Numerical Renormalization Group Results Research Articles

Generalized atomic limit of a double quantum dot coupled to superconducting leads

We present an exactly solvable effective model of a double quantum dot coupled to superconducting leads. This model is a generalization of the well-known superconducting atomic limit approximation of the paradigmatic superconducting impurity Anderson model. However, in contrast to the standard atomic limit and other effective models, it gives quantitatively correct predictions for the quantum phase transition boundaries, subgap bound states as well as Josephson supercurrent in a broad range of parameters including experimentally relevant regimes. The model allows fast and reliable parameter scans important for the preparation and analysis of experiments, which are otherwise inaccessible by more precise but computational heavy methods such as quantum Monte Carlo or the numerical renormalization group. The scans also allowed us to identify and investigate new previously unnoticed phase diagram regimes. We provide a thorough analysis of the strengths and limitations of the effective model and benchmark its predictions against numerical renormalization group results.

Level occupation switching with density functional theory

The charge transport properties of zero-temperature multiorbital quantum dot systems with one dot coupled to leads and the other dots coupled only capacitatively are studied within density functional theory. It is shown that the setup is equivalent to an effective single impurity Anderson model. This allows to understand the level occupation switching effect as transitions between ground states of different integer occupations in the uncoupled dots. Level occupation switching is very sensitive to small energy differences and therefore also to the details of the parametrized exchange-correlation functionals. An existing functional already captures the effect on a qualitative level, but we also provide an improved parametrization which is very accurate when compared to reference numerical renormalization group results.

Universal Thermal Entanglement of Multichannel Kondo Effects.

Quantum entanglement between an impurity and its environment is expected to be central in quantum impurity problems. We develop a method to compute the entanglement in spin-1/2 impurity problems, based on the entanglement negativity and the boundary conformal field theory (BCFT). Using the method, we study the thermal decay of the entanglement in the multichannel Kondo effects. At zero temperature, the entanglement has the maximal value independent of the number of the screening channels. At low temperature, the entanglement exhibits a power-law thermal decay. The power-law exponent equals two times of the scaling dimension of the BCFT boundary operator describing the impurity spin, and it is attributed to the energy-dependent scaling behavior of the entanglement in energy eigenstates. These agree with numerical renormalization group results, unveiling quantum coherence inside the Kondo screening length.

Universality and thermoelectric transport properties of quantum dot systems

We discuss the temperature-dependent thermoelectric transport properties of semiconductor nanostructures comprising a quantum dot coupled to quantum wires: the thermal dependence of the electrical conductance, thermal conductance, and thermopower. We explore the universality of the thermoelectric properties in the temperature range associated with the Kondo crossover. In this thermal range, general arguments indicate that any equilibrium property's temperature dependence should be a universal function of the ratio ${T}^{*}=T/{T}_{K}$, where ${T}_{K}$ is the Kondo temperature. Considering the particle-hole symmetric, spin-degenerate Anderson model, the zero-bias electrical conductance has already been shown to map linearly onto a universal conductance through a quantum dot embedded or side-coupled to a quantum wire. Employing rigorous renormalization-group arguments, we calculate universal thermoelectric transport coefficients that allow us to extend this result to the thermopower and the thermal conductance. We present numerical renormalization-group results to illustrate the physics in our findings. Applying the universal thermoelectric coefficients to recent experimental results of the electrical conductance and thermovoltages versus ${V}_{\mathrm{gate}}$, at different temperatures in the Kondo regime, we calculate all the thermoelectric properties and obtain simple analytical fitting functions that can be used to predict the experimental results of these properties. However, we cannot check all of them, due to the lack of available experimental results over a broad temperature range.

Uncovering Non-Fermi-Liquid Behavior in Hund Metals: Conformal Field Theory Analysis of an SU(2)×SU(3) Spin-Orbital Kondo Model

Hund metals have attracted attention in recent years due to their unconventional superconductivity, which supposedly originates from non-Fermi-liquid (NFL) properties of the normal state. When studying Hund metals using dynamical mean-field theory, one arrives at a self-consistent "Hund impurity problem" involving a multiorbital quantum impurity with nonzero Hund coupling interacting with a metallic bath. If its spin and orbital degrees of freedom are screened at different energy scales, $T_\mathrm{sp} < T_\mathrm{orb}$, the intermediate energy window is governed by a novel NFL fixed point, whose nature had not yet been clarified. We resolve this problem by providing an analytical solution of a paradigmatic example of a Hund impurity problem, involving two spin and three orbital degrees of freedom. To this end, we combine a state-of-the-art implementation of the numerical renormalization group, capable of exploiting non-Abelian symmetries, with a generalization of Affleck and Ludwig's conformal field theory (CFT) approach for multichannel Kondo models. We characterize the NFL fixed point of Hund metals in detail for a Kondo model with an impurity forming an SU(2)$\times$SU(3) spin-orbital multiplet, tuned such that the NFL energy window is very wide. The impurity's spin and orbital susceptibilities then exhibit striking power-law behavior, which we explain using CFT arguments. We find excellent agreement between CFT predictions and numerical renormalization group results. Our main physical conclusion is that the regime of spin-orbital separation, where orbital degrees of freedom have been screened but spin degrees of freedom have not, features anomalously strong local spin fluctuations: the impurity susceptibility increases as $\chi_\mathrm{sp}^\mathrm{imp} \sim \omega^{-\gamma}$, with $\gamma > 1$.

Local moment approach as a quantum impurity solver for the Hubbard model

The local moment approach (LMA) has presented itself as a powerful semi-analytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we attempt to complete the phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum-rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. We also show that, in the metallic phase, the low-energy scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong-coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. Finally we study optical properties and find universal features such as absorption peak position governed by the low-energy scale and a doping independent crossing point, often dubbed as the "isosbestic point" in experiments.

Spin versus charge noise from Kondo traps

Magnetic and charge noise have common microscopic origin in solid state devices, as described by a universal electron trap model. In spite of this common origin, magnetic (spin) and charge noise spectral densities display remarkably different behaviours when many-particle correlations are taken into account, leading to the emergence of the Kondo effect. Our numerical renormalization group results indicate that while spin noise is a universal function of the Kondo temperature, charge noise remains well described by single-particle theory even when the trap is deep in the Kondo regime. This difference survives even in the presence of disorder, showing that noise can be more manageable in devices that are sensitive to magnetic (rather than charge) fluctuations and that the signature of the Kondo effect can be observed in spin noise spectroscopy experiments.

Shiba states and zero-bias anomalies in the hybrid normal-superconductor Anderson model

We determine the phase diagram of an Anderson impurity in contact with superconducting and normal-state leads for arbitrary ratio of the gap $\Delta$ to the Kondo temperature $T_K$. We observe a considerable effect of even very weak coupling to the normal lead that is usually considered as a non-perturbing tunneling probe. The numerical renormalization group results are analyzed in the context of relevant experimental scenarios such as parity crossing (doublet-singlet) quantum phase transitions induced by a gap reduction as well as novel Kondo features induced by the normal lead. We point out the important role of finite temperatures and magnetic fields. Overall, we find a very rich behavior of spectral functions with zero-bias anomalies which can emerge irrespective of whether the ground state is a doublet or a singlet. Our findings are pertinent to the tunnelling-spectroscopy experiments aiming at detecting Majorana modes in nanowires.

Iron impurities in gold and silver: Comparison of transport measurements to numerical renormalization group calculations exploiting non-Abelian symmetries

We consider iron impurities in the noble metals gold and silver and compare experimental data for the resistivity and decoherence rate to numerical renormalization group results. By exploiting non-Abelian symmetries, we show improved numerical data for both quantities as compared to previous calculations [Costi et al., Phys. Rev. Lett. 102, 056802 (2009).], using the discarded weight as criterion to reliably judge the quality of convergence of the numerical data. In addition, we also carry out finite-temperature calculations for the magnetoresistivity of fully screened Kondo models with $S=\frac{1}{2}$, 1, and $\frac{3}{2}$, and compare the results with available measurements for iron in silver, finding excellent agreement between theory and experiment for the spin-$\frac{3}{2}$ three-channel Kondo model. This lends additional support to the conclusion of Costi et al. that the latter model provides a good effective description of the Kondo physics of iron impurities in gold and silver.

Admittance of the SU(2) and SU(4) Anderson quantumRCcircuits

We study the Anderson model as a description of the quantum $RC$ circuit for spin-$1/2$ electrons and a single level connected to a single lead. Our analysis relies on the Fermi liquid nature of the ground state, which fixes the form of the low-energy effective model. The constants of this effective model are extracted from a numerical solution of the Bethe ansatz equations for the Anderson model. They allow us to compute the charge relaxation resistance ${R}_{q}$ in different parameter regimes. In the Kondo region, the peak in ${R}_{q}$ as a function of the magnetic field is recovered and proven to be in quantitative agreement with previous numerical renormalization group results. In the valence-fluctuation region, the peak in ${R}_{q}$ is shown to persist, with a maximum value of $h/2{e}^{2}$, and an analytical expression is obtained using perturbation theory. We extend our analysis to the SU(4) Anderson model where we also derive the existence of a giant peak in the charge relaxation resistance.

Alternative functional renormalization group approach to the single impurity Anderson model

We present an alternative functional renormalization group (fRG) approach to the single-impurity Anderson model at finite temperatures. Starting with the exact self-energy and interaction vertex of a small system ('core') containing a correlated site, we switch on the hybridization with a non-interacting bath in the fRG-flow and calculate spectra of the correlated site. Different truncations of the RG-flow-equations and choices of the core are compared and discussed. Furthermore we calculate the linear conductance and the magnetic susceptibility as functions of temperature and interaction strength. The signatures of Kondo physics arising in the flow are compared with numerical renormalization group results.

Functional renormalization group approach to the singlet–triplet transition in quantum dots

We present a functional renormalization group approach to the zero bias transport properties of a quantum dot with two different orbitals and in the presence of Hund’s coupling. Tuning the energy separation of the orbital states, the quantum dot can be driven through a singlet–triplet transition. Our approach, based on the approach by Karrasch et al (2006 Phys. Rev. B 73 235337), which we apply to spin-dependent interactions, recovers the key characteristics of the quantum dot transport properties with very little numerical effort. We present results on the conductance in the vicinity of the transition and compare our results both with previous numerical renormalization group results and with predictions of the perturbative renormalization group.

Dynamical magnetic anisotropy and quantum phase transitions in a vibrating spin-1 molecular junction

We study the electronic transport through a spin-1 molecule in which mechanical stretching produces a magnetic anisotropy. In this type of device, a vibron mode along the stretching axis will couple naturally to the molecular spin. We consider a single molecular vibrational mode and find that the electron-vibron interaction induces an effective correction to the magnetic anisotropy that shifts the ground state of the device toward a non-Fermi-liquid phase. A transition into a Fermi-liquid phase could then be achieved, by means of mechanical stretching, passing through an underscreened spin-1 Kondo regime. We present numerical renormalization-group results for the differential conductance, the spectral density, and the magnetic susceptibility across the transition.

Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

For phase transitions in dissipative quantum impurity models, the existence of a quantum-to-classical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-Ohmic spin-boson model with a power-law bath spectrum ∝ω(s), we confirm classical mean-field behavior for s<1/2, correcting earlier numerical renormalization-group results. We also provide the first results for an XY-symmetric model of a spin coupled to two competing bosonic baths, where we find a rich phase diagram, including both critical and strong-coupling phases for s<1, different from that of classical spin chains. This illustrates that symmetries are decisive for whether or not a quantum-to-classical correspondence exists.

Ground State of the Parallel Double Quantum Dot System

We resolve the controversy regarding the ground state of the parallel double quantum dot system near half filling. The numerical renormalization group predicts an underscreened Kondo state with residual spin-1/2 magnetic moment, ln2 residual impurity entropy, and unitary conductance, while the Bethe ansatz solution predicts a fully screened impurity, regular Fermi-liquid ground state, and zero conductance. We calculate the impurity entropy of the system as a function of the temperature using the hybridization-expansion continuous-time quantum Monte Carlo technique, which is a numerically exact stochastic method, and find excellent agreement with the numerical renormalization group results. We show that the origin of the unconventional behavior in this model is the odd-symmetry "dark state" on the dots.

Two-impurity Anderson model revisited: Competition between Kondo effect and reservoir-mediated superexchange in double quantum dots

We study a series-coupled double quantum dot in the Kondo regime modeled by the two-impurity Anderson model and find a new conduction-band mediated superexchange interaction that competes with Kondo physics in the strong Coulomb interaction limit. Our numerical renormalization group results, complemented with the higher-order Rayleigh-Schr\"odinger perturbation theory, show that the novel exchange mechanism leads to clear experimental consequences that can be checked in transport measurements through double quantum dots.

Universal conductance for the Anderson model

We discuss the thermal dependence of the zero-bias electrical conductance for a quantum dot embedded in a quantum wire, or side-coupled to it. In the Kondo regime, the temperature-dependent conductances map linearly onto the conductance for the symmetric Anderson Hamiltonian. The mapping fits accurately numerical renormalization-group results for the conductance in each geometry. In the side-coupled geometry, the conductance is markedly affected by a gate potential applied to the wire; in the embedded geometry, it is not.

Universal zero-bias conductance for the single-electron transistor

The thermal dependence of the electrical conductance of the single-electron transistor (SET) in the zero-bias Kondo regime is discussed. An exact mapping to the universal curve for the symmetric Anderson model is established. Linear, the mapping is parametrized by the Kondo temperature and the charge in the Kondo cloud. Illustrative numerical renormalization-group results, in excellent agreement with the mapping, are presented.

Dynamical correlations in the spin-half two-channel Kondo model

Dynamical correlations of various local operators are studied in the spin-half two-channel Kondo (2CK) model in the presence of channel anisotropy or external magnetic field. A conformal field theory-based scaling approach is used to predict the analytic properties of various spectral functions in the vicinity of the two-channel Kondo fixed point. These analytical results compare well with highly accurate density matrix numerical renormalization group results. The universal cross-over functions interpolating between channel-anisotropy or magnetic field-induced Fermi liquid regimes and the two-channel Kondo, non-Fermi liquid regimes are determined numerically. The boundaries of the real 2CK scaling regime are found to be rather restricted, and to depend both on the type of the perturbation and on the specific operator whose correlation function is studied. In a small magnetic field, a universal resonance is observed in the local fermion's spectral function. The dominant superconducting instability appears in the composite superconducting channel.

Matrix-product-state comparison of the numerical renormalization group and the variational formulation of the density-matrix renormalization group

Wilson's numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White's density-matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model with a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an ``unfolded'' Wilson chain, which brings about a significant reduction in numerical costs compared to those of NRG. We show that all NRG eigenstates (kept and discarded) can be reproduced using VMPS, and compare the difference in truncation criteria, sharp vs smooth in energy space, of the two approaches. Finally, we demonstrate that NRG results can be improved upon systematically by performing a variational optimization in the space of variational matrix product states, using the states produced by NRG as input.