Numerical Renormalization Group Approach Research Articles

Open Wilson chain numerical renormalization group approach to Green's functions

By combining Wilson's numerical renormalization group (NRG) with a modified Bloch-Redfield approach (BRA), we are able to eliminate the artificial broadening of the Lehmann representation of quantum impurity spectral functions required by the standard NRG algorithm. Our approach is based on the exact reproduction of the continuous coupling function in the original quantum impurity model. It augments each chain site of the Wilson chain by a coupling to an additional reservoir. This open Wilson chain is constructed by a continuous fraction expansion, and the coupling function is treated in second order in the context of the BRA. The eigenvalues of the resulting Bloch-Redfield tensor generate a finite lifetime of the NRG excitations that leads to a natural broadening of the spectral functions. We combine this approach with z-averaging and an analytical exact expression for the correlation part of the self-energy to obtain an accurate representation of the spectral function of the original continuum model in the absence and presence of an external magnetic field.

Dynamical quantum phase transitions in a mesoscopic superconducting system

We inspect signatures of dynamical quantum phase transitions driven by two types of quenches acting on a correlated quantum dot embedded between superconducting and metallic reservoirs. Under stationary conditions the proximity induced on-dot pairing combined with the strong Coulomb repulsion prefers the quantum dot to be either in the singly occupied (spinful) or BCS-type (spinless) ground state configuration. We study the time evolution upon traversing such a phase boundary due to quantum quenches by means of the time-dependent numerical renormalization group approach, revealing non-analytic features in the low-energy return rate. Quench protocols can be realized in a controllable manner and we are confident that detection of this dynamical singlet-doublet phase transition would be feasible by the charge tunnelling spectroscopy.

Nonlinear Fermi liquid transport through a quantum dot in asymmetric tunnel junctions

We study the nonlinear conductance through a quantum dot, specifically its dependence on the asymmetries in the tunnel couplings and bias voltages $V$, at low energies. Extending the microscopic Fermi liquid theory for the Anderson impurity model, we obtain an exact formula for the steady current $I$ up to terms of order ${V}^{3}$ in the presence of these asymmetries. The coefficients for the nonlinear terms are described in terms of a set of the Fermi liquid parameters: the phase shift, static susceptibilities, and three-body correlation functions of electrons in the quantum dots, defined with respect to the equilibrium ground state. We calculate these correlation functions, using the numerical renormalization group approach (NRG), over a wide range of impurity-electron filling that can be controlled by a gate voltage in real systems. The NRG results show that the order ${V}^{2}$ nonlinear current is enhanced significantly in the valence fluctuation regime. It is caused by the order $V$ energy shift of the impurity level, induced in the presence of the tunneling or bias asymmetry. Furthermore, in the valence fluctuation regime, we also find that the order ${V}^{3}$ nonlinear current exhibits a shoulder structure, for which the three-body correlations that evolve for large asymmetries play an essential role.

Self-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation of motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.

Computing Local Multipoint Correlators Using the Numerical Renormalization Group

Local three- and four-point correlators yield important insight into strongly correlated systems and have many applications. However, the nonperturbative, accurate computation of multipoint correlators is challenging, particularly in the real-frequency domain for systems at low temperatures. In the accompanying paper, we introduce generalized spectral representations for multipoint correlators. Here, we develop a numerical renormalization group (NRG) approach, capable of efficiently evaluating these spectral representations, to compute local three- and four-point correlators of quantum impurity models. The key objects in our scheme are partial spectral functions, encoding the system's dynamical information. Their computation via NRG allows us to simultaneously resolve various multiparticle excitations down to the lowest energies. By subsequently convolving the partial spectral functions with appropriate kernels, we obtain multipoint correlators in the imaginary-frequency Matsubara, the real-frequency zero-temperature, and the real-frequency Keldysh formalisms. We present exemplary results for the connected four-point correlators of the Anderson impurity model, and for resonant inelastic x-ray scattering (RIXS) spectra of related impurity models. Our method can treat temperatures and frequencies -- imaginary or real -- of all magnitudes, from large to arbitrarily small ones.

Kondo screening regimes in multi-Dirac and Weyl systems

We have investigated the Kondo physics of a single magnetic impurity embedded in multi-Dirac (Weyl) node fermionic systems. By using a generic effective model for the host material and employing a numerical renormalization group approach we access the low temperature behavior of the system, identifying the existence of Kondo screening in single-, double-, and triple-Dirac (Weyl) node models. We find that in any multi-Dirac node systems the low-energy regime lies within one of the known classes of pseudogap Kondo problem, extensively studied in the literature. Kondo screening is also observed for time reversal symmetry broken Weyl systems. This is, however, possible only in the particle-hole symmetry broken regime obtained for finite chemical potential $\mu$. Although weakly, breaking time-reversal symmetry suppresses the Kondo resonance, especially in the single-node Weyl semimetals. More interesting Kondo screening regimes are obtained for inversion symmetry broken multi-Weyl fermions. In these systems the Kondo regimes of double- and triple-Weyl node models are much richer than in the single-Weyl node model. While in the single-Weyl node model the Kondo temperature increases monotonically with $|\mu|$ regardless the value of the inversion symmetry breaking parameter $Q_0$, in double- and triple-Weyl node models there are two distinct regimes: (i) For $Q_0< |\mu|$ the Kondo temperature depends strongly on $\mu$, while (ii) for $Q_0 > |\mu|$ the Kondo temperature depends very weakly on $\mu$, resembling the flat-band single impurity Anderson model.

Restoring the continuum limit in the time-dependent numerical renormalization group approach

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.

Inelastic electron tunneling spectroscopy for probing strongly correlated many-body systems by scanning tunneling microscopy

We present an extension of the tunneling theory for scanning tunneling microcopy (STM) to include different types of vibrational-electronic 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 electronic 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 anti-adiabatic 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 NTCDA 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 experiment scanning tunneling spectra can be resolved by including two mechanisms: (i) a strong unconventional Holstein term on the local substrate orbital leads to reduction of the Kondo temperature and (ii) a different electron-vibrational coupling to the tunneling matrix element is responsible for inelastic steps in the $dI/dV$ curve at finite frequencies.

Modeling of the gate-controlled Kondo effect at carbon point defects in graphene

We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound $\sigma$ orbital and the vacancy induced bound $\pi$ 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.

From Kondo to local singlet state in graphene nanoribbons with magnetic impurities

A detailed analysis of the Kondo effect of a magnetic impurity in a zigzag graphene nanoribbon is addressed. An adatom is coupled to the graphene nanoribbon via a hybridization amplitude $\Gamma_{imp}$ in a hollow or top site configuration. In addition, the adatom is also weakly coupled to a metallic STM tip by a hybridization function $\Gamma_{tip}$ that provides a Kondo screening of its magnetic moment. The entire system is described by an Anderson-like Hamiltonian whose low-temperature physics is accessed by employing the numerical renormalization group approach, which allows us to obtain the thermodynamic properties used to compute the Kondo temperature of the system. We find two screening regimes when the adatom is close to the edge of the zigazag graphene nanoribbon: (1) a weak coupling regime ($\Gamma_{imp}\ll \Gamma_{tip}$), in which the edge states produce an enhancement of the Kondo temperature $T_K$ and (2) a strong coupling regime ($\Gamma_{imp}\gg \Gamma_{tip}$), in which a local singlet is formed, in detriment to the Kondo screening by the STM tip. These two regimes can be clearly distinguished by the dependence of their characteristic temperature $T^*$ on the coupling between the adatom and the carbon sites of the graphene nanoribon ($V_{imp}$). We observe that in the weak coupling regime $T^*$ increases exponentially with $V_{imp}^2$. Differently, in the strong coupling regime, $T^*$ increases linearly with $V_{imp}^2$.

SU(4) Kondo effect in double quantum dots with ferromagnetic leads

We investigate the spin-resolved transport properties, such as the linear conductance and the tunnel magnetoresistance, of a double quantum dot device attached to ferromagnetic leads and look for signatures of SU(4) symmetry in the Kondo regime. We show that the transport behavior greatly depends on the magnetic configuration of the device, and the spin-SU(2) as well as the orbital and spin-SU(4) Kondo effects become generally suppressed when the magnetic configuration of the leads varies from the antiparallel to the parallel one. Furthermore, a finite spin polarization of the leads lifts the spin degeneracy and drives the system from the SU(4) to an orbital-SU(2) Kondo state. We analyze in detail the crossover and show that the Kondo temperature between the two fixed points has a non-monotonic dependence on the degree of spin polarization of the leads. In terms of methods used, we characterize transport by using a combination of analytical and numerical renormalization group approaches.

Influence of phonon-assisted tunneling on the linear thermoelectric transport through molecular quantum dots

We investigate the effect of vibrational degrees of freedom on the linear thermoelectric transport through a single-level quantum dot described by the spinless Anderson-Holstein impurity model. To study the effects of strong electron-phonon coupling, we use the nonperturbative numerical renormalization group approach. We also compare our results, at weak to intermediate coupling, with those obtained by employing the functional renormalization group method, finding good agreement in this parameter regime. When applying a gate voltage at finite temperatures, the inelastic scattering processes, induced by phonon-assisted tunneling, result in an interesting interplay between electrical and thermal transport. We explore different parameter regimes and identify situations for which the thermoelectric power as well as the dimensionless figure of merit are significantly enhanced via a Mahan-Sofo type of mechanism. We show, in particular, that this occurs at strong electron-phonon coupling and in the antiadiabatic regime.

Time Evolution of the Kondo Resonance in Response to a Quench.

We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function G(t+t^{'},t), and its associated time-dependent spectral function, A(ω,t), for times t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times t→0^{+}, only fully develops at very long times t≳1/T_{K}, where T_{K} is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale 1/Γ during the time interval -1/Γ≲t≲+1/Γ, where Γ is the hybridization strength. Initial and final state spectral functions are recovered in the limits t→-∞ and t→+∞, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.

Kondo screening in two-dimensional p -type transition-metal dichalcogenides

Systems with strong spin orbit coupling support a number of new phases of matter and novel phenomena. This work focuses on the interplay of spin orbit coupling and interactions in yielding correlated phenomena in two dimensional transition metal dichalcogenides. In particular we explore the physics of Kondo screening resulting from the lack of centro-symmetry, large spin splitting and spin valley locking in hole doped systems. The key ingredients are i) valley dependent spin-momentum locking perpendicular to the two dimensional crystal; ii) single nondegenerate Fermi surface per valley, and iii) nontrivial Berry curvature associated with the low energy bands. The resulting Kondo resonance has a finite triplet component and nontrivial momentum space structure which facilitates new approaches to both probe and manipulate the correlated state. Using a variational wave function and the numerical renormalization group approaches we study the nature of the Kondo resonance both in the absence and presence of circularly polarized light. The latter induces an imbalance in the population of the two valleys leading to novel magnetic phenomena in the correlated state.

Local Kondo entanglement and symmetry protected local criticality in the two-impurity Kondo problem

The local Kondo entanglement is defined as the concurrence of a short-ranged Kondo singlet state consisting of a localized magnetic moment and a nearby conduction electron. We derive the entanglement phase diagram of the Rasul-Schlottmann model, the effective spin-only Hamiltonian for the two-impurity Kondo model in the numerical renormalization group approach. We show that the local Kondo entanglement vanishes exactly at the two-impurity Kondo critical point, associated concomitantly with a jump in the inter-impurity entanglement. We discuss how to generalize this result to a Kondo lattice model preserving the same enhanced spin symmetry.

Magnetic-field-induced mixed-level Kondo effect in two-level systems

We consider a two-orbital impurity system with intra and inter-level Coulomb repulsion that is coupled to a single conduction channel. This situation can generically occur in multilevel quantum dots or in systems of coupled quantum dots. For finite energy-spacing between spin-degenerate orbitals, an in-plane magnetic field drives the system from a local singlet ground state to a "mixed-level" Kondo regime, where the Zeeman-split levels are degenerate for opposite spin states. We use the numerical renormalization group approach to fully characterize this mixed level Kondo state and discuss its properties in terms of the applied Zeeman field, temperature and system parameters. Under suitable conditions, the total spectral function is shown to develop a Fermi level resonance, so that the linear conductance of the system peaks at a finite Zeeman field while it decreases as function of temperature. These features, as well as the local moment and entropy contribution of the impurity system are commensurate with Kondo physics, which can be studied in suitably tuned quantum dot systems.

Fingerprints of Majorana fermions in spin-resolved subgap spectroscopy

When a strongly correlated quantum dot is tunnel-coupled to a superconductor, it leads to the formation of Shiba bound states inside the superconducting gap. They have been measured experimentally in a superconductor-quantum dot-normal lead setup. Side coupling the quantum dot to a topological superconducting wire that supports Majorana bound states at its ends, drastically affects the structure of the Shiba states and induces supplementary in-gap states. The anomalous coupling between the Majorana bound states and the quantum dot gives rise to a characteristic imbalance in the spin resolved spectral functions for the dot operators. These are clear fingerprints for the existence of Majorana fermions and they can be detected experimentally in transport measurements. In terms of methods employed, we have used analytical approaches combined with the numerical renormalization group approach.

Dynamics of the two-spin spin-boson model with a common bath.

Dynamics of the two-spin spin-boson model in the presence of Ohmic and sub-Ohmic baths is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Its accuracy in dynamics simulations of the two-spin SBM is improved significantly over the single D1 Ansatz, especially in the weak to moderately strong coupling regime. Validity of the multi-D1 Ansatz for various coupling strengths is also systematically examined by making use of the deviation vector which quantifies how faithfully the trial state obeys the Schrödinger equation. The time evolution of population difference and entanglement has been studied for various initial conditions and coupling strengths. Careful comparisons are carried out between our approach and three other methods, i.e., the time-dependent numerical renormalization group (TD-NRG) approach, the Bloch-Redfield theory, and a method based on a variational master equation. For strong coupling, the multi-D1 trial state yields consistent results as the TD-NRG approach in the Ohmic regime while the two disagree in the sub-Ohmic regime, where the multi-D1 trial state is shown to be more accurate. For weak coupling, the multi-D1 trial state agrees with the two master-equation methods in the presence of both Ohmic and sub-Ohmic baths, but shows considerable differences with the TD-NRG approach in the presence of a sub-Ohmic bath, calling into question the validity of the TD-NRG results at long times in the literature.

Transmission of a microwave cavity coupled to localized Shiba states

We consider a strongly correlated quantum dot, tunnel coupled to two superconducting leads and capacitively coupled to a single mode microwave cavity. When the superconducting gap is the largest energy scale, multiple Shiba states are formed inside the gap. The competition of these states for the ground state signals a quantum phase transition. We demonstrate that photonic measurements can be used to probe such localized Shiba states. Moreover, the quantum phase transition can be pinpointed exactly from the sudden change in the transmission signal. Calculations were performed using the numerical renormalization group approach.

Quantum criticality of the two-channel pseudogap Anderson model: universal scaling in linear and non-linear conductance

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, (0 < r < 1) near the Fermi energy . At a fixed lead-impurity 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 . Surprisingly, in the 2CK phase, different power-law scalings from the well-known or 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.