Anderson Impurity Model Research Articles

A molecular shift register made using tunable charge patterns in one-dimensional molecular arrays on graphene

The ability to tune the electronic properties of molecular arrays is an important step in the development of molecule-scale electronic devices. However, control over internal device charge distributions by tuning interactions between molecules has proved challenging. Here, we show that gate-tunable charge patterning can occur in one-dimensional molecular arrays on graphene field-effect transistors. One-dimensional molecular arrays are fabricated using an edge-templated self-assembly process that allows organic molecules (F4TCNQ) to be precisely positioned on graphene devices. The charge configurations of the molecular arrays can be reversibly switched between different collective charge states by tuning the graphene Fermi level via a back-gate electrode. Charge pinning at the ends of the molecular arrays allows the charge state of the entire array to be controlled by adding or removing an edge molecule and changing the total number of molecules in an array between odd and even integers. Charge patterns altered in this way propagate down the array in a cascade effect, allowing the array to function as a charge-based molecular shift register. An extended multi-site Anderson impurity model is used to quantitatively explain this behaviour. One-dimensional molecular arrays on graphene field-effect transistors can be reversibly switched between different periodic charge states by tuning the graphene Fermi level via a back-gate electrode and by manipulating individual molecules, allowing them to function as a nanoscale shift register.

Collision of protons with carbon atoms of a graphene surface in the presence of adsorbed potassium

In this work we study the frontal collision of protons with the carbon atoms of a graphene surface with a low coverage of adsorbed potassium. It is aimed at the analysis of the effect of the adsorbates in both charge exchange and electron emission processes, when the binary collision occurs between the proton and a carbon atom of the surface. The frontal collision with the K adsorbate, already analyzed and discussed in a previous work, is compared with the frontal collision with different carbon neighbors. In the present work we studied the signals, due to the localized structures in the density matrix of the composed graphene plus potassium surface, that can be distinguished when the collision occurs either with the adsorbate, a nearby carbon atom, or a carbon atom that does not feel the presence of the adsorbate. The interacting system is described by the Anderson Hamiltonian which takes into account the electronic repulsion on the projectile site; the charge fractions, the energy distribution of electrons in the solid, and the electron emission after the collision are calculated by using the nonequilibrium Green-Keldysh functions formalism solved by the equation of motion method. In the binary collision with a carbon atom, the extended features of the band structure of graphene smooth the dependence of the projectile charge fractions on the incoming energy and notably decrease the negative ions formation. The localized structures of the density of matrix caused by the presence of the adsorbate are perceptible for scattered carbon atoms close to K. The intense emission of low energy electrons obtained in the case of the scattering by potassium is fundamentally associated with the very localized K-$4s$ empty band. This characteristic, although less marked, remain in the scattering by nearby carbon atoms, due to both the interaction with K along the projectile trajectory and the perturbed local density of states on the carbon atoms due to the adsorbate presence. In addition, the extended nature of the electronic structure of graphene allows for the emission of more energetic electrons.

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.

Magnetic anisotropy of a Dy atom on a graphene/Cu(111) surface

The electronic structure and magnetism of individual Dy atoms adsorbed on the graphene/Cu(111) surface is investigated using the combination of the density functional theory with the Hubbard-I approximation to the Anderson impurity model ($\mathrm{DFT}+\mathrm{U}+\mathrm{HIA}$). We find that the results of the $\mathrm{DFT}+\mathrm{U}+\mathrm{HIA}$ depend on the choice of the double-counting term. For fully localized limit, the divalent ${\mathrm{Dy}}^{2+}$ adatom is found, with the total magnetic moment of $9.71\phantom{\rule{4pt}{0ex}}{\ensuremath{\mu}}_{B}$. The spin and orbital magnetic moments are evaluated, and compared with the x-ray magnetic circular dichroism data. The calculated positive magnetic anisotropy energy determines the out-of-plane orientation of the Dy adatom magnetic moment, in agreement with available experimental data.

High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation

Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet-equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and non-local static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.

Kondo effect under the influence of spin–orbit coupling in a quantum wire

The analysis of the impact of spin–orbit coupling (SOC) on the Kondo state has generated considerable controversy, mainly regarding the dependence of the Kondo temperature TK on SOC strength. Here, we study the one-dimensional (1D) single impurity Anderson model (SIAM) subjected to Rashba (α) and Dresselhaus (β) SOC. It is shown that, due to time-reversal symmetry, the hybridization function between impurity and quantum wire is diagonal and spin independent (as it is the case for the zero-SOC SIAM), thus the finite-SOC SIAM has a Kondo ground state similar to that for the zero-SOC SIAM. This similarity allows the use of the Haldane expression for TK, with parameters renormalized by SOC, which are calculated through a physically motivated change of basis. Analytic results for the parameters of the SOC-renormalized Haldane expression are obtained, facilitating the analysis of the SOC effect over TK. It is found that SOC acting in the quantum wire exponentially decreases TK while SOC at the impurity exponentially increases it. These analytical results are fully supported by calculations using the numerical renormalization group (NRG), applied to the wide-band regime, and the projector operator approach, applied to the infinite-U regime. Literature results, using quantum Monte Carlo, for a system with Fermi energy near the bottom of the band, are qualitatively reproduced, using NRG. In addition, it is shown that the 1D SOC SIAM for arbitrary α and β displays a persistent spin helix SU(2) symmetry similar to the one for a 2D Fermi sea with the restriction α = β.

Coherent transport through a resonant level coupled to random-matrix leads

We study the transport through a resonant level coupled to two leads with the latter being described by Wigner’s random matrices. By taking appropriate thermodynamic limit before taking the long time limit, we obtain the stationary current as a function of voltage bias. The I–V curve is similar to that of single impurity Anderson model. On the other hand, the current matrix and initial density matrix in our model look like random matrices in the eigenbasis of Hamiltonian. They satisfy the description of eigenstate thermalization hypothesis (ETH) and nonequilibrium steady state hypothesis (NESSH), respectively. A statistical formula of current has been derived based on ETH and NESSH (Wang 2017 J. Stat. Mech. 093105). We check this formula in our model and find it to predict the stationary current to a high precision. The shape of I–V curve is explained by the peak structure in the characteristic function of NESSH, which is reminiscent of the transmission coefficient.

Machine learning effective models for quantum systems

The construction of good effective models is an essential part of understanding and simulating complex systems in many areas of science. It is a particular challenge for correlated many body quantum systems displaying emergent physics. We propose a machine learning approach that optimizes an effective model based on an estimation of its partition function. The success of the method is demonstrated by application to the single impurity Anderson model and double quantum dots, where non-perturbative results are obtained for the old problem of mapping to effective Kondo models. For quantum impurity parent Hamiltonians, we derive an alternative approach based on learning from observables. When mapping to minimal effective models, emergent scales may not be captured by observable learning, while partition function learning may not reproduce all observables.

Stochastic equation of motion approach to fermionic dissipative dynamics. II. Numerical implementation

This paper provides a detailed account of the numerical implementation of the stochastic equation of motion (SEOM) method for the dissipative dynamics of fermionic open quantum systems. To enable direct stochastic calculations, a minimal auxiliary space (MAS) mapping scheme is adopted, with which the time-dependent Grassmann fields are represented by c-number noises and a set of pseudo-operators. We elaborate on the construction of the system operators and pseudo-operators involved in the MAS-SEOM, along with the analytic expression for the particle current. The MAS-SEOM is applied to study the relaxation and voltage-driven dynamics of quantum impurity systems described by the single-level Anderson impurity model, and the numerical results are benchmarked against those of the highly accurate hierarchical equations of motion method. The advantages and limitations of the present MAS-SEOM approach are discussed extensively.

Universal approach to quantum thermodynamics of strongly coupled systems under nonequilibrium conditions and external driving

We present an approach based on a density matrix expansion to study thermodynamic properties of a quantum system strongly coupled to two or more baths. For slow external driving of the system, we identify the adiabatic and nonadiabatic contributions to thermodynamic quantities, and we show how the first and second laws of thermodynamics are manifested in the strong coupling regime. Particularly, we show that the entropy production is positive up to second order in the driving speed. The formulation can be applied both for Bosonic and Fermionic systems, and recovers previous results for the equilibrium case (Phys. Rev. B 98, 134306 [2018]). The approach is then demonstrated for the driven resonant level model as well as the driven Anderson impurity model, where the hierarchical quantum master equation method is used to accurately simulate the nonequilibrium quantum dynamics.

Theoretical Study on Four-fold Symmetric Anisotropic Magnetoresistance Effect in Cubic Single-crystal Ferromagnetic Model

In this study, we present a theoretical interpretation of the experimental results that the anisotropic magnetoresistance (AMR) effect has a four-fold symmetric component, $c_4$, in cubic ferromagnetic metals. The theoretical model that we employ is based on the Anderson impurity model that includes a four-fold symmetric crystalline electric field, and we assume that the impurities have 3d electron orbitals and spin--orbit interaction (SOI). We describe the DC conductivity on the basis of the Kubo formula, and we investigate $c_4$ by analyzing the magnetization direction dependence of the resultant AMR ratio. Analytical and numerical calculations are performed; the analytical calculation reveals that $c_4$ arises from the fourth-order contribution of the SOI, and the numerical calculation provides the parameter dependencies of $c_{4}$ in our model. From the calculation results, we observe that the splitting of impurity 3d levels due to SOI is responsible for the existence of $c_{4}$ in cubic ferromagnetic metals.

Reentrant Kondo effect for a quantum impurity coupled to a metal-semiconductor hybrid contact

Using NRG, we show that a system containing a quantum impurity (QI), strongly coupled to a semiconductor (gap $2 \Delta$) and weakly coupled to a metal, displays a 'reentrant' Kondo stage at low temperatures. The NRG analysis of the corresponding Single Impurity Anderson Model (SIAM) shows that the reentrant stage is characterized by a second sequence of SIAM fixed points: free orbital (FO) > local moment (LM) > strong coupling (SC). In the first stage, the SC fixed point (Kondo temperature $T_{K1}$) is unstable, while the second stage exhibits a much lower Kondo temperature $T_{K2}$, associated to a stable SC fixed point. The results indicate that the reentrant Kondo screening is associated to an effective SIAM, with an effective repulsion $U_{eff}$. This low temperature effective SIAM, which we dub as 'reentrant' SIAM, behaves as a 'replica' of the high temperature (bare) SIAM. The intuitive picture that emerges is that the first Kondo state develops through impurity screening by semiconducting electrons, while the second Kondo state involves screening by metallic electrons, once the semiconducting electrons are out of reach to thermal excitations ($T < \Delta$) and only the metallic spectral weight inside the gap is available for impurity screening. In addition, we analyze a hybrid system formed by a QI `sandwiched' between an armchair graphene nanoribbon (AGNR) and a scanning tunneling microscope (STM) tip, with respective couplings set to reproduce the generic model described above. The energy gap in the AGNR can be externally tuned by an electric-field-induced Rashba spin-orbit interaction. We analyzed this system for realistic parameter values, using NRG, and concluded that the reentrant SIAM, with its associated second stage Kondo, is worthy of experimental investigation.

LDA+DMFTapproach to resonant inelastic x-ray scattering in correlated materials

We present a computational study of $L$-edge resonant inelastic x-ray scattering (RIXS) in correlated 3$d$ transition-metal oxides using an $ab$ $initio$ method based on local density approximation + dynamical mean-field theory (DMFT). The present method, building on Anderson impurity model with an optimized continuum bath within DMFT, is an extension of the cluster model to include unbound electron-hole pair excitations as well as material-specific charge-transfer excitations with less empirical parameters. We find a good agreement with available experimental data. The relationship between correlated bands and fluorescence-like feature in the RIXS spectra is discussed.

Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy

We investigate several definitions of the time-dependent spectral function $A(\omega,t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group method. In terms of the two-time retarded Green function $G^r(t_1,t_2)$, the definitions differ in the choice of the time variable $t$ with respect to $t_1$ and/or $t_2$. In a previous study [Nghiem {\it et al.} Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function, obtained from the Fourier transform of ${\rm Im}[G^r(t_1,t_2)]$ w.r.t. the time difference $t'=t_1-t_2$, with $t=t_2$. Here, we derivie expressions for the retarded Green function for the choices $t=t_1$ and the average time $t=(t_1+t_2)/2$, within the TDNRG approach. We compare and contrast the resulting $A(\omega,t)$ for the different choices of time reference. Expressions for the lesser, greater and advanced Green functions are also derived within TDNRG for all choices of time reference. The average time lesser Green function $G^<(\omega,t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(\omega,t)=G^<(\omega,t)/(2\pi i)$, a quantity that determines the photoemission current in time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(\omega,t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy peaks. We also discuss the issue of thermalization at long times for $N(\omega,t)$. Finally, we use the results for $N(\omega,t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.

Configuration interaction approaches for solving quantum impurity models.

We develop several configuration interaction approaches for characterizing the electronic structure of an adsorbate on a metal surface (at least in model form). When one can separate the adsorbate from the substrate, these methods can achieve a reasonable description of adsorbate on-site electron-electron correlation in the presence of a continuum of states. While the present paper is restricted to the Anderson impurity model, there is hope that these methods can be extended to ab initio Hamiltonians and provide insight into the structure and dynamics of molecule-metal surface interactions.

Hierarchical equations of motion method based on Fano spectrum decomposition for low temperature environments.

The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.

Time dependent variational principle for tree Tensor Networks

We present a generalization of the Time Dependent Variational Principle (TDVP) to any finite sized loop-free tensor network. The major advantage of TDVP is that it can be employed as long as a representation of the Hamiltonian in the same tensor network structure that encodes the state is available. Often, such a representation can be found also for long-range terms in the Hamiltonian. As an application we use TDVP for the Fork Tensor Product States tensor network for multi-orbital Anderson impurity models. We demonstrate that TDVP allows to account for off-diagonal hybridizations in the bath which are relevant when spin-orbit coupling effects are important, or when distortions of the crystal lattice are present.

Generalized quadrature for finite temperature Green’s function methods

In electronic structure and quantum transport calculations, many physical quantities are integrations over the electron energy, weighted by the Fermi–Dirac distribution function. Green’s function based approaches commonly circumvent the numerically difficult real energy integration by extending the integrand analytically into the complex energy plane, and using a Gaussian quadrature integration over a complex energy contour for zero temperature. For finite temperatures, a much slower convergent sum over the Matsubara frequencies is necessary. We present a generalized quadrature method that uses orthogonal polynomials on the manifold of Matsubara frequencies to enable rapid convergence. Both Gaussian quadrature integration and Matsubara frequency summation methods are shown to be limiting cases of the generalized method. Tests on an all-electron ab initio code and total energy calculation of interacting Anderson impurity model show convergence with a small number of energy mesh points.

Nonequilibrium Kondo effect in a graphene-coupled quantum dot in the presence of a magnetic field.

Background: Quantum dots connected to larger systems containing a continuum of states like charge reservoirs allow the theoretical study of many-body effects such as the Coulomb blockade and the Kondo effect.Results: Here, we analyze the nonequilibrium Kondo effect and transport phenomena in a quantum dot coupled to pure monolayer graphene electrodes under external magnetic fields for finite on-site Coulomb interaction. The system is described by the pseudogap Anderson Hamiltonian. We use the equation of motion technique to determine the retarded Green’s function of the quantum dot. An analytical formula for the Kondo temperature is derived for electron and hole doping of the graphene leads. The Kondo temperature vanishes in the vicinity of the particle–hole symmetry point and at the Dirac point. In the case of particle–hole asymmetry, the Kondo temperature has a finite value even at the Dirac point. The influence of the on-site Coulomb interaction and the magnetic field on the transport properties of the system shows a tendency similar to the previous results obtained for quantum dots connected to metallic electrodes. Most remarkably, we find that the Kondo resonance does not show up in the density of states and in the differential conductance for zero chemical potential due to the linear energy dispersion of graphene. An analytical method to calculate self-energies is also developed which can be useful in the study of graphene-based systems.Conclusion: Our graphene-based quantum dot system provides a platform for potential applications of nanoelectronics. Furthermore, we also propose an experimental setup for performing measurements in order to verify our model.

Magnetic States of the Zigzag Edge of a Graphene Nanoribbon

Using a simple structural model and the multicenter Anderson Hamiltonian, Green’s functions are obtained for the atoms of the zigzag edge of an epitaxial graphene nanoribbon. The electronic structure of the free nanoribbon is discussed in detail. Specifically, expressions for the band spectrum and density of states are found and estimates of the occupation numbers and magnetic moments are given. For a nanoribbon strongly bonded to a metal substrate, the criteria for the appearance of magnetic moments are determined. As it is shown for both free and epitaxial nanoribbons, the probability of the appearance of magnetic moments and their magnitude for zigzag edge atoms that have two nearest neighbors is higher than for atoms with three nearest neighbors.