Numerical Renormalization Group Technique Research Articles

Yu-Shiba-Rusinov states assisted by asymmetric Coulomb repulsion in a bipartite molecular device

Interfacing magnetism with superconducting condensates are promising candidates holding Majorana bound states with which fault-tolerant quantum computation could be implemented. Within this field, understanding the detailed dynamics is important both for fundamental reasons and for the development of innovative quantum technologies. Herein, motivated by a molecular magnet Tb2Pc3 interacting with a superconducting Pb(111) substrate, which results in spin-orbital Yu-Shiba-Rusinov (YSR) states, as is affirmed by a theoretical simulation with the aid of the numerical renormalization group technique (see Xia et al 2022 Nat. Commun. 13 6388), we study the YSR states and quantum phase transitions (QPTs) in a bipartite molecular device adsorbed on an wave superconducting substrate. We highlight the effect of asymmetric Coulomb repulsion by computing the spectrum function and spin correlation function in various parameter regimes. We demonstrate that if one impurity is non-interacting, there are no YSR states in both impurities with any repulsion value in the other impurity. Whereas if the repulsion in one impurity is strong, the YSR states are observed in both impurities, and a QPT arises as the repulsion in the other impurity sweeps, assisted by the competition between the superconducting singlet (Cooper pair) and the Kondo singlet. The evolution of YSR states distinguishes from the single impurity case and can be well interpreted by the energy scales of the isotropic superconducting gap parameter, as well as the two Kondo temperatures. Our findings provide theoretical insights into the phase diagram of two magnetic impurities on a superconducting host and shine light on the effects induced by asymmetric Coulomb repulsion on many-body interactions.

Subgap states spectroscopy in a quantum dot coupled to gapped hosts: Unified picture for superconductor and semiconductor bands

We present a unified theory of quantum phase transitions for half-filled quantum dots (QDs) coupled to gapped host bands. We augment the bands by an additional weakly coupled metallic lead, which allows us to analyze the system by using standard numerical renormalization-group techniques. The ground-state properties of the systems without the additional metallic lead are then extrapolated in a controlled way from the broadened subgap spectral functions. We show that a broad class of narrow-gap-semiconductor tunneling densities of states (TDOSs) support the existence of two distinct phases known from their superconducting counterpart: the 0 phase, which is marked by the singlet ground state, and the $\ensuremath{\pi}$ phase regime with the doublet ground state. To keep a close analogy with the superconducting case, we focus on the influence of particle-hole asymmetry of the TDOS on the subgap spectral features. Nevertheless, we also discuss the possibility of inducing singlet-doublet quantum phase transitions in experimental setups by varying the filling of the QD. In addition, for gapped TDOS functions with smoothed gap edges, we demonstrate that all subgap peaks may leak out of the gap into the continuous part of the spectrum, an effect that has no counterpart in the superconducting Anderson model.

Scenarios of local spectral property and multifunctional spin selecting for a triple quantum dot molecule: How do the bonding and antibonding orbitals contribute to the spin currents?

Molecular electronic device is considered as a promising candidate for next generation electronic component, where the basic challenge involves understanding the charge and spin transports through molecular objects. In this paper, with the help of the sophisticated numerical renormalization group technique, we study theoretically the spin selective transport in a parallel triple quantum dot molecular device pierced by a local magnetic field along z axis. Based on simplified parameters of real molecular system, we find such device acts as a multifunctional spin selector when the inter-molecule tunneling couplings are asymmetric, including two 100% polarized spin-up summits, and one 100% spin-down summit in the linear conductance. We show in detail the local density of states for the bonding and anti-boding orbitals, and attribute the spin selection to the orbital polarized Coulomb blockade effect. We demonstrate our numerical results are consistent with those estimated by the analytical techniques, including the Friedel sum rule and the energy level crossings of the isolated orbitals.

Dynamical effects on superconductivity in a BCS-BEC crossover: Dynamical mean field theory and numerical renormalization group study of the Holstein model in infinite dimensions

We investigate the dynamical effects of pairing interaction on superconductivity in BCS-BEC crossover by studying the Holstein model at half-filling where the electron-phonon coupling $g$ controls the crossover. The dynamical mean-field theory was employed in combination with the numerical renormalization group technique. The dynamical effects induce distinct features such as absenceof the dispersion back-bending of Bogoliubov quasi-particles, non-monotonous coupling dependence of the pairing gap, and the soft phonon spectrum due to the Goldstone mode of local pair phase fluctuations. Also interesting is the maximum critical temperature being at the normal state phase boundary. Some of these features have intriguing similarities with the recent observations in the FeSe$_{1-x}$S$_x$ and Fe$_{1+y}$Se$_x$Te$_{1-x}$ iron-chalcogenides in the BCS-BEC crossover.

Kondo Signatures of a Quantum Magnetic Impurity in Topological Superconductors.

We study the Kondo physics of a quantum magnetic impurity in two-dimensional topological superconductors (TSCs), either intrinsic or induced on the surface of a bulk topological insulator, using a numerical renormalization group technique. We show that, despite sharing the p+ip pairing symmetry, intrinsic and extrinsic TSCs host different physical processes that produce distinct Kondo signatures. Extrinsic TSCs harbor an unusual screening mechanism involving both electron and orbital degrees of freedom that produces rich and prominent Kondo phenomena, especially an intriguing pseudospin Kondo singlet state in the superconducting gap and a spatially anisotropic spin correlation. In sharp contrast, intrinsic TSCs support a robust impurity spin doublet ground state and an isotropic spin correlation. These findings advance fundamental knowledge of novel Kondo phenomena in TSCs and suggest experimental avenues for their detection and distinction.

Bidirectional spin filter in a triple orbital molecule junction by tuning the magnetic field along a single direction.

Metal-molecule-metal junction is considered the basing block and key element of molecular spintronic devices, within which to generate spin polarized currents is one of the most fundamental issues for quantum computation and quantum information. In this paper, by employing a parallel triple orbital molecule junction with large inter-orbital tunneling couplings, we propose theoretically a bidirectional spin filter where both spin-up and spin-down currents could be obtained by simply adjusting the external magnetic field to different regimes along a single direction, and the filtered efficiencies could reach almost 100%. The Zeeman effect and the occupancy switching for the bonding and anti-bonding states are found to be responsible for the spin selective transport. We demonstrate that our scheme is robust for large parameter spaces of the orbital energy level, except the particle-hole symmetric point, and is widely suitable for the strong-, weak-, and non-interacting cases. To implement these problems, we use the Wilson's numerical renormalization group technique to treat such systems.

Phase transition and charge transport through a triple dot device beyond the Kondo regime**Project supported by the National Natural Science Foundation of China (Grant No. 11504102), the Scientific Research Items Foundation of Hubei Educational Committee (Grant Nos. Q20161803 and D20171803), the Doctoral Scientific Research Foundation of Hubei University of Automotive Technology (Grant No. BK201407).

Semiconductor quantum dot structure provides a promising basis for quantum information processing, within which to reveal the quantum phase and charge transport is one of the most important issues. In this paper, by means of the numerical renormalization group technique, we study the quantum phase transition and the charge transport for a parallel triple dot device in the strongly correlated limit, focusing on the effect of inter-dot hopping t beyond the Kondo regime. We find the quantum behaviors depend closely on the initial electron number on the dots, and the present model may map to single, double, and side-coupled impurity models in different parameter spaces. An orbital spin-1/2 Kondo effect between the conduction leads and the bonding orbital, and several magnetic-frustration phases are demonstrated when t is adjusted to different regimes. To understand these phenomena, a canonical transformation of the energy levels is given, and important physical quantities with respect to increasing t and necessary theoretical discussions are shown.

Spectral properties and the Kondo effect of cobalt adatoms on silicene

In terms of the state-of-the-art first principle computational methods combined with the numerical renormalization group technique the spectroscopic properties of Co adatoms deposited on silicene are analyzed. By establishing an effective low-energy Hamiltonian based on first principle calculations, we study the behavior of the local density of states of Co adatom on external parameters, such as magnetic field and gating. It is shown that the Kondo resonance with the Kondo temperature of the order of a few Kelvins can emerge by fine-tuning the chemical potential. The evolution and splitting of the Kondo peak with external magnetic field is also analyzed. Furthermore, it is shown that the spin polarization of adatom's spectral function in the presence of magnetic field can be relatively large, and it is possible to tune the polarization and its sign by electrical means.

Voltage-controlled Kosterlitz–Thouless transitions and various kinds of Kondo behaviors in a triple dot device**Project supported by the National Natural Science Foundation of China (Grant No. 11504102), the Scientific Research Items Foundation of Hubei Educational Committee, China (Grant Nos. Q20161803 and B2016091), the Doctoral Scientific Research Foundation (Grant No. BK201407), and the Major Scientific Research Project Pre-funds of Hubei University of

The transport property and phase transition for a parallel triple dot device are studied by adopting Wilson's numerical renormalization group technique, focusing on the effects of level spacings between neighboring dot sites. By keeping dot 2 at the half-filled level and tuning the level differences, it is demonstrated that the system transits from local spin quadruplet to triplet and doublet sequently, and three kinds of Kondo peaks at the Fermi surface could be found, which are separated by two Kosterlitz–Thouless type quantum phase transitions and correspond to spin–3/2, spin–1, and spin–1/2 Kondo effect, respectively. To obtain a detailed understanding of these problems, the charge occupation, the spin—spin correlation, the transmission coefficient, and the temperature-dependent magnetic moment are shown, and necessary physical arguments are given.

Effect of Dot Geometry on Various Kinds of Kondo Behaviors for Parallel Triple Dot System

With the help of the Wilson’s numerical renormalization group technique, we study the dynamical and magnetic properties for a parallel triple dot system with five different dot geometries in the strong correlation limit. We reveal that when the dot configuration changes, several kinds of Kondo effect could be found, i.e., the partially screened spin − 3/2, the totally screened spin − 1, the spin − 1/2, and the spin − 0 Kondo effects, implying numerically that the manners in which the dots and leads are organized play important roles in the Kondo behaviors. To understand these quantum phenomena, the local density of states, the transmission coefficient, the spin-spin correlation function, and the temperature-dependent magnetic moment for each configuration are shown, and necessary theoretical discussions are given.

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

By means of the numerical renormalization group (NRG) technique, we study the low temperature transport property and the phase transition for a triangular triple quantum dot system, including two centered dots (dot 1 and 2) and one side dot (dot 3). We focus on the effect of interdot repulsion V between two centered dots in a wide range of the interdot hopping tij (i,j = 1,2,3). When the hoppings between the centered dot and the side dot are symmetric, i.e., t13 = t23, and that between two centered dots t12 is small, two centered dots form a spin triplet when V is absent, and a totally screened spin−1 Kondo effect is observed. In this case, one has a spin 1 that is partially screened by the leads as in the usual spin−1 Kondo model, and the remaining spin 1/2 degree of freedom forms a singlet with the side dot. As V is large enough, one of the centered dots is singly occupied, while the other one is empty. The spin−1/2 Kondo effect is found when t13 is small. For large t12, two centered dots form a spin singlet when V = 0, leading to zero conductance. As V is large enough, the spin−1/2 Kondo effect is recovered in the case of small t13. For asymmetric t13≠t23 and small t12, a crossover is found as V increases in comparison with a first order quantum phase transition for the symmetric case. In the regime of large V, the spin−1/2 Kondo effect could also be found when both t13 and t23 are small. We demonstrate the present model is similar to the side-coupled double dot system in some appropriate regimes, and it appears as a possible realization of side-controllable molecular electronics and spintronics devices.

Strong spin Seebeck effect in Kondo T-shaped double quantum dots

We investigate, taking a theoretical approach, the thermoelectric and spin thermoelectric properties of a T-shaped double quantum dot strongly coupled to two ferromagnetic leads, focusing on the transport regime in which the system exhibits the two-stage Kondo effect. We study the dependence of the (spin) Seebeck coefficient, the corresponding power factor and the figure of merit on temperature, leads’ spin polarization and dot level position. We show that the thermal conductance fulfills a modified Wiedemann–Franz law, also in the regime of suppression of subsequent stages of the Kondo effect by the exchange field resulting from the presence of ferromagnets. Moreover, we demonstrate that the spin thermopower is enhanced at temperatures corresponding to the second stage of Kondo screening. Very interestingly, the spin-thermoelectric response of the system is found to be highly sensitive to the spin polarization of the leads. In some cases spin polarization of the order of 1% is sufficient for a strong spin Seebeck effect to occur. This is explained as a consequence of the interplay between the two-stage Kondo effect and the exchange field induced in the double quantum dot. Due to the possibility of tuning the exchange field by the choice of gate voltage, the spin thermopower may also be tuned to be maximal for desired spin polarization of the leads. All calculations are performed with the aid of the numerical renormalization group technique.

Zero-field splitting of the Kondo resonance and quantum criticality in triple quantum dots

We consider a triple-quantum-dot (TQD) system composed by an interacting quantum dot connected to two effectively non-interacting dots, which in turn are both connected in parallel to metallic leads. As we show, this system can be mapped onto a single-impurity Anderson model with a non-trivial density of states. The TQD's transport properties are investigated under a continuous tuning of the non-interacting dots' energy-levels, employing the Numerical Renormalization Group technique. Interference between single and many-particle resonances splits the Kondo peak, fulfilling a generalized Friedel sum rule. In addition, a particular configuration in which one of the non-interacting dots is held out of resonance with the leads allows to access a pseudogap regime where a Kosterlitz-Thouless type quantum-phase-transition (QPT) occur, separating the Kondo and non-Kondo behavior. Within this same configuration, the TQD exhibits traces of the Fano-Kondo effect, which is in turn, strongly affected by the QPT. Signatures of all these phenomena are neatly displayed by the calculated linear conductance.

Field-Controllable Spin Filter Based on Parallel Quantum Dot Systems

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.

Synthetic-gauge-field-induced resonances and Fulde-Ferrell-Larkin-Ovchinnikov states in a one-dimensional optical lattice

Coherent coupling generated by laser light between the hyperfine states of atoms, loaded in a 1D optical lattice, gives rise to the "synthetic dimension" system which is equivalent to a Hofstadter model in a finite strip of square lattice. An SU(M) symmetric attractive interaction in conjunction with the synthetic gauge field present in this system gives rise to unusual effects. We study the two- body problem of the system using the T-matrix formalism. We show that the two-body ground states pick up a finite momentum and can transform into two-body resonance like features in the scattering continuum with a large change in the phase shift. As a result, even for this 1D system, a critical amount of attraction is needed to form bound states. These phenomena have spectacular effects on the many body physics of the system analyzed using the numerical density matrix renormalization group technique. We show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states form in the system even for a "balanced" gas and the FFLO momentum of the pairs scales linearly with flux. Considering suitable measures, we investigate interesting properties of these states. We also discuss a possibility of realization of a generalized interesting topological model, called the Creutz ladder.

Kondo Resonance of a Co Atom Exchange Coupled to a Ferromagnetic Tip.

The Kondo effect of a Co atom on Cu(100) was investigated with a low-temperature scanning tunneling microscope using a monoatomically sharp nickel tip. Upon a tip-Co contact, the differential conductance spectra exhibit a spin-split asymmetric Kondo resonance. The computed ab initio value of the exchange coupling is too small to suppress the Kondo effect, but sufficiently large to produce the splitting observed. A quantitative analysis of the line shape using the numerical renormalization group technique indicates that the junction spin polarization is weak.

Kondo Impurities in the Kitaev Spin Liquid: Numerical Renormalization Group Solution and Gauge-Flux-Driven Screening.

Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.

Tunable spin selective transport and quantum phase transition in parallel double dot system

We study theoretically the spin selective transport and the quantum phase transition (QPT) in a double dot device by means of the numerical renormalization group technique. When the gate voltage ε is in the Kondo regime and the interdot hopping t is large enough, a first order QPT between local spin singlet and Sz=1 of the triplet is observed as the magnetic field B increases. Beyond the Kondo regime, the QPTs depend closely on ε and t, and perfect spin filter is found, where the effect of spin filtering could easily be manipulated by tuning external parameters. We show that the interplay between the Zeeman effect and the antiferromagnetic interdot hopping, and occupancy switching are responsible for the QPT and the spin selective transport.

Phase diagram and low temperature scenario for a triangular triple dot system

We present a triangular triple quantum dot (TTQD) system with two dots connected parallelly to one conduction lead, and investigate the phase diagram, the electric transport, and the temperature-dependent magnetic moment at half filling. When the hopping between two connected dots t12 = 0, and those between the connected dots and the side dot are symmetric t13 = t23, two connected dots form a spin triplet due to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the dot-lead coupling and/or the hopping t13 (t23). For t13 = 0, the triplet is partially screened by the conduction leads at low temperature. Both the connected dots and the side dot contribute to the magnetic moment of the system. For any definite t13, the triplet is totally screened by the conduction leads and the side dot, and the two-stage Kondo effect occurs. When t12 increases beyond a critical t12c, two connected dots form a spin singlet and decouple from the side dot. In this case, the Kondo peak is strongly suppressed, indicating zero conductance, and only the localized side dot contribute to the magnetic moment at low temperature. When t13 ≠ t23, we find a crossover as t12 increases, contrast to the first order transition of the symmetric case. Numerical renormalization group technique and physical arguments are used to obtain a detailed understanding of these problems.

RKKY interaction and local density of states for a triangular triple quantum dot system

By means of the numerical renormalization group technique, we study the local density of states (LDOS) for a triangular triple quantum dot system, with two dots connected in parallel to the conduction leads. We find the location of the Ruderman–Kittel–Kasuya–Yosida (RKKY) peak identified in the LDOS could be illustrated as JRKKY=aΓ2/U+bt22/U, with U being the on-site Coulomb repulsion, Γ the dot-lead coupling, and t2 the hopping between the connected dots and the side dot. When the hopping between two connected dots t1 turns on, the spectrum weight of the RKKY peaks decreases due to the competition between the direct and the RKKY interactions. As t1 increases beyond a critical point t1c, two connected dots form a spin singlet, and decouple from both the side dot and the conduction leads, thus the Kondo and RKKY peaks could not be found. For t1<t1c, the conductance reaches to the unitary limit, while for t1≥t1c, it drops to zero.