Coulomb blockade in quantum boxes

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.

We study the Kondo model --a magnetic impurity coupled to a one dimensional wire via exchange coupling-- by using Wilson's numerical renormalization group (NRG) technique. By applying an approach similar to which was used to compute the two impurity problem we managed to improve the bad spatial resolution of the numerical renormalization group method. In this way we have calculated the impurity spin - conduction electron spin correlation function which is a measure of the Kondo compensation cloud whose existence has been a long standing problem in solid state physics. We also present results on the temperature dependence of the Kondo correlations.

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 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.

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.

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 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

A superconductor (SC) in proximity to a ferromagnetic insulator (FMI) is predicted to exhibit mixed singlet and triplet correlations. The magnetic proximity effect of FMI spin splits the energy of Bogoliubov excitations and leads to a spin polarization at the surface for superconducting films that are thinner than the superconducting coherence length. In this work, we study manifestations of these phenomena in the properties of a magnetic impurity coupled via Kondo coupling to this FMI/SC system. Using the numerical renormalization group (NRG) method, we compute the properties of the ground state and low-lying excited states of a model that incorporates the Kondo interaction and a Ruderman-Kittel-Kasuya-Yosida (RKKY)-like interaction with the surface spin polarization. Our main finding is an energy splitting of the lowest even fermion-parity states caused by the proximity to the FMI. As the Kondo coupling increases, the splitting grows and saturates to a universal value equal to twice the exchange field of the FMI. We introduce a two-site model that can be solved analytically and provides a qualitative understanding of this and other NRG results. In addition, using perturbation theory, we demonstrate that the mechanism behind the splitting involves the RKKY field and the triplet correlations of the spin-split superconductor. A scaling analysis combined with NRG shows that the splitting can be written as a single-parameter scaling function of the ratio of the Kondo temperature and the superconducting gap, which is also numerically obtained. Published by the American Physical Society 2024

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 effect of a magnetic field on the spectral density of a S = 1/2 Kondo impurity is investigated at zero and finite temperatures by using Wilson's numerical renormalization group method. A splitting of the total spectral density is found for fields larger than a critical value H(c)(T = 0) approximately 0.5T(K), where T(K) is the Kondo scale. The splitting correlates with a peak in the magnetoresistivity of dilute magnetic alloys which we calculate and compare with the experiments on CexLa1-xAl2,x = 0.0063. The linear magnetoconductance of quantum dots exhibiting the Kondo effect is also calculated.

We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy $E_C \ll \Delta$. We consider a small superconducting grain of finite capacity connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is supposed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system may be described as quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima which become degenerate at the phase difference on the leads equal to $\pi$, the tunneling amplitude between them being controlled by the gate voltage at the grain. We find the Josephson current and its low-frequency fluctuations and predict their periodic dependence on the induced charge $Q_x=C V_g$ with period $2e$.

Fractionally charged quasiparticles in the quantum Hall state with a filling factor nu=5/2 are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a nonanalytic dependence on the tunneling amplitudes which can be controlled by gate voltages.

We investigate spin-charge mixing effect on resonant tunneling in spin-polarized Tomonaga-Luttinger liquid with double impurities. The mixing arises from Fermi velocity difference between two spin species due to Zeeman effect. Zero bias conductance is calculated as a function of gate voltage $V_{\rm g}$, gate magnetic field $B_{\rm g}$, temperature and magnetic field applied to the system. Mixing effect is shown to cause rotation of the lattice pattern of the conductance peaks in $(V_{\rm g},B_{\rm g})$ plane, which can be observed in experiments. At low temperatures, the contour shapes are classified into three types, reflecting the fact that effective barrier potential is renormalized towards ``perfect reflection'', ``perfect transmission'' and magnetic field induced ``spin-filtering'', respectively.

Quantum dots attached to BCS superconducting leads exhibit a $0-\pi$ impurity quantum phase transition, which can be experimentally controlled either by the gate voltage or by the superconducting phase difference. For the pertinent superconducting single-impurity Anderson model, we newly present two simple analytical formulae describing the position of the phase boundary in parameter space for the weakly correlated and Kondo regime, respectively. Furthermore, we show that the two-level approximation provides an excellent description of the low temperature physics of superconducting quantum dots near the phase transition. We discuss reliability and mutual agreement of available finite temperature numerical methods (Numerical Renormalization Group and Quantum Monte Carlo) and suggest a novel approach for efficient determination of the quantum phase boundary from measured finite temperature data. Our results enable fast and efficient, yet reliable characterization and design of such nanoscopic tunable Josephson junction devices.