Numerical Renormalization Group Research Articles

Manipulating Majorana zero modes in double quantum dots

Majorana zero modes (MZMs) emerging at the edges of topological superconducting wires have been proposed as the building blocks of novel, fault-tolerant quantum computation protocols. Coherent detection and manipulation of such states in scalable devices are, therefore, essential in these applications. Recent detection proposals include semiconductor quantum dots (QDs) coupled to the end of these wires, as changes in the QD electronic spectral density due to the MZM coupling could be detected in transport experiments. Here, we propose that multi-QD systems can also be used to manipulate MZMs through precise control over the QDs' parameters. The simplest case where Majorana manipulation is possible is in a double quantum dot (DQD) geometry. By using exact analytical methods and numerical renormalization-group calculations, we show that the QDs' spectral functions can be used to characterize the presence or not of MZMs "leaking" into the DQD. More importantly, we find that these signatures respond to changes in the DQD parameters such as gate-voltages and couplings in a consistent fashion. Additionally, we show that different MZM-DQD coupling geometries ("symmetric" , "in-series" and "T-shaped" junctions) offer distinct ways in which MZMs can be switched from dot to dot. These results highlight the interesting possibilities that DQDs offer for all-electrical MZM control in scalable devices.

Interaction-dependent anisotropy of fractional quantum Hall states

A fractional quantum Hall (FQH) system with broken rotational symmetry exploits its geometric degree of freedom to minimize its ground state energy. The mass anisotropy of bare particles interacting isotropically is partially inherited by the many-body FQH state, and the extent to which it does so depends on the type of interaction, filling fraction and ground state phase. Using numerical infinite density matrix renormalization group simulations, we investigate the transference of elliptical ($C_2$-symmetric) anisotropy from the band mass of the bare particles to the FQH states, for various power law interactions. We map out the response of FQH states to small anisotropy as a function of power law exponent, filling, and statistics (bosonic or fermionic) of the constituents. Interestingly, we find a non-analyticity in the linear response of the FQH state at a special filling-dependent value of the power law exponent, above which the interaction effectively becomes zero-range (point-like). We also investigate the the effect of $C_4$-symmetric band distortions, where we observe a strikingly different dependence on filling.

Topological Mott insulator with bosonic edge modes in one-dimensional fermionic superlattices

We investigate topological phase transitions driven by interaction and identify a novel topological Mott insulator state in one-dimensional fermionic optical superlattices through numerical density matrix renormalization group (DMRG) method. Remarkably, the low-energy edge excitations change from spin-1/2 fermionic single-particle modes to spin-1 bosonic collective modes across the phase transition. Due to spin-charge separation, the low-energy theory is governed by an effective spin superexchange model, whereas the charge degree of freedom is fully gapped out. Such topological Mott state can be characterized by a spin Chern number and gapless magnon modes protected by a finite spin gap. The proposed experimental setup is simple and may pave the way for the experimental observation of exotic topological Mott states.

Voltage-controllable multifunctional spin polarizer based on side-coupled quantum dots

Abstract Quantum dot system is considered as the prime candidate for nanospintronics devices, within which to generate spins one by one is the long term goal for quantum information and quantum computation. In this paper, with the aid of the Wilson’s numerical renormalization group method, we find a side-coupled double quantum dot molecule could be used to implement a multifunctional spin polarizer in the strongly correlated regime without switching the direction of the magnetic field. The spin polarizer reveals almost 100 % efficiencies, and the polarizing scenarios could be controlled easily by the initial charge state through external gate voltage, leading to various kinds of spin polarizers, e.g., unidirectional, bidirectional, and dual spin polarizers. We demonstrate the Zeeman effect and the inter-dot transport processes are responsible for the spin-polarized transport. Our scheme is robust for any inter-dot coupling and may also be useful for quantum dots connected with magnetic leads.

Quench dynamics of spin in quantum dots coupled to spin-polarized leads

We investigate the quench dynamics of a quantum dot strongly coupled to spin-polarized ferromagnetic leads. The real-time evolution is calculated by means of the time-dependent density-matrix numerical renormalization group method implemented within the matrix product states framework. We examine the system's response to a quench in the spin-dependent coupling strength to ferromagnetic leads as well as in the position of the dot's orbital level. The spin dynamics is analyzed by calculating the time-dependent expectation values of the quantum dot's magnetization and occupation. Based on these, we determine the time dependence of a ferromagnetic-contact-induced exchange field and predict its nonmonotonic buildup. In particular, two timescales are identified, describing the development of the exchange field and the dot's magnetization sign change. Finally, we study the effects of finite temperature on the dynamical behavior of the system.

Observation of Coexistence of Yu-Shiba-Rusinov States and Spin-Flip Excitations.

We investigate the spectral evolution in different metal phthalocyanine molecules on NbSe2 surface using scanning tunnelling microscopy (STM) as a function of the coupling with the substrate. For manganese phthalocyanine (MnPc), we demonstrate a smooth spectral crossover from Yu-Shiba-Rusinov (YSR) bound states to spin-flip excitations. This has not been observed previously and it is in contrast to simple theoretical expectations. We corroborate the experimental findings using numerical renormalization group calculations. Our results provide fundamental new insight on the behavior of atomic scale magnetic/SC hybrid systems, which is important, for example, for engineered topological superconductors and spin logic devices.

Sublattice symmetry breaking and Kondo-effect enhancement in strained graphene

Kondo physics in doped monolayer graphene is predicted to exhibit unusual features due to the linear vanishing of the pristine material's density of states at the Dirac point. Despite several attempts, conclusive experimental observation of the phenomenon remains elusive. One likely obstacle to identification is a very small Kondo temperature scale $T_K$ in situations where the chemical potential lies near the Dirac point. We propose tailored mechanical deformations of monolayer graphene as a means of revealing unique fingerprints of the Kondo effect. Inhomogeneous strains are known to produce specific alternating changes in the local density of states (LDOS) away from the Dirac point that signal sublattice symmetry breaking effects. Small LDOS changes can be amplified in an exponential increase or decrease of $T_K$ for magnetic impurities attached at different locations. We illustrate this behavior in two deformation geometries: a circular 'bubble' and a long fold, both described by Gaussian displacement profiles. We calculate the LDOS changes for modest strains and analyze the relevant Anderson impurity model describing a magnetic atom adsorbed in either a 'top-site' or a 'hollow-site' configuration. Numerical renormalization-group solutions of the impurity model suggest that higher expected $T_K$ values, combined with distinctive spatial patterns under variation of the point of graphene attachment, make the top-site configuration the more promising for experimental observation of signatures of the Kondo effect. The strong strain sensitivity of $T_K$ may lift top-site Kondo physics into the range experimentally accessible using local probes such as scanning tunneling microscopy.

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.

Temperature-dependent spectral function of a Kondo impurity in ans-wave superconductor

Using the numerical renormalization group method, the effect due to a Kondo impurity in an $s$-wave superconductor is examined at finite temperature ($T$). The $T$-behaviors of the spectral function and the magnetic moment at the impurity site are calculated. At $T$=0, the spin due to the impurity is in singlet state when the ratio between the Kondo temperature $T_k$ and the superconducting gap $\Delta$ is larger than 0.26. Otherwise, the spin of the impurity is in a doublet state. We show that the separation of the double Yu-Shiba-Rusinov peaks in the spectral function shrinks as $T$ increases if $T_k/\Delta<0.26$ while it is expanding if $T_k/\Delta>0.26$ and $\Delta$ remains to be a constant. These features could be measured by experiments and thus provide a unique way to determine whether the spin of the single Kondo impurity is in singlet or doublet state at zero temperature.

Practical Guide to Quantum Phase Transitions in Quantum-Dot-Based Tunable Josephson Junctions

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

Two-dimensional quantum-spin-1/2 XXZ magnet in zero magnetic field: Global thermodynamics from renormalisation group theory

ABSTRACTPhase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.

Magnetically Tuned Kondo Effect in a Molecular Double Quantum Dot: Role of the Anisotropic Exchange

We investigate theoretically and experimentally the singlet-triplet Kondo effect induced by a magnetic field in a molecular junction. Temperature dependent conductance, $G(T)$, is calculated by the numerical renormalization group, showing a strong imprint of the relevant low energy scales, such as the Kondo temperature, exchange and singlet-triplet splitting. We demonstrate the stability of the singlet-triplet Kondo effect against weak spin anisotropy, modeled by an anisotropic exchange. Moderate spin anisotropy manifests itself by lowering the Kondo plateaus, causing the $G(T)$ to deviate from a standard temperature dependence, expected for a spin-half Kondo effect. We propose this scenario as an explanation for anomalous $G(T)$, measured in an organic diradical molecule coupled to gold contacts. We uncover certain new aspects of the singlet-triplet Kondo effect, such as coexistence of spin-polarization on the molecule with Kondo screening and non-perturbative parametric dependence of an effective magnetic field induced by the leads.

Controllable precision of the projective truncation approximation for Green’s functions**Project supported by the National Key Basic Research Program of China (Grant No. 2012CB921704), the National Natural Science Foundation of China (Grant No. 11374362), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).

Recently, we developed the projective truncation approximation for the equation of motion of two-time Green’s functions (Fan et al., Phys. Rev. B 97, 165140 (2018)). In that approximation, the precision of results depends on the selection of operator basis. Here, for three successively larger operator bases, we calculate the local static averages and the impurity density of states of the single-band Anderson impurity model. The results converge systematically towards those of numerical renormalization group as the basis size is enlarged. We also propose a quantitative gauge of the truncation error within this method and demonstrate its usefulness using the Hubbard-I basis. We thus confirm that the projective truncation approximation is a method of controllable precision for quantum many-body systems.

Cavity-induced spin-orbit coupling in an interacting bosonic wire

We consider theoretically ultra-cold interacting bosonic atoms confined to a wire geometry and coupled to the field of an optical cavity. A spin-orbit coupling is induced via Raman transitions employing a cavity mode and a transverse running wave pump beam, the transition imprints a spatial dependent phase onto the atomic wavefunction. Adiabatic elimination of the cavity field leads to an effective Hamiltonian for the atomic degrees of freedom, with a self-consistency condition. We map the spin-orbit coupled bosonic wire to a bosonic ladder in a magnetic field, by discretizing the spatial dimension. Using the numerical density matrix renormalization group method, we show that in the continuum limit the dynamical stabilization of a Meissner superfluid is possible, for parameters achievable by nowadays experiments.

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.

Numerical study of the Kondo cloud using finite-U slave bosons

In this work, using the finite-U slave boson mean-field approximation to solve the single-impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ${\ensuremath{\xi}}_{K}$ through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ${\ensuremath{\xi}}_{K}^{L})$, while the second relies on the analysis of spin correlations $({\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}})$. Our calculations show that there is exact quantitative agreement, in the way ${\ensuremath{\xi}}_{K}$ depends on $U/\mathrm{\ensuremath{\Gamma}}$, between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and the results obtained through the heuristic expression ${\ensuremath{\xi}}_{K}\ensuremath{\propto}{v}_{F}/{T}_{K}$, while there is very close quantitative agreement between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and ${\ensuremath{\xi}}_{K}^{L}$. The use of the slave boson technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the density matrix renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous numerical renormalization group calculations [Phys. Rev. B 84, 115120 (2011)].

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.

Exact equilibrium results in the interacting resonant level model

We present exact results for the susceptibility of the interacting resonant level model in equilibrium. Detailed simulations using both the Numerical Renormalization Group and Density Matrix Renormalization Group were performed in order to compare with closed analytical expressions. By first bosonizing the model and then utilizing the integrability of the resulting boundary sine-Gordon model, one finds an analytic expression for the relevant energy scale $T_K$ with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level model does not correctly reproduce $T_K$ - however if the bare parameters in the model are renormalised, then quantities obtained via the direct Bethe ansatz such as the occupation of the resonant level as a function of the local chemical potential do match the numerical results. The case of one lead is studied in the most detail, with many results also extending to multiple leads, although there still remain open questions in this case.

Quantum phase transitions in the dimerized extended Bose-Hubbard model

We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges $c=1$ and $c=1/2$, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with $c=7/10$. At even stronger Coulomb interactions the transition becomes first order.

Towards verified numerical renormalization group calculations

Numerical approaches are an important tool to study strongly correlated quantum systems. However, their fragility with respect to rounding errors is not well studied and numerically verified enclosures of the results are not available. In this work we apply interval arithmetic to the well established numerical renormalization group scheme. This extension enables us to provide a numerically verified NRG excitation spectrum.