Symmetric Anderson Impurity Model Research Articles

Improved strong-coupling perturbation theory of the symmetric Anderson impurity model

In a previous work [N. H. Tong, Phys. Rev. B 92 (2015) 165126], an equation-of-motion-based series expansion formalism was used to do the second-order strong-coupling expansion for the single-particle Green function of the Anderson impurity model (AIM). In this paper, we improve this theory in two aspects. We first use a more accurate scheme to self-consistently calculate the averages that appear in [Formula: see text]. In the resummation process, we use updated coefficients for the continued fraction (CF), guided by the formally exact CF from the Mori–Zwanzig theory. These changes lead to more accurate impurity spin responses to the magnetic bias of the bath. Combined with the dynamical mean-field theory, our theory gives improved description for the antiferromagnetism of Hubbard model at half-filling.

Absence of Coulomb Blockade in the Anderson Impurity Model at the Symmetric Point

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 Coulomb blockade regime for both high and low temperatures, where either the approximate master equation approach and the Green's function method provide accurate results for the current. We conclude that the steady-state current in the symmetric Anderson model at this regime does not encode characteristics of a many-body interacting system.

Symmetric Anderson impurity model: Magnetic susceptibility, specific heat and Wilson ratio

We extend the spin-polarized effective-interaction approximation of the parquet renormalization scheme from Refs. [1,2] applied on the symmetric Anderson model by adding the low-temperature asymptotics of the total energy and the specific heat. We calculate numerically the Wilson ratio and determine analytically its asymptotic value in the strong-coupling limit. We demonstrate in this way that the exponentially small Kondo scale from the strong-coupling regime emerges in qualitatively the same way in the spectral function, magnetic susceptibility and the specific heat.

Renormalized perturbation theory flow equations for the Anderson impurity model

We apply the renormalized perturbation theory (RPT) to the symmetric Anderson impurity model. Within the RPT framework exact results for physical observables such as the spin and charge susceptibility can be obtained in terms of the renormalized values $\tilde{\boldsymbol\mu} = (\tilde{\Delta}, \tilde{U})$ of the hybridization $\Delta$ and Coulomb interaction $U$ of the model. The main difficulty in the RPT approach usually lies in the calculation of the renormalized values themselves. In the present work we show how this can be accomplished by deriving differential flow equations describing the evolution of $\tilde{\boldsymbol \mu} (\Delta)$ with $\Delta$. By exploiting the fact that $\tilde{\boldsymbol \mu} (\Delta)$ can be determined analytically in the limit $\Delta \rightarrow \infty$ we solve the flow equations numerically to obtain estimates for the renormalized parameters in the range $0<U/\pi\Delta<3.5$.

Reply to “Comment on ‘Conductance scaling in Kondo-correlated quantum dots: Role of level asymmetry and charging energy’”

The Comment of A. A. Aligia claims that the superperturbation theory (SPT) approach [E. Mu\~noz, C. J. Bolech, and S. Kirchner, Phys. Rev. Lett. 110, 016601 (2013)] formulated using dual fermions [A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 77, 033101 (2008)] and used by us to compare with numerical renormalization group (NRG) results for the conductance [L. Merker, S. Kirchner, E. Mu\~noz, and T. A. Costi, Phys. Rev. B 87, 165132 (2013)], fails to correctly extend the results of the symmetric Anderson impurity model (SIAM) for general values of the local level ${E}_{d}$ in the Kondo regime. We answer this criticism. We also compare new NRG results for ${c}_{B}$, with ${c}_{B}$ calculated directly from the low-field conductance, with new higher-order SPT calculations for this quantity, finding excellent agreement for all ${E}_{d}$ and for $U/\ensuremath{\pi}\ensuremath{\Delta}$ extending into the strong coupling regime.

Sub-Ohmic two-level system representation of the Kondo effect

It has been recently shown that the particle-hole symmetric Anderson impurity model can be mapped onto a ${Z}_{2}$ slave-spin theory without any need of additional constraints. Here, we prove by means of numerical renormalization group that the slave-spin behaves in this model like a two-level system coupled to a sub-Ohmic dissipative environment. It follows that the ${Z}_{2}$ symmetry gets spontaneously broken at zero temperature, which we find can be identified with the onset of Kondo coherence, being the Kondo temperature proportional to the square of the order parameter. Since the model is numerically solvable, the results are very enlightening on the role of quantum fluctuations beyond mean field in the context of slave-boson approaches to correlated electron models, an issue that has been attracting interest since the 80${}^{\ensuremath{'}}$s. Finally, our results suggest as a byproduct that the paramagnetic metal phase of the Hubbard model at half-filling, in infinite coordination lattices and at zero temperature, as described for instance by dynamical mean-field theory, corresponds to a slave-spin theory with a spontaneous breakdown of a local ${Z}_{2}$ gauge symmetry.

Anderson impurity model with a narrow-band host: From orbital physics to the Kondo effect

A particle-hole symmetric Anderson impurity model with a metallic host of narrow bandwidth is studied within the framework of the local moment approach. The resultant single-particle spectra are compared to unrestricted Hartree-Fock, second order perturbation theory about the noninteracting limit, and Lanczos spectra by Hofstetter and Kehrein. Rather accurate analytical results explain the spectral evolution over almost the entire range of interactions. These encompass, in particular, a rationale for the four-peak structure observed in the low-energy sector of the Lanczos spectra in the moderate-coupling regime. In weak coupling, the spectral evolution is governed by orbital effects, while in the strong coupling Kondo limit, the model is shown to connect smoothly to the generic Anderson impurity with a flat and infinitely wide hybridization band.

Weak-coupling quantum Monte Carlo calculations on the Keldysh contour: Theory and application to the current-voltage characteristics of the Anderson model

We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the "interaction quench" and the "voltage quench", which are found to be suitable for large and small voltage biasses, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson impurity model is presented for interactions U up to 10 times the intrinsic level width Gamma. We compare the Monte Carlo results to fourth order perturbation theory, finding that perturbation theory begins to fail at U/Gamma>4. Within the parameter range studied we find no evidence for a splitting of the Kondo resonance due to the applied voltage. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.

The soft-gap Anderson model:comparison of renormalization group and local moment approaches

The symmetric Anderson impurity model with a hybridization vanishingat the Fermi level, ΔI∝|ω|r, isstudied via the numerical renormalization group (NRG) at T = 0; anddetailed comparison made with predictions arising from the localmoment approach (LMA), a recently developed many-body theory whichis found to provide a remarkably successful description of theproblem. Results for the `normal' (r = 0) impurity model areobtained as a specific case, and likewise compared. Particularemphasis is given both to single-particle excitation dynamics, andto the transition between the strong-coupling (SC) and local moment(LM) phases of the model. Scaling characteristics and asymptoticbehaviour of the SC/LM phase boundaries are considered.Single-particle spectra D(ω) are investigated in some detail, for the SC phase in particular. Here, in accordance with a recentlyestablished result, the modified spectral functions ℱ(ω)∝|ω|rD(ω) are found to contain ageneralized Kondo resonance that is ubiquitously pinned at the Fermilevel; and which exhibits a characteristic low-energy Kondo scale, ωK(r), that narrows progressively upon approach to theSC→LM transition, where it vanishes.Universal scaling of the spectra as the transition is approachedthus results. The scaling spectrum characteristic of the normalAnderson model is recovered as a particular case, that exemplifiesbehaviour characteristic of the SC phase generally, and which iscaptured quantitatively by the LMA. In all cases the r-dependentscaling spectra are found to possess characteristic low-energyasymptotics, but to be dominated by generalizedDoniach-Sunjic tails, in agreement with LMA predictions.

A local moment approach to magnetic impurities in gapless Fermi systems

A local moment approach is developed for single-particle excitations of a symmetric Anderson impurity model (AIM) with a soft-gap hybridization vanishing at the Fermi level: I||r, with r>0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which single-particle excitations are coupled dynamically to low-energy transverse spin fluctuations. The resultant theory is applicable on all energy scales, and captures both the spin-fluctuation regime of strong coupling (large U), as well as the weak-coupling regime where it is perturbatively exact for those r-domains in which perturbation theory in U is non-singular. While the primary emphasis is on single-particle dynamics, the quantum phase transition between strong-coupling (SC) and local moment (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained, notably for the behaviour of the critical Uc(r) separating SC/LM states and the Kondo scale m(r) characteristic of the SC phase. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies; and a number of further testable predictions are made. Single-particle spectra are examined systematically for both SC and LM states; in particular, for all 0 r<½, spectra characteristic of the SC state are predicted to exhibit an r-dependent universal scaling form as the SC/LM phase boundary is approached and the Kondo scale vanishes. Results for the `normal' r = 0 AIM are moreover recovered smoothly from the limit r0, where the resultant description of single-particle dynamics includes recovery of Doniach-Sunjic tails in the wings of the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour and the exponential diminution with U of the Kondo scale itself. The normal AIM is found to represent a particular case of more generic behaviour characteristic of the r>0 SC phase which, in agreement with conclusions drawn from recent NRG work, may be viewed as a non-trivial but natural generalization of Fermi liquid physics.

Magnetic impurities in gapless Fermi systems: perturbation theory

We consider a symmetric Anderson impurity model, with a soft-gap hybridization vanishing at the Fermi level with a power law r > 0. Three facets of the problem are examined. First the non-interacting limit, which despite its simplicity contains much physics relevant to the U > 0 case: it exhibits both strong coupling (SC) states (for r < 1) and local moment (LM) states (for r > 1), with characteristic signatures in both spectral properties and thermodynamic functions. Second, we establish general conditions upon the interaction self-energy for the occurence of a SC state for U > 0. This leads to a pinning theorem, whereby the modified spectral function is pinned at the Fermi level for any U where a SC state exists; it generalizes to arbitrary r the familiar pinning condition for the normal r = 0 Anderson model. Finally, we consider explicitly spectral functions at the simplest level: second order perturbation theory in U, which we conclude is applicable for r < 1/2 and r > 1 but not for 1/2 < r < 1. Characteristic spectral features observed in numerical renormalization group calculations are thereby recovered, for both SC and LM phases; and for the SC state the modified spectral functions are found to contain a generalized Abrikosov-Suhl resonance exhibiting a characteristic low-energy Kondo scale with increasing interaction strength.

Symmetric Anderson impurity model with a narrow band

The single-channel Anderson impurity model is a standard model for the description of magnetic impurities in metallic systems. Usually, the bandwidth represents the largest energy scale of the problem. In this paper, we analyze the limit of a narrow band, which is relevant for the Mott-Hubbard transition in infinite dimensions. For the symmetric model we discuss two different effects. (i) The impurity contribution to the density of states at the Fermi surface always turns out to be negative in such systems. This leads to a new crossover in the thermodynamic quantities that we investigate using the numerical renormalization group. (ii) Using the Lanczos method, we calculate the impurity spectral function and demonstrate the breakdown of the skeleton expansion on an intermediate energy scale. Luttinger's theorem, as an example of the local Fermi liquid property of the model, is shown to still be valid.

Non-Fermi-liquid theory of a compactified Anderson single-impurity model.

We consider a version of the symmetric Anderson impurity model (compactified) which has a non-Fermi liquid weak coupling regime. We find that in the Majorana fermion representation, perturbation theory can be conveniently developed in terms of Pfaffian determinants and we use this formalism to calculate the impurity free energy, self energies, and vertex functions. In the second-order perturbation theory, a linear temperature dependence of electrical resistivity is obtained, and the leading corrections to the impurity specific heat are found to behave as $T\ln T$. The impurity susceptibilities have terms in $\ln T$ to zero, first, and second order, and corrections of $\ln^2 T$ to second order as well. The singlet superconducting paired susceptibility at the impurity site, is found to have second-order corrections $\ln T$, which we interpret as an indication that a singlet conduction electron pairing resonance forms at the Fermi level. When the perturbation theory is extended to third order logarithmic divergences are found in the vertex function $\Gamma_{0,1,2,3}(0,0,0,0)$. Multiplicative renormalization-group method is then used to sum all the leading order logarithmic contributions, giving rise to a new weak-coupling low-temperature energy scale $T_c=\Delta{\rm exp}\left[-\frac{1}{9}\left(\frac{\pi\Delta}{U} \right )^{2}\right]$. The scaling equation shows the dimensionless coupling constant $\frac{U}{\pi\Delta}$ is increased as the energy scale $\Delta$ reduces. Our perturbational results can be justified only in the regime $T>T_c$.

Dynamical approach to analytic continuation of quantum Monte Carlo data.

A new method of analytic continuation from a Matsubara single-particle Green's function to a spectral function is presented. We recast this problem onto a new one where we dynamically minimize a suitably defined potential. Our method allows the imposition of physical constraints such as smoothness and adherence to the sum rule. The method is applied to the symmetric Anderson impurity model. We show how the spectral function changes with the Kondo temperature ${T}_{K}$, the hybridization witdth \ensuremath{\Delta}, and the Coulomb potential U.

Monte Carlo study of the symmetric Anderson-impurity model.

Using a recently proposed quantum Monte Carlo technique, we consider moment formation and magnetic properties of the symmetric Anderson-impurity model for a wide range of parameters and temperatures. We parametrize the general behavior, determining the approach towards universality and establishing the range of validity of various approximations.