Kondo screening cloud in a one-dimensional wire: Numerical renormalization group study

The interplay between the Kondo screening of quantum impurities (by the electronic channels to which they couple) and the interimpurity RKKY interactions (mediated by the same channels) has been extensively studied. However, the effect of unidirectional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated interimpurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. Here we take Wilson's numerical renormalization group (NRG), the most established numerical method for treating quantum impurity models, and extend it to systems consisting of two impurities coupled at different locations to unidirectional channel(s). This is challenging due to the incompatibility of unidirectionality with one of the main ingredients in NRG---the mapping of the channel(s) to a Wilson chain---a tight-binding chain with the impurity at one end and hopping amplitudes which decay exponentially with the distance. We bridge this gap by introducing a ``Wilson ladder'' consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the unidirectionality of the channel(s), as well as the distance between the two impurities. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, showing that all local properties and thermodynamic quantities are indifferent to the interimpurity distance, and correspond to two separate single-impurity models. Extensions to more impurities and/or helical channels are possible.

Using numerical renormalization group method the temperature dependences of the magnetic susceptibility $\chi(T)$ and specific heat $C(T)$ are obtained in the single-impurity Anderson model with inclusion of the $d-f$ Coulomb interaction. It is shown that the exciton effects owing to this interaction can change considerably the dependence $C(T)$ in comparison with the standard Anderson model at not too low temperatures, whereas the dependence $\chi (T)$ remains universal. The renormalization of the effective hybridization parameter and $f$-level position, which is connected with the $d-f$ interaction, is calculated, a satisfactory agreement with the Hartree-Fock approximation being derived.

We study the temperature dependence of the linear conductance and tunnel magnetoresistance of quantum dots coupled to ferromagnetic leads. Using the numerical renormalization group method, we determine transport properties for a wide range of temperatures $T$, ranging from zero through the Kondo temperature ${T}_{K}$ up to the high-temperature regime. We show that tunnel magnetoresistance in the local moment regime displays a nonmonotonic dependence on $T$ and vanishes when $T~{T}_{K}$. In addition, we also analyze the spin polarization $\mathcal{P}$ of the linear conductance in the parallel configuration and show that $\mathcal{P}$ is suppressed in the odd electron Coulomb blockade valley and can be enhanced above the spin polarization of the leads in the even Coulomb valley.

We discuss the scale-free property of Wilson's numerical renormalization group (NRG) for the Kondo-impurity problem. The single-particle state of the effective Hamiltonian with a cutoff $\ensuremath{\Lambda}$ is described by the wave-packet basis having the scale-free property. The energy scale of the system can be controlled by the lattice translation of the wave-packet basis with no reference of rescaling of the lattice space. We also analyze the role of the Kondo interaction in the context of wave-packet basis and then discuss the scaling and renormalization of the Kondo coupling. In addition, we clarify the role of the edge state in the lowest energy scale of Wilson NRG.

We study the Hubbard-Holstein model, which includes both the electron-electron and electron-phonon interactions characterized by $U$ and $g$, respectively, employing the dynamical mean-field theory combined with Wilson's numerical renormalization group technique. A zero temperature phase diagram of metal-insulator and insulator-insulator transitions at half-filling is mapped out which exhibits the interplay between $U$ and $g$. As $U$ ($g$) is increased, a metal to Mott-Hubbard insulator (bipolaron insulator) transition occurs, and the two insulating states are distinct and can not be adiabatically connected. The nature of and transitions between the three states are discussed.

The transport coefficients of the Anderson model are calculated by extending Wilson's numerical renormalization group method to finite-temperature Green functions. Accurate results for the frequency and temperature dependence of the single-particle spectral densities and transport time tau ( omega , T) are obtained and used to extract the temperature dependence of the transport coefficients in the strong-correlation limit of the Anderson model. Results are obtained for values of the local level position ranging from the Kondo regime to the mixed valency and empty orbital regimes. The low-temperature anomalies in the resistivity, rho (T), thermopower, S(T), thermal conductivity, kappa (T), and Hall coefficient, RH(T), are discussed in terms of the behaviour of the spectral densities. At low temperature all quantities exhibit the expected Fermi liquid behaviour, rho (T)= rho 0(1-c(T/TK)2), S(T) approximately gamma T, kappa (T)/ alpha T=1+ beta (T/TK)2, RH(T)=-Rinfinity (1- delta (T/TK)2). Analytic results based on Fermi liquid theory are derived here for the first time for beta and the numerical results are shown to be consistent with this coefficient. The range of temperatures over which universal behaviour extends is also discussed. Scattering of conduction electrons in higher-angular-momentum, l>0, channels is also considered and an expression is derived for the corresponding transport time and used to discuss the influence of the interference terms between the resonant l=0 and non-resonant l=1 channels on the transport properties. The presence of non-resonant scattering is shown to be particularly important for the thermopower at half filling, where the sign of the thermopower can depend sensitively on the non-resonant phase shift. Finally the relation of the results to experiment is discussed.

The temperature dependence of electric dipolar susceptibility \chi_P is discussed on the basis of the Anderson-Holstein model with the use of a numerical renormalization group (NRG) technique. Note that P is related with phonon Green's function D. In order to obtain correct temperature dependence of P at low temperatures, we propose a method to evaluate P through the Dyson equation from charge susceptibility \chi_c calculated by the NRG, in contrast to the direct NRG calculation of D. We find that the irreducible charge susceptibility estimated from \chi_c agree with the perturbation calculation, suggesting that our method works well.

The charging of a quantum box connected to a lead by a single-mode point contact is solved for arbitrary temperatures, tunneling amplitudes, and gate voltages, using a variant of Wilson's numerical renormalization group. The charge inside the box and the capacitance of the junction are calculated on equal footing for all physical regimes, including weak tunneling, near perfect transmission, and the crossover regime in between. At the charge plateaus, perturbation theory is found to break down at fairly small tunneling amplitudes. Near perfect transmission, we confirm Matveev's scenario for the smearing of the Coulomb-blockade staircase. A surprising reentrance of the Coulomb-blockade staircase is found for large tunneling amplitudes. At the degeneracy points, we obtain two-channel Kondo behavior directly from the Coulomb-blockade Hamiltonian, without the restriction to two charge configurations or the introduction of an effective cutoff.

Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.

We investigate theoretically the possibility of observing two-channel Kondo (2CK) physics in tunnel-coupled double quantum dots (TCDQDs), at both zero and finite magnetic fields; taking the two-impurity Anderson model (2AIM) as the basic TCDQD model, together with effective low-energy models arising from it by Schrieffer-Wolff transformations to second and third order in the tunnel couplings. The models are studied primarily using Wilson's numerical renormalization group. At zero-field our basic conclusion is that while 2CK physics arises in principle provided the system is sufficiently strongly-correlated, the temperature window over which it could be observed is much lower than is experimentally feasible. This finding disagrees with recent work on the problem, and we explain why. At finite field, we show that the quantum phase transition known to arise at zero-field in the two-impurity Kondo model (2IKM), with an essentially 2CK quantum critical point, persists at finite fields. This raises the prospect of access to 2CK physics by tuning a magnetic field, although preliminary investigation suggests this to be even less feasible than at zero field.

In this paper, we investigate the spin-polarized transport through parallel N−dot (N = 1, 2, 3) systems in the strongly correlated regime. We focus our attention on the responses of the Ntot = integer states to an increasing magnetic field B, where Ntot is the total charge number on the dots. We show that when the charge level 𝜖 is chosen at the particle-hole (p-h) symmetric case, spin filtering is difficult to occur. While if 𝜖 is beyond the p-h symmetric point, perfect spin-polarized currents could be achieved, and the spin directions can be easily manipulated by tuning external electric and/or magnetic fields, making it easy to be realized in future experiments of the ideas. To approach these problems, the celebrated numerical renormalization group (NRG) technique is implemented, the dynamical properties and the quantum fluctuations are shown.

Screened spin−1 and −1/2 Kondo effect in a triangular quantum dot system with interdot Coulomb repulsion

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 non-equilibrium 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 $\Lambda > 1$. 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 quantum-impurity systems. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to non-trivial models, such as the interacting resonant-level model and the single impurity Anderson model.

We investigate f-electron charge susceptibility in a two-impurity Anderson model on the basis of Wilson's numerical renormalization group method. The f-electron charge susceptibility diverges logarithmically at the critical point of this model when conduction-electron bands exhibit particle–hole symmetry. Although the critical point disappears without the particle–hole symmetry, the f-electron charge fluctuation is much more enhanced near the crossover regime between the Kondo–Yosida singlet and intersite spin-singlet states than that in the single-impurity case. This result shows that charge fluctuations are enhanced owing to the competition between intersite and Kondo–Yosida spin singlets. A possible scenario for the enhanced residual resistivity near the region where the Kondo temperature becomes comparable with the Néel temperatures under pressure in some heavy-fermion compounds is proposed.

The thermoelectric properties of strongly correlated quantum dots, described\nby a single level Anderson model coupled to conduction electron leads, is\ninvestigated using Wilson's numerical renormalization group method. We\ncalculate the electronic contribution, $K_{\\rm e}$, to the thermal conductance,\nthe thermopower, $S$, and the electrical conductance, $G$, of a quantum dot as\na function of both temperature, $T$, and gate voltage, ${\\rm v}_g$, for strong,\nintermediate and weak Coulomb correlations, $U$, on the dot. For strong\ncorrelations and in the Kondo regime, we find that the thermopower exhibits two\nsign changes, at temperatures $T_{1}({\\rm v}_g)$ and $T_{2}({\\rm v}_g)$ with\n$T_{1}< T_{2}$. Such sign changes in $S(T)$ are particularly sensitive\nsignatures of strong correlations and Kondo physics. The relevance of this to\nrecent thermopower measurements of Kondo correlated quantum dots is discussed.\nWe discuss the figure of merit, power factor and the degree of violation of the\nWiedemann-Franz law in quantum dots. The extent of temperature scaling in the\nthermopower and thermal conductance of quantum dots in the Kondo regime is also\nassessed.\n