Counting statistics for the Anderson impurity model: Bethe ansatz and Fermi liquid study
Imperial College London（帝国理工学院）；Brookhaven National Laboratory（布鲁克海文国家实验室）；University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；CEA Saclay（法国原子能和替代能源委员会萨克雷中心）；University of Southern California（南加州大学）
A. O. Gogolin, R. M. Konik, A. W. W. Ludwig, H. Saleur
We study the counting statistics of charge transport in the Anderson impurity model (AIM) employing both Keldysh perturbation theory in a Fermi liquid picture and the Bethe ansatz. In the Fermi liquid approach, the object of our principal interest is the generating function for the cumulants of the charge current distribution. We derive an exact analytic formula relating the full counting statistic (FCS) generating function to the self-energy of the system in the presence of a measuring field. We first check that our approach reproduces correctly known results in simple limits, like the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM perturbatively in the Coulomb interaction. By comparing this perturbative analysis with a strong coupling expansion, we arrive at a conjecture for an expression for the FCS generating function at O(V^3) (V is the voltage across the impurity) valid at all orders in the interaction.
In the second part of the article, we examine a Bethe ansatz analysis of the current noise for the AIM. Unlike the Fermi liquid approach, here the goal is to obtain qualitative, not quantitative, results for a wider range of voltages both in and out of a magnetic field. Particularly notable are finite field results showing a double peaked structure in the current noise for voltages satisfying eV ~ mu H$. This double peaked structure is the ``smoking gun'' of Kondo physics in the current noise and is directly analogous to the single peak structure predicted for the differential conductance of the AIM.
We investigate the effect of assisted hopping in the Anderson impurity model. We use the flow equation method, which, by means of unitary transformations, generates a sequence of Hamiltonians in order to eliminate the assisted hopping terms. This approach yields a renormalized on-site energy $ε_d^*$, a renormalized correlation energy $U^*$ and other terms, which include pair hopping. For some parameter values, the initial Hamiltonian flows towards an attractive Anderson model. We argue that this result implies a tendency towards local pairing fluctuations.
Based on the finite $U$ slave boson method, we have investigated the effect of Rashba spin-orbit(SO) coupling on the persistent charge and spin currents in mesoscopic ring with an Anderson impurity. It is shown that the Kondo effect will decrease the magnitude of the persistent charge and spin currents in this side-coupled Anderson impurity case. In the presence of SO coupling, the persistent currents change drastically and oscillate with the strength of SO coupling. The SO coupling will suppress the Kondo effect and restore the abrupt jumps of the persistent currents. It is also found that a …
We consider a single impurity Anderson model (SIAM) in which the quantum dot(QD) is strongly coupled to a phonon bath in polaron regime. This coupling results in an effective e-e attraction. By computing the self energies using a current conserving approximation which is up to second order in this effective attraction, we show that if the interaction is strong enough, in non particle-hole(PH) symmetric case, the system would be bi-stable and we have hysteresis loop in the I-V characteristic. Moreover, the system shows negative differential conductance in some bias voltage intervals.
We theoretically investigate the spin resolved Friedel oscillation (FO) and quasiparticle interference (QPI) in graphene induced by an Anderson impurity. Once the impurity becomes magnetic, the resulted FO becomes spin dependent, which gives rise to a local magnetic moment oscillation with an envelop decaying as $r^{-2}$ in real space in the doping cases. Meanwhile, at half filling, the charge density and local magnetic moment will not oscillate but decay as $r^{-3}$. Such spin resolved FO has both sublattice and spin asymmetry. Interestingly, the local magnetic moment decay at half filling on…
We investigate the mean-field phase diagram of the Anderson impurity model out of equilibrium. Generalising the unrestricted Hartree-Fock approach to the non-equilibrium situation we derive and analyse the system of equations defining the critical surface separating the magnetic regime from the non-magnetic one. An exact analytic solution for the phase boundary as a function of the applied voltage is found in the symmetric case. Surprisingly, we find that as soon as there is an asymmetry, even small, between the contacts, no finite voltage is able to destroy the magnetic regime which persists …
An approximate groundstate of the Anderson-Friedel impurity problem is presented in a very compact form. It requires solely the optimization of two localized electron states and consists of four Slater states (Slater determinants). The resulting singlet ground state energy lies far below the Anderson mean field solution and agrees well with the numerical results by Gunnarsson and Schoenhammer, who used an extensive 1/N_{f}-expansion for a spin 1/2 impurity with double occupancy of the impurity level. PACS: 85.20.Hr, 72.15.Rn
We study spin-entanglement of the quasiparticles of the local Fermi liquid excited in nonlinear current through a quantum dot described by the Anderson impurity model with two degenerate orbitals coupled to each other via an exchange interaction. Applying the renormalized perturbation theory, we obtain the precise form of the cumulant generating function and cross-correlations for the currents with spin angled to arbitrary directions, up to third order in the applied bias voltage. It is found that the exchange interaction gives rise to spin-angle dependency in the cross-correlation between the…
We discuss the transient effects in the Anderson impurity model that occur when two fermionic continua with finite bandwidths are instantaneously coupled to a central level. We present results for the analytically solvable noninteracting resonant level system first and then consistently extend them to the interacting case using the conventional perturbation theory and recently developed nonequilibrium Monte Carlo simulation schemes. The main goal is to gain an understanding of the full time-dependent nonlinear current-voltage characteristics and the population probability of the central level.…
We analyze the consequences of a first-order (in the effective Kondo coupling) inter-impurity correlation present in the Two-Impurity Anderson Hamiltonian. This correlation generates a pair of Kondo Doublet states which are responsible for the first (higher energy) Kondo stage of these systems. Applied to systems of quantum dots in a semiconductor heterolayer, these Doublets predict a mainly ferromagnetic dot-dot spin correlation which is different than the expected from the second-order RKKY interaction. In order to effectively screen both dots, a parallel alignment of the dot-spins is favore…
We study the nonequilibrium spectral function of the single-impurity Anderson model connecting with multi-terminal leads. The full dependence on frequency and bias voltage of the nonequilibrium self-energy and spectral function is obtained analytically up to the second-order perturbation regarding the interaction strength $U$. High and low bias voltage properties are analyzed for a generic multi-terminal dot, showing a crossover from the Kondo resonance to the Coulomb peaks with increasing bias voltage. For a dot where the particle-hole symmetry is not present, we construct a current-preservin…
The time dependent full counting statistics of charge transport through an interacting quantum junction is evaluated from its generating function, controllably computed with the inchworm Monte Carlo method. Exact noninteracting results are reproduced; then, we continue to explore the effect of electron--electron interactions on the time-dependent charge cumulants, first-passage time distributions and $n$-electron transfer distributions. We observe a crossover in the noise from Coulomb blockade- to Kondo-dominated physics as the temperature is decreased. In addition, we uncover long-tailed spin…
The strongly correlated two-electron transport in one-dimensional channel coupled with an Anderson-type impurity is solved exactly via a Bethe ansatz approach. We show that the transport properties are fundamentally different for spin singlet and triplet states, thus the impurity acts as a novel filter that operates based on the total spin angular momentum of the electron pairs, but not individual spins. The filter provides a deterministic generation of electron entanglement in spin, as well as energy and momentum space.
In a recent Comment, Kolf et al. (cond-mat/0503669) state that our analysis of the Fano resonance for Anderson impurity systems [Luo et al., Phys. Rev. Lett 92, 256602 (2004)] is incorrect. Here we want to point out that their comments are not based on firm physical results and their criticisms are unjustified and invalid.
Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models (see A. Weichselbaum and J. von Delft) by using the completeness of the eliminated states introduced by F. B. Anders and A. Schiller (2005). While these developments prove particularly useful in the calculation of transient response and finite temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to t…
We develop an exact non-perturbative framework to compute steady-state properties of quantum-impurities subject to a finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates which satisfy an appropriate Lippman-Schwinger equation. Introducing a generalization of the equilibrium Bethe-Ansatz - the Nonequilibrium Bethe-Ansatz (NEBA), we explicitly construct the scattering eigenstates for the Interacting Resonance Level model and derive exact, nonperturbative results for the steady-state properties of the system.
We compute exactly the low-energy single-electron Green's function, the impurity and electron self-energies, and the resistivity for the two-channel Anderson impurity model. These results are obtained by exploiting the boundary conformal field theory identified from the Bethe Ansatz solution of the model. Using that solution we can make contact with the parameters of the original Hamiltonian and provide the detailed crossover between the two integer valence limits. Our results generalize those obtained previously in the context of the two-channel Kondo model.
Transient and steady state currents through dc-biased quantum impurity models beyond the linear response regime are of considerable interest, both from an experimental and a theoretical point of view. Here we present a new analytical approach for the calculation of such currents based on the flow equation method (method of infinitesimal unitary transformations). Specifically, we analyze the Anderson impurity model in its mixed valence regime where the coupling to the leads is switched on suddenly at time $t=0$. We observe the real time buildup of the current until it reaches its steady state l…
We reobtain the Kondo-Doublet interaction by means of the Brillouin-Wigner Perturbation theory. By applying the same method to the single impurity case we show that the Kondo-Doublet interaction is a direct consequence of the Kondo screening cloud. We fully confirm our previous results on this novel interaction. In particular, we found that for values of the system parameters typical of semiconductor Quantum Dots heterostructures, the ferromagnetic Kondo-Doublet interaction dominates over the RKKY one.
We study a series-coupled double quantum dot in the Kondo regime modeled by the two-impurity Anderson model and find a new conduction-band mediated superexchange interaction that competes with Kondo physics in the strong Coulomb interaction limit. Our numerical renormalization group results, complemented with the higher-order Rayleigh-Schrödinger perturbation theory, show that the novel exchange mechanism leads to clear experimental consequences that can be checked in transport measurements through double quantum dots.
We study the Anderson model in which a configuration with a doublet is hybridized with another with a singlet and a triplet. We calculate the conductance through the system as a function of temperature and bias voltage, near the quantum critical line for which the system is exactly solvable. The results explain recent transport measurements in a single-molecule quantum dot.
We derive the transport properties of a quantum dot subject to a source-drain bias voltage at zero temperature and magnetic field. Using the Scattering Bethe Anstaz, a generalization of the traditional Thermodynamic Bethe Ansatz to open systems out of equilibrium, we derive exact results for the quantum dot occupation out of equilibrium and, by introducing phenomenological spin- and charge-fluctuation distribution functions in the computation of the current, obtain the differential conductance for large U/Γ. The Hamiltonian to describe the quantum dot system is the Anderson impurity Hamiltonia…
In this work, we investigate the characteristics of the electric current in the so-called symmetric Anderson impurity model. We study the nonequilibrium model using two complementary approximate methods, the perturbative quantum master equation approach to the reduced density matrix, and a self-consistent equation of motion approach to the nonequilibrium Green's function. We find that at a particular symmetry point, an interacting Anderson impurity model recovers the same steady-state current as an equivalent non-interacting model, akin a two-band resonant level model. We show this in the Coul…
The magnetic moment of a single impurity atom in a finite free electron gas is studied in a combined x-ray magnetic circular dichroism spectroscopy and density functional theory study of size-selected free chromium-doped gold clusters. The observed size-dependence of the local magnetic moment can essentially be understood in terms of the Anderson impurity model. Electronic shell closure in the host metal minimizes the interaction of localized impurity states with the confined free electron gas and preserves the full magnetic moment of $\unit[5]{μ_B}$ in $\mathrm{CrAu}_{2}^{+}$ and $\mathrm{CrA…
We study the interacting, symmetrically coupled single impurity Anderson model. By employing the nonequilibrium Green's function formalism, we establish an exact relationship between the steady-state charge current flowing through the impurity (dot) and its occupation. We argue that the steady state current-occupation relation can be used to assess the consistency of simulation techniques, and identify spurious transport phenomena. We test this relation in two different model variants: First, we study the Anderson-Holstein model in the strong electron-vibration coupling limit using the polaron…
Quantum transport is studied for the nonequilibrium Anderson impurity model at zero temperature employing the multilayer multiconfiguration time-dependent Hartree theory within the second quantization representation (ML-MCTDH-SQR) of Fock space. To adress both linear and nonlinear conductance in the Kondo regime, two new techniques of the ML-MCTDH-SQR simulation methodology are introduced: (i) the use of correlated initial states, which is achieved by imaginary time propagation of the overall Hamiltonian at zero voltage and (ii) the adoption of the logarithmic discretization of the electronic …
It is shown that the calculation of the magnetic moment of a Friedel-Anderson impurity in mean-field theory is unreliable. A class of approximate solutions, which contains the mean-field solution as an element, is expressed in rotated Hilbert space and optimized. The optimal state has considerably lower energy than the mean field solution and requires almost twice the Coulomb exchange U to become magnetic. Since most moment calculations of magnetic impurities, for example the spin-density-functional theory, use the mean-field approximation the resulting magnetic moments have to be critically r…
We analyze the spectrum of the electric-current autocorrelation function (noise power) in the Anderson impurity model biased by a finite transport voltage. Special emphasis is placed on the interplay of non-equilibrium effects and electron-electron interactions. Analytic results are presented for a perturbation expansion in the interaction strength $U$. Compared to the non-interacting setup we find a suppression of noise for finite frequencies in equilibrium and an amplification in non-equilibrium. Furthermore, we use a diagrammatic resummation scheme to obtain non-perturbative results in the …
We present an extension of the non-crossing approximation (NCA), which is widely used to calculate properties of Anderson impurity models in the limit of infinite Coulomb repulsion $U\to\infty$, to the case of finite $U$. A self-consistent conserving pseudo-particle representation is derived by symmetrizing the usual NCA diagrams with respect to empty and doubly occupied local states. This requires an infinite summation of skeleton diagrams in the generating functional thus defining the ``Symmetrized finite-U NCA'' (SUNCA). We show that within SUNCA the low energy scale $T_K$ (Kondo temperatur…
We propose a realization of the two-impurity Anderson model in a double quantum-dot device. When charge transfer between the dots is suppressed the system exhibits a quantum phase transition, controlled by a surface of non-Fermi liquid fixed points parameterized by the charge valences of the dots. Employing conformal field theory techniques, we identify the scaling exponents that govern transport and thermodynamics close to criticality. We also determine the dynamical exponents that set the time scale for buildup of the non-Fermi liquid state after the system is suddenly shifted into the criti…
The Distributional Exact Diagonalization (DED) scheme is applied to the description of Kondo physics in the Anderson impurity model. DED maps Anderson's problem of an interacting impurity level coupled to an infinite bath onto an ensemble of finite Anderson models, each of which can be solved by exact diagonalization. An approximation to the self-energy of the original infinite model is then obtained from the ensemble averaged self-energy. Using Friedel's sum rule, we show that the particle number constraint, a central ingredient of the DED scheme, ultimately imposes Fermi liquid behavior on t…
We investigate the dynamical and steady-state spin response of the nonequilibrium Anderson model to magnetic fields, bias voltage, and temperature using a numerically exact method combining a bold-line quantum Monte Carlo technique with the memory function formalism. We obtain converged results in a range of previously inaccessible regimes, in particular the crossover to the Kondo domain. We provide detailed predictions for novel nonequilibrium phenomena, including non-monotonic temperature dependence of observables at high bias voltage and oscillatory quench dynamics at high magnetic fields.
We present a microscopic Fermi-liquid view on the low-energy transport through an Anderson impurity with $N$ discrete levels, at arbitrary electron filling $N_d$. It is applied to nonequilibrium current fluctuations, for which the two-quasiparticle collision integral and the three-body correlations that determine the quasiparticle energy shift play important roles. Using the numerical renormalization group up to $N=6$, we find that for strong interactions the three-body fluctuations are determined by a single parameter other than the Kondo energy scale in a wide filling range $1 \lesssim N_d \…
The bosonic single-impurity Anderson model (B-SIAM) is studied to understand the local dynamics of an atomic quantum dot (AQD) coupled to a Bose-Einstein condensation (BEC) state, which can be implemented to probe the entanglement and the decoherence of a macroscopic condensate. Our recent approach of the numerical renormalization group (NRG) calculation for the B-SIAM revealed a zero-temperature phase diagram, where a Mott phase with local depletion of normal particles is separated from a BEC phase with enhanced density of the condensate. As an extension of the previous work, we present the c…
We show how the density-matrix numerical renormalization group (DM-NRG) method can be used in combination with non-Abelian symmetries such as SU(N), where the decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.
An exact relation between the conductance maximum $G_0$ at zero temperature and a ratio of lead densities is derived within the framework of the single impurity Anderson model: $G_0={\mathfrak R}[n] \frac{2e^2}{h}$, where ${\mathfrak R}[n]=4ΔN_{{\cal L},x} ΔN_{{\cal R},x}/(ΔN_{{\cal L},x}+ΔN_{{\cal R},x})^2$ and $ΔN_{{\cal L},x}$, $ΔN_{{\cal R},x}$ denote the excess density in the left/right lead at distance $x$ due to the presence of the impurity at the origin, $x=0$. The relation constitutes a parameter-free expression of the conductance of the model in terms of the ground state density that…
We investigate a modified Anderson model at the large-$N$ limit, where Coulomb interaction is replaced by Sachdev-Ye-Kitaev random interaction. The resistivity of conduction electron $ρ_{c}$ has a minimum value around temperature $\widetilde{T}_{K}$, which is similar to the Kondo system, but the impurity electron's density of state $A_{d}(ω)$ elucidates no sharp-peak like Kondo resonance around the Fermi surface. The impurity electron's entropy $S_{d}$ and specific heat capacity $C_{\textrm{v}}$ illustrate a crossover from Fermi liquid to the non-Fermi liquid. The system is a non-Fermi liquid …
We have developed a self-consistent conserving pseudo particle approximation for the Anderson impurity model with finite Coulomb interaction, derivable from a Luttinger Ward functional. It contains an infinite series of skeleton diagrams built out of fully renormalized Green's functions. The choice of diagrams is motivated by the Schrieffer Wolff transformation which shows that singly and doubly occupied states should appear in all bare diagrams symmetrically. Our numerical results for $T_K$ are in excellent agreement with the exact values known from the Bethe ansatz solution. The low energy p…
We study the single impurity Anderson model by means of cluster perturbation theory and the variational cluster approach (VCA). An expression for the VCA grand potential for a system in a non interacting bath is presented. Results for the single-particle dynamics in different parameter regimes are shown to be in good agreement with established renormalization group results. We aim at a broad and comprehensive overview of the capabilities and shortcomings of the methods. We address the question to what extent the elusive low energy properties of the model are reproducible within the framework o…
Although the Kondo effect and the Kondo ground state of a magnetic impurity have been investigated for more than forty years it was until recently difficult if not impossible to calculate spatial properties of the ground state. In particular the calculation of the spatial distribution of the so-called Kondo cloud or even its existence have been elusive. In recent years a new method has been introduced to investigate the properties of magnetic impurities, the FAIR method, where the abbreviation stands for Friedel Artificially Inserted Resonance. The FAIR solution of the Friedel-Anderson and the…
An Anderson impurity in a Hubbard model on chains with finite length is studied using the density-matrix renormalization group (DMRG) technique. In the first place, we analyzed how the reduction of electron density from half-filling to quarter-filling affects the Kondo resonance in the limit of Hubbard repulsion U=0. In general, a weak dependence with the electron density was found for the local density of states (LDOS) at the impurity except when the impurity, at half-filling, is close to a mixed valence regime. Next, in the central part of this paper, we studied the effects of finite Hubbard…
The M-channel Anderson impurity model (M=1,2) is studied in the Kondo limit with a finite voltage bias applied to the conduction electron reservoirs. Using the Non-Crossing Approximation (NCA), we calculate the local spectral functions, the differential conductance, and susceptibility at non-zero bias for symmetric as well as asymmetric coupling of the impurity to the leads. We describe an effective procedure to solve the NCA integral equations which enables us to reach temperatures far below the Kondo scale. This allows us to study the scaling regime where the conductance depends on the bias …
We analyze the effects of high energy configurations on the Kondo-doublet interaction between two Anderson impurities. We found that the Kondo doublet states are robust and that their coherence energy is incremented by the inclusion of high energy configurations. Analytic expressions are obtained for the corrections near the Kondo limit. Analysis of the system in the intermediate valence regime shows that the behavior of the system can be changed from ferromagnetic to sligthy antiferromagnetic by tuning the system parameters; this regime can also be used to study the interplay between hole-dri…
We apply the recently proposed quasi-classical approach for a second quantized many-electron Hamiltonian in Cartesian coordinates [J. Chem. Phys. 137, 154107 (2012)] to correlated nonequilibrium quantum transport. The approach provides accurate results for the resonant level model for a wide range of temperatures, bias and gate voltages, correcting for the flaws of our recently proposed mapping using action-angle variables. When electron-electron interactions are included, higher order schemes are required to map the two-electron integrals, leading to semi-quantitative results for the Anderson…
The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a co…
We study the single impurity Anderson model (SIAM) using the equations of motion method (EOM), the non-crossing approximation (NCA), the one-crossing approximation (OCA), and Wilson's numerical renormalization group (NRG). We calculate the density of states and the linear conductance focusing on their dependence on the chemical potential and on the temperature paying special attention to the Kondo and Coulomb blockade regimes for a large range of model parameters. We report that some standard approximations based on the EOM technique display a rather unexpected poor behavior in the Coulomb blo…
We derive and apply a general scheme for mapping a setup consisting of a half-filled single level quantum dot coupled to one normal metallic and two superconducting phase-biased leads onto an ordinary half-filled single impurity Anderson model with single modified tunneling density of states. The theory allows for the otherwise unfeasible application of the standard numerical renormalization group and enables to obtain phase-dependent local spectral properties as well as phase-dependent induced pairing and Josephson current. The resulting transport properties match well with the numerically ex…
We study a pseudogap Anderson-Holstein model of a magnetic impurity level that hybridizes with a conduction band whose density of states vanishes in power-law fashion at the Fermi energy, and couples, via its charge, to a nondispersive bosonic mode (e.g., an optical phonon). The model, which we treat using poor-man's scaling and the numerical renormalization group, exhibits quantum phase transitions of different types depending on the strength of the impurity-boson coupling. For weak impurity-boson coupling, the suppression of the density of states near the Fermi energy leads to quantum phase …
We analyze the spectral function of the single-impurity two-terminal Anderson model at finite voltage using the recently developed diagrammatic quantum Monte Carlo technique as well as perturbation theory. In the (particle-hole-)symmetric case we find an excellent agreement of the numerical data with the perturbative results of second order up to interaction strengths $U/Γ\approx 2$, where $Γ$ is the transparency of the impurity-electrode interface. The analytical results are obtained in form of the nonequilibrium self-energy for which we present explicit formulas in the closed form at arbitra…
We analyze the full counting statistics of a multiorbital Kondo effect in a quantum dotwith the SU(N) symmetry in the framework of the renormalized perturbation theory. The current probability distribution function is calculated for an arbitrary dot-site Coulomb repulsion $U$ in the particle-hole symmetric case. The resulting cumulant up to the leading nonlinear term of applied bias voltages indicates two types of electron transfer, respectively carrying charge $e$ and $2e$, with different $N$-dependences. The cross correlation between different orbital currents shows exponential enhancement w…
