Numerical Renormalization Group Method Research Articles

Probing magnetic and triplet correlations in spin-split superconductors with magnetic impurities

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

Atomic-scale magnetic doping of monolayer stanene by revealing Kondo effect from self-assembled Fe spin entities

Atomic-scale spin entity in a two-dimensional topological insulator lays the foundation to manufacture magnetic topological materials with single atomic thickness. Here, we have successfully fabricated Fe monomer, dimer and trimer doped in the monolayer stanene/Cu(111) through a low-temperature growth and systematically investigated Kondo effect by combining scanning tunneling microscopy/spectroscopy (STM/STS) with density functional theory (DFT) and numerical renormalization group (NRG) method. Given high spatial and energy resolution, tunneling conductance (dI/dU) spectra have resolved zero-bias Kondo resonance and resultant magnetic-field-dependent Zeeman splitting, yielding an effective spin Seff = 3/2 with an easy-plane magnetic anisotropy on the self-assembled Fe atomic dopants. Reduced Kondo temperature along with attenuated Kondo intensity from Fe monomer to trimer have been further identified as a manifestation of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between Sn-separated Fe atoms. Such magnetic Fe atom assembly in turn constitutes important cornerstones for tailoring topological band structures and developing magnetic phase transition in the single-atom-layer stanene.

Prediction of impurity spectrum function by deep learning algorithm

By using the numerical renormalization group (NRG) method, we construct a large dataset with about one million spectral functions of the Anderson quantum impurity model. The dataset contains the density of states (DOS) of the host material, the strength of Coulomb interaction between on-site electrons (U), and the hybridization between the host material and the impurity site (Γ). The continued DOS and spectral functions are stored with Chebyshev coefficients and wavelet functions, respectively. From this dataset, we build seven different machine learning networks to predict the spectral function from the input data, DOS, U, and Γ. Three different evaluation indexes, mean absolute error (MAE), relative error (RE) and root mean square error (RMSE), are used to analyze the prediction abilities of different network models. Detailed analysis shows that, for the two kinds of widely used recurrent neural networks (RNNs), gate recurrent unit (GRU) has better performance than the long short term memory (LSTM) network. A combination of bidirectional GRU (BiGRU) and GRU has the best performance among GRU, BiGRU, LSTM, and BiLSTM. The MAE peak of BiGRU+GRU reaches 0.00037. We have also tested a one-dimensional convolutional neural network (1DCNN) with 20 hidden layers and a residual neural network (ResNet), we find that the 1DCNN has almost the same performance of the BiGRU+GRU network for the original dataset, while the robustness testing seems to be a little weak than BiGRU+GRU when we test all these models on two other independent datasets. The ResNet has the worst performance among all the seven network models. The datasets presented in this paper, including the large data set of the spectral function of Anderson quantum impurity model, are openly available at https://doi.org/10.57760/sciencedb.j00113.00192.

Quantum correlations in the frustrated XY model on the honeycomb lattice

We have investigated the spin-1/2 XY frustrated antiferromagnetic Heisenberg honeycomb model, which features an intermediate region in its ground state phase diagram that is not well understood. The two dominant phases in the diagram are the quantum spin-liquid (QSL) and the antiferromagnetic Ising order. Quantum correlations suggest that the QSL phase is likely to exhibit entanglement. To explore this possibility, we utilized numerical Lanczos and density matrix renormalization group (DMRG) methods to calculate concurrence, quantum discord (QD), and entanglement entropy. The results of our study indicate the existence of quantum entanglement within the intermediate region, implying a greater probability for the dominance of the quantum spin-liquid (QSL) phase over the antiferromagnetic Ising order. This discovery underscores the importance of considering quantum correlations in comprehending the model’s behavior and provides insight into the complex nature of quantum systems.

Manipulating the optimal transport orbital and virtual states in a supramolecular transistor

Supramolecular electronics bridges the gap between single-molecule electronics and bulk, plastic electronics, and is embodied as one of the biggest promises of nanometer electronics made by supramolecular functional materials. Within this field, the way in which electrons propagate through the supramolecule plays a profound role both in designing supramolecular devices and in understanding the structure–property relationships at the molecular level. Herein, we consider a supramolecular transistor, which contains bipartite molecular monomers, with one molecular orbital is non-interacting, whereas the other one is strongly correlated. Modeling this device using the two-impurity Anderson model, we unravel the orbital selections and the virtual states in the electronic transport process with the aid of the celebrated numerical renormalization group method. We demonstrate that if both orbitals are in their particle-hole symmetric regimes, electrons favor to transmit through the non-interacting molecular orbital via the resonant tunneling process, rather than through the strongly correlated molecular orbital via the Kondo effect, despite the energy of the strongly correlated orbital is lower. This phenomenon indicates that if the Kondo effect occurs, it experiences a temporarily doubly occupied virtual state. Interestingly, the situation reverses if the inter-molecule hopping integral is applied, where the strongly correlated orbital becomes optimal and the virtual state turns to be an empty state. To understand these behaviors, both the static and the dynamical properties, as well as those thermodynamic quantities are studied in detail, and necessary physical arguments are given. Our findings afford deep insights into the manipulations of the optimal transport orbital and the Kondo behaviors within quantum impurity devices.

Field-induced local Fermi-liquid states in the underscreened Kondo model for Eu compounds

We have studied an underscreened Kondo model for the impurity spin S=7/2S=7/2 capturing the 4f4f characteristics of a nearly divalent Eu ion. By using the numerical renormalization group method, it is shown that an unusual heavy-fermion state is induced from the underscreened spin state when a weak crystal field is introduced. It is also found that the application of the magnetic field enhances the effective mass of the heavy fermions, reflecting the field-induced level crossing of the spin states. We will discuss the mechanism of these mass enhancements in terms of a realization of the two-stage Kondo effect.

Kondo effect in twisted bilayer graphene

The emergence of flat bands in twisted bilayer graphene at the magic angle can be understood in terms of a vanishing Fermi velocity of the Dirac cone. This is associated with van Hove singularities approaching the Fermi energy and becoming higher-order. In the density of states, this is reflected by flanking logarithmic van Hove divergences pinching off the central Dirac cone in energy space. The low-energy pseudogap of the Dirac cone away from the magic angle is replaced by a power-law divergence due to the higher-order van Hove singularity at the magic angle. This plays an important role in the exotic phenomena observed in this material, such as superconductivity and magnetism, by amplifying electronic correlation effects. Here we investigate one such correlation effect---the Kondo effect due to a magnetic impurity embedded in twisted bilayer graphene. We use the Bistritzer-MacDonald model to extract the low-energy density of states of the material as a function of twist angle and study the resulting quantum impurity physics using perturbative and numerical renormalization group methods. Although at zero temperature the impurity is only Kondo screened precisely at the magic angle, we find highly nontrivial behavior at finite temperatures relevant to experiments, due to the complex interplay between Dirac, van Hove, and Kondo physics.

Nonequilibrium Seebeck effect and thermoelectric efficiency of Kondo-correlated molecular junctions

We theoretically study the nonequilibrium thermoelectric transport properties of a strongly-correlated molecule (or quantum dot) embedded in a tunnel junction. Assuming that the coupling of the molecule to the contacts is asymmetric, we determine the nonlinear current driven by the voltage and temperature gradients by using the perturbation theory. However, the subsystem consisting of the molecule strongly coupled to one of the contacts is solved by using the numerical renormalization group method, which allows for accurate description of Kondo correlations. We study the temperature gradient and voltage dependence of the nonlinear and differential Seebeck coefficients for various initial configurations of the system. In particular, we show that in the Coulomb blockade regime with singly occupied molecule, both thermopowers exhibit sign changes due to the Kondo correlations at nonequilibrium conditions. Moreover, we determine the nonlinear heat current and thermoelectric efficiency, demonstrating that the system can work as a heat engine with considerable efficiency, depending on the transport regime.

Strongly correlated electrons in superconducting islands with fluctuating Cooper pairs

We present a particle-number-conserving theory for many-body effects in mesoscopic superconducting islands connected to normal electrodes, which explicitly includes quantum fluctuations of Cooper pairs in the condensate. Beyond previous BCS mean-field descriptions, our theory can precisely treat the pairing and Coulomb interactions over a broad range of parameters by using the numerical renormalization group method. On increasing the ratio of pairing interactions to Coulomb interactions, the low-energy physics of the system evolves from the spin Kondo regime to the mixed-valence regime and eventually reaches an anisotropic charge Kondo phase, while a crossover from $1e$- to $2e$-periodic Coulomb blockade of transport is revealed at high temperatures. For weak pairing, the superconducting condensate is frozen in the local spin-flip processes but fluctuates in the virtual excitations, yielding an enhanced spin Kondo temperature. For strong pairing, massive fluctuations of Cooper pairs are crucial for establishing charge Kondo correlations whose Kondo temperature rapidly decreases with the pairing interaction. Surprisingly, a charge-exchange-induced local field may occur even at the charge degenerate point, thereby destroying the charge Kondo effect. These are demonstrated in the spectral and transport properties of the island.

Unified formulations for RKKY interaction, side Kondo behavior, and Fano antiresonance in a hybrid tripartite quantum dot device with filtered density of states

Hybrid quantum dot structures are potential building blocks for spintronic devices and quantum logic gates, within which the understanding of many-body correlations is a prerequisite for implementing quantum information processing and generating controlled entanglement. Herein, we consider a hybrid tripartite quantum dot device, one of which is embedded directly between the source and drain electrodes, while the other two dots are side coupled to the first one. Modeling the system using the three-impurity Anderson model, we concentrate on the Ruderman-Kittel-Kasuya-Yosida interaction, the suppression of the Kondo behavior in the side dots, the destructive quantum interference, the gate-controlled quantum phase transitions, the Dicke effect, the ferromagnetic Kondo effect, and the thermodynamical properties in various parameter regimes using the numerical renormalization group method. Unified formulations are established, which are associated with the effective width of the zero-energy peak of the central dot without side dots, the central-side exchange coupling, as well the on-site electron-electron repulsion. We demonstrate that these formulas are well suitable for the non-, weak-, and strong-interacting cases.

Numerical renormalization group study of the Loschmidt echo in Kondo systems

We study the dynamical properties of the one-channel and two-channel spin-1/2 Kondo models after quenching in Hamiltonian variables. Eigen spectrum of the initial and final Hamiltonians is calculated by using the numerical renormalization group method implemented within the matrix product states formalism. We consider multiple quench protocols in the considered Kondo systems, also in the presence of external magnetic field of different intensities. The main emphasis is put on the analysis of the behavior of the Loschmidt echo L(t), which measures the ability of the system’s revival to its initial state after a quench. We show that the decay of the Loschmidt echo strongly depends on the type of quench and the ground state of the system. For the one-channel Kondo model, we show that L(t) decays as, L(t)sim (tcdot T_K)^{-1.4}, where T_K is the Kondo temperature, while for the two-channel Kondo model, we demonstrate that the decay is slower and given by L(t)sim (tcdot T_K)^{-0.7}. In addition, we also determine the dynamical behavior of the impurity’s magnetization, which sheds light on identification of the relevant time scales in the system’s dynamics.

Two-Channel Charge-Kondo Physics in Graphene Quantum Dots.

Nanoelectronic quantum dot devices exploiting the charge-Kondo paradigm have been established as versatile and accurate analogue quantum simulators of fundamental quantum impurity models. In particular, hybrid metal–semiconductor dots connected to two metallic leads realize the two-channel Kondo (2CK) model, in which Kondo screening of the dot charge pseudospin is frustrated. In this article, a two-channel charge-Kondo device made instead from graphene components is considered, realizing a pseudogapped version of the 2CK model. The model is solved using Wilson’s Numerical Renormalization Group method, uncovering a rich phase diagram as a function of dot–lead coupling strength, channel asymmetry, and potential scattering. The complex physics of this system is explored through its thermodynamic properties, scattering T-matrix, and experimentally measurable conductance. The strong coupling pseudogap Kondo phase is found to persist in the channel-asymmetric two-channel context, while in the channel-symmetric case, frustration results in a novel quantum phase transition. Remarkably, despite the vanishing density of states in the graphene leads at low energies, a finite linear conductance is found at zero temperature at the frustrated critical point, which is of a non-Fermi liquid type. Our results suggest that the graphene charge-Kondo platform offers a unique possibility to access multichannel pseudogap Kondo physics.

Band Polarization Effect on the Kondo State in a Zigzag Silicene Nanoribbon.

Using the Numerical Renormalization Group method, we study the properties of a quantum impurity coupled to a zigzag silicene nanoribbon (ZSNR) that is subjected to the action of a magnetic field applied in a generic direction. We propose a simulation of what a scanning tunneling microscope will see when investigating the Kondo peak of a magnetic impurity coupled to the metallic edge of this topologically non-trivial nanoribbon. This system is subjected to an external magnetic field that polarizes the host much more strongly than the impurity. Thus, we are indirectly analyzing the ZSNR polarization through the STM analysis of the fate of the Kondo state subjected to the influence of the polarized conduction electron band. Our numerical simulations demonstrate that the spin-orbit-coupling-generated band polarization anisotropy is strong enough to have a qualitative effect on the Kondo peak for magnetic fields applied along different directions, suggesting that this contrast could be experimentally detected.

Spin-orbit coupling effects on thermoelectric transport properties in quantum dots

We study the effects caused by Rashba and Dresselhaus spin-orbit coupling over the thermoelectric transport properties of a single-electron transistor, viz., a quantum dot connected to one-dimensional leads. Using linear response theory and employing the numerical renormalization group method, we calculate the thermopower, electrical and thermal conductances, dimensionless thermoelectric figure of merit, and study the Wiedemann-Franz law, showing their temperature maps. Our results for all those properties indicate that spin-orbit coupling drives the system into the Kondo regime. We show that the thermoelectric transport properties, in the presence of spin-orbit coupling, obey the expected universality of the Kondo strong coupling fixed point. In addition, our results show a notable increase in the thermoelectric figure of merit, caused by the spin-orbit coupling in the one-dimensional quantum dot leads.

Fractional spin excitations and conductance in the spiral-staircase Heisenberg ladder

We investigate theoretically the spiral staircase Heisenberg spin-$1/2$ ladder in the presence of antiferromagnetic long-range spin interactions and a uniform magnetic field. As a special case we also consider the Kondo necklace model. If the magnetizations of the two chains forming the ladder satisfy a certain resonance condition, involving interchain couplings as perturbations, the system is in a partially gapped magnetic phase hosting excitations characterized by fractional spins, whose values can be changed by the magnetic field. We show that these fractional spin excitations can be probed by spin currents in a transport setup with a spin conductance that reveals the fractionalized spin. In some special cases, the spin conductance reaches universal values in units of $(g\mu_B)^2/h$, where $g$ is the $g$-factor, $\mu_B$ the Bohr magneton, and $h$ the Planck constant. We obtain our results with the help of bosonization and numerical density matrix renormalization group methods.

Frustration-controlled quantum phase transition between multiple singular two-stage Kondo behaviors in a tetrahedral quadruple quantum dot structure

Semiconductor quantum dots are considered to be promising candidates for the hardware of quantum information technology and optoelectronic devices. Herein, motivated by a tetrahedrally shaped colloidal quadruple quantum dot structure made from In-based III--V semiconductors, which has been synthesized very recently by Leemans et al. [J. Am. Chem. Soc., 143, 4290 (2021)], we provide timely insight into the electronic transport and the quantum phase transition (QPT) for such an architecture. When the interdot hopping between different side dots (${t}_{2}$) is absent, a singular two-stage Kondo effect is revealed for small central-side coupling ${t}_{1}$. The two screening processes are separated by an energy scale of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, and the fitting parameters deviate from the regular ones of the side-coupled double dot system. The RKKY interaction and temperature are well illustrated by functions of ${t}_{1}^{4}/(U{T}_{K1}^{2})$, where $U$ and ${T}_{K1}$ are the on-site electron-electron repulsion and the first Kondo temperature, respectively. When ${t}_{2}$ turns on, the ground state of the side dots transits from a spin quadruplet to a magnetic frustration phase, and then to a singlet, through two first-order QPTs. In the frustration phase, another new two-stage Kondo effect is demonstrated, which includes the process of screening the local spin on the central dot firstly, and then that on the neighborless side dot is screened at a lower temperature. Both the Kondo temperatures are found to be rather sensitive to ${t}_{2}$. When ${t}_{2}$ is large enough, the reappearance of the regular Kondo effect is found. With fixed ${t}_{2}=0$, charging the central dot triggers a transition from an antiferromagnetic correlation among the side dots to a ferromagnetic one, accompanied by a Kondo behavior in the central dot. We adopt the state-of-the-art numerical renormalization group method to implement the above behaviors, combined with analytical arguments.

Spin-resolved thermal signatures of Majorana-Kondo interplay in double quantum dots

We investigate theoretically the thermoelectric transport properties of a T-shaped double quantum dot side-coupled to a topological superconducting nanowire hosting Majorana zero-energy modes. The calculations are performed using the numerical renormalization group method focusing on the transport regime, where the system exhibits the two-stage Kondo effect. It is shown that the leakage of Majorana quasiparticles into the double dot system results in a half-suppression of the second stage of the Kondo effect, which is revealed through fractional values of the charge and heat conductances and gives rise to new resonances in the Seebeck coefficient. The heat conductance is found to satisfy a modified Wiedemann-Franz law. Finally, the interplay of Majorana-induced interference with strong electron correlations is discussed in the behavior of the spin Seebeck effect, a unique feature of the considered setup.

Majorana-Kondo competition in a cross-shaped double quantum dot-topological superconductor system

We examine the transport properties of a double quantum dot system coupled to a topological superconducting nanowire hosting Majorana quasiparticles at its ends, with the central quantum dot attached to the left and right leads. We focus on the behavior of the local density of states and the linear conductance, calculated with the aid of the numerical renormalization group method, to describe the influence of the Majorana coupling on the low-temperature transport properties induced by the Kondo correlations. In particular, we show that the presence of Majorana quasiparticles in the system affects both the spin-up and spin-down transport channels, affecting the energy scales associated with the first-stage and second-stage Kondo temperatures, respectively, and modifying the low-energy behavior of the system.

Spintronic transport through a double quantum dot-based spin valve with noncollinear magnetizations

We study the magnetoresistive properties of a spin valve based on a double quantum dot attached to ferromagnetic leads with noncollinear alignment of magnetic moments. It is assumed that each dot is strongly coupled to its own ferromagnetic electrode, while the hopping between the dots is relatively weak. The calculations are performed by using the perturbation theory in the coupling between the dots, while the local density of states of a quantum dot attached to a given external lead is determined with the aid of the numerical renormalization group method. We demonstrate that the examined device can exhibit considerable positive or inverse tunnel magnetoresistance. It can be also a source of highly spin-polarized current. Importantly, the spin-resolved transport properties can be controlled by gate and bias voltages and depend on the angle between the magnetizations of the ferromagnets.

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.