstract 10.1103/PhysRevB.87.115115 **Influence of vibrational modes on quantum transport through a nanodevice**

Jovchev, A. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*87 (2013)We employ the recently proposed scattering states numerical renormalization group (SNRG) approach to calculate I(V) and the differential conductance through a single molecular level coupled to a local molecular phonon using the spinless Anderson-Holstein model. We also discuss the equilibrium physics of the model and demonstrate that the low-energy Hamiltonian is given by an effective interacting resonant level model. From the NRG level flow, we directly extract the effective charge transfer scale Γeff and the dynamically induced capacitive coupling Ueff between the molecular level and the lead electrons, which turns out to be proportional to the polaronic energy shift Ep for the regimes investigated here. The equilibrium spectral functions for the different parameter regimes are discussed. The additional phonon peaks at multiples of the phonon frequency Ï0 correspond to additional maxima in the differential conductance. Nonequilibrium effects, however, lead to significant deviations between a symmetric junction and a junction in the tunnel regime. The suppression of the current for particle-hole asymmetric junctions with increasing electron-phonon coupling, the hallmark of the Franck-Condon blockade, is discussed. © 2013 American Physical Society.view abstract 10.1103/PhysRevB.87.195112 **Spectral properties of a molecular wire in the kondo regime**

Greuling, A. and Temirov, R. and Lechtenberg, B. and Anders, F.B. and Rohlfing, M. and Tautz, F.S.*Physica Status Solidi (B) Basic Research*250 (2013)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. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/pssb.201349238 **Coherent control of correlated nanodevices: A hybrid time-dependent numerical renormalization-group approach to periodic switching**

Eidelstein, E. and Schiller, A. and Güttge, F. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*85 (2012)The time-dependent numerical renormalization-group approach (TD-NRG), originally devised for tracking the real-time dynamics of quantum-impurity systems following a single quantum quench, is extended to multiple switching events. This generalization of the TD-NRG encompasses the possibility of periodic switching, allowing for coherent control of strongly correlated systems by an external time-dependent field. To this end, we have embedded the TD-NRG in a hybrid framework that combines the outstanding capabilities of the numerical renormalization group to systematically construct the effective low-energy Hamiltonian of the system with the prowess of complementary approaches for calculating the real-time dynamics derived from this Hamiltonian. We demonstrate the power of our approach by hybridizing the TD-NRG with the Chebyshev expansion technique in order to investigate periodic switching in the interacting resonant-level model. Although the interacting model shares the same low-energy fixed point as its noninteracting counterpart, we surprisingly find the gradual emergence of damped oscillations as the interaction strength is increased. Focusing on a single quantum quench and using a strong-coupling analysis, we reveal the origin of these interaction-induced oscillations and provide an analytical estimate for their frequency. The latter agrees well with the numerical results. © 2012 American Physical Society.view abstract 10.1103/PhysRevB.85.075118 **Ab initio study of a mechanically gated molecule: From weak to strong correlation**

Greuling, A. and Rohlfing, M. and Temirov, R. and Tautz, F.S. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)The electronic spectrum of a chemically contacted molecule in the junction of a scanning tunneling microscope can be modified by tip retraction. We analyze this effect by a combination of density-functional, many-body perturbation, and numerical renormalization-group theories taking into account both the nonlocality and the dynamics of electronic correlation. Our findings, in particular the evolution from a broad quasiparticle resonance below to a narrow Kondo resonance at the Fermi energy, correspond to the experimental observations. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.125413 **Comment on "scaling feature of magnetic field induced Kondo-peak splittings"**

Schmitt, S. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*83 (2011)In a recent work, Zhang and coworkers [Phys. Rev. BPLRBAQ1098-012110.1103/ PhysRevB.82.075111 82, 075111 (2010)] studied the Zeeman splitting of the Kondo resonance for the single impurity Anderson model in a finite magnetic field B with the numerical renormalization group (NRG) method. They report a discrepancy between the position of the Kondo resonance in the total spectral function and the position in the spin-resolved spectral function at large magnetic fields. Additionally, the position of the Kondo maximum exceeded the Zeeman energy for B/TK5-10, where TK is the low-energy Kondo scale of the model. In this comment, we argue that both these findings are a result of the specific choice of NRG parameter values. However, we reproduce the crossover in the splitting from Kondo-like behavior to a nonuniversal splitting larger than the Zeeman energy, but this crossover occurs at much larger fields of the order of the charge scale. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.83.197101 **Kinks in the electronic dispersion of the Hubbard model away from half filling**

Grete, P. and Schmitt, S. and Raas, C. and Anders, F.B. and Uhrig, G.S.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)We study kinks in the electronic dispersion of a generic strongly correlated system by dynamic mean-field theory (DMFT). The focus is on doped systems away from particle-hole symmetry where valence fluctuations matter potentially. Three different algorithms are compared to asses their strengths and weaknesses, as well as to clearly distinguish physical features from algorithmic artifacts. Our findings extend a view previously established for half-filled systems where kinks reflect the coupling of the fermionic quasiparticles to emergent collective modes, which are identified here as spin fluctuations. Kinks are observed when strong spin fluctuations are present and, additionally, a separation of energy scales for spin and charge excitations exists. Both criteria are met by strongly correlated systems close to a Mott-insulator transition. The energies of the kinks and their doping dependence fit well to the kinks in the cuprates, which is surprising in view of the spatial correlations neglected by DMFT. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.205104 **Nonequilibrium Zeeman splitting in quantum transport through nanoscale junctions**

Schmitt, S. and Anders, F.B.*Physical Review Letters*107 (2011)We calculate the differential conductance G(V) through a quantum dot in an applied magnetic field. We use a Keldysh conserving approximation for weakly correlated and the scattering-states numerical renormalization group for the intermediate and strongly correlated regime out of equilibrium. In the weakly correlated regime, the Zeeman splitting observable in G(V) strongly depends on the asymmetry of the device. In contrast, in the strongly correlated regime the position ΔK of the Zeeman-split zero-bias anomaly is almost independent of such asymmetries and of the order of the Zeeman energy Δ0. We find a crossover from the purely spin-fluctuation driven Kondo regime at small magnetic fields with ΔK < Δ 0 to a regime at large fields where the contribution of charge fluctuations induces larger splittings with ΔK > Δ0 as it was observed in recent experiments. © 2011 American Physical Society.view abstract 10.1103/PhysRevLett.107.056801 **Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids**

Schneider, I. and Fritz, L. and Anders, F.B. and Benlagra, A. and Vojta, M.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)We discuss the two-channel Kondo problem with a pseudogap density of states ρ(ω) α |ω|r of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r = 0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for 0 < r < rmax ≈ 0.23. For r > rmax, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively.We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ Ïµ expansion techniques to calculate critical properties near the critical dimensions r = 0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene.We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.125139 **Comparison between scattering-states numerical renormalization group and the kadanoff-baym-keldysh approach to quantum transport: Crossover from weak to strong correlations**

Schmitt, S. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*81 (2010)The quantum transport through nanoscale junctions is governed by the charging energy U of the device. We employ the recently developed scattering-states numerical renormalization-group approach to open quantum systems to study nonequilibrium Green's functions and current-voltage characteristics of such junctions for small and intermediate values of U. We establish the accuracy of the approach by a comparison with diagrammatic Kadanoff-Baym-Keldysh results which become exact in the weak-coupling limit U → 0. We demonstrate the limits of the diagrammatic expansions at intermediate values of the charging energy. While the numerical renormalization-group approach correctly predicts only one single, universal low-energy scale at zero bias voltage, some diagrammatic expansions yield two different low-energy scales for the magnetic and the charge fluctuations. At large voltages, however, the self-consistent second Born as well as the GW approximation reproduce the scattering-states renormalization-group spectral functions for symmetric junctions while for asymmetric junctions the voltage-dependent redistribution of spectral weight differs significantly in the different approaches. The second-order perturbation theory does not capture the correct single-particle dynamics at large bias and violates current conservation for asymmetric junctions. © 2010 The American Physical Society.view abstract 10.1103/PhysRevB.81.165106 **Interplay between Kondo physics and spin-orbit coupling in carbon nanotube quantum dots**

Galpin, M.R. and Jayatilaka, F.W. and Logan, D.E. and Anders, F.B.*Physical Review B - Condensed Matter and Materials Physics*81 (2010)We investigate the influence of spin-orbit coupling on the Kondo effects in carbon nanotube quantum dots, using the numerical renormalization group technique. A sufficiently large spin-orbit coupling is shown to destroy the SU(4) Kondo effects at zero magnetic field, leaving only two SU(2) Kondo effects in the one- and three-electron Coulomb-blockade valleys. On applying a finite magnetic field, two additional, spin-orbit induced SU(2) Kondo effects arise in the three- and two-electron valleys. Using physically realistic model parameters, we calculate the differential conductance over a range of gate voltages, temperatures, and fields. The results agree well with measurements from two different experimental devices in the literature, and explain a number of observations that are not described within the standard framework of the SU(4) Anderson impurity model. © 2010 The American Physical Society.view abstract 10.1103/PhysRevB.81.075437 **Mass-flow error in the numerical renormalization-group method and the critical behavior of the sub-Ohmic spin-boson model**

Vojta, M. and Bulla, R. and Güttge, F. and Anders, F.*Physical Review B - Condensed Matter and Materials Physics*81 (2010)We discuss a particular source of error in the numerical renormalization group (NRG) method for quantum impurity problems, which is related to a renormalization of impurity parameters due to the bath propagator. At any step of the NRG calculation, this renormalization is only partially taken into account, leading to systematic variation in the impurity parameters along the flow. This effect can cause qualitatively incorrect results when studying quantum-critical phenomena, as it leads to an implicit variation in the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power-law bath spectrum, J (Ï) Ïs, namely, the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for s<1/2, consistent with mean-field behavior as expected from quantum-to-classical mapping. © 2010 The American Physical Society.view abstract 10.1103/PhysRevB.81.075122 **Multiple-charge transfer and trapping in DNA dimers**

Tornow, S. and Bulla, R. and Anders, F.B. and Zwicknagl, G.*Physical Review B - Condensed Matter and Materials Physics*82 (2010)We investigate the charge transfer characteristics of one and two excess charges in a DNA base-pair dimer using a model Hamiltonian approach. The electron part comprises diagonal and off-diagonal Coulomb matrix elements such a correlated hopping and the bond-bond interaction, which were recently calculated by Starikov for different DNA dimers. The electronic degrees of freedom are coupled to an ohmic or a superohmic bath serving as dissipative environment. We employ the numerical renormalization group method in the nuclear tunneling regime and compare the results to Marcus theory for the thermal activation regime. For realistic parameters, the rate that at least one charge is transferred from the donor to the acceptor in the subspace of two excess electrons significantly exceeds the rate in the single charge sector. Moreover, the dynamics is strongly influenced by the Coulomb matrix elements. We find sequential and pair transfer as well as a regime where both charges remain self-trapped. The transfer rate reaches its maximum when the difference of the on-site and intersite Coulomb matrix element is equal to the reorganization energy which is the case in a guanine/cytosine (GC)-dimer. Charge transfer is completely suppressed for two excess electrons in adenine/thymine (AT)-dimer in an ohmic bath and replaced by damped coherent electron-pair oscillations in a superohmic bath. A finite bond-bond interaction W alters the transfer rate: it increases as function of W when the effective Coulomb repulsion exceeds the reorganization energy (inverted regime) and decreases for smaller Coulomb repulsion. © 2010 The American Physical Society.view abstract 10.1103/PhysRevB.82.195106 **Quantum transport through a quantum dot: Combining the scattering-states numerical renormalization group with nonequilibrium Green functions**

Anders, F.B. and Schmitt, S.*Journal of Physics: Conference Series*220 (2010)Scattering states fulfill the correct boundary conditions of a current carrying open quantum system. Discretizing the energy continuum of these states allows for employing Wilson's numerical renormalization group approach without violating the boundary conditions by using a finite size system. We evolve the analytically known steady-state density operator for a non-interacting quantum-system at finite bias to the full interacting problem by the time-dependent numerical renormalization group after switching on the local charging energy. Using a newly developed algorithm for steady-state nonequilibrium Green functions, we can calculate the current I as function of bias voltage V for arbitrary temperature and magnetic field. A comparison with second-order and GW Kadanoff-Baym-Keldysh results shows excellent agreement for weak interaction strength U. © 2010 IOP Publishing Ltd.view abstract 10.1088/1742-6596/220/1/012021

#### magnetism

#### molecular electronics

#### numerical methods

#### statistical mechanics