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arXiv:1503.03718 · Physical Review B

High-order terms in the renormalized perturbation theory for the Anderson impurity model

Vassilis Pandis, Alex C. Hewson

Published Sep 14, 2015• 3 citations
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Abstract

We study the renormalized perturbation theory of the single-impurity Anderson model, particularly the high-order terms in the expansion of the self-energy in powers of the renormalized coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$. Though the presence of counterterms in the renormalized theory may appear to complicate the diagrammatics, we show how these can be seamlessly accommodated by carrying out the calculation order-by-order in terms of skeleton diagrams. We describe how the diagrams pertinent to the renormalized self-energy and four vertex can be automatically generated, translated into integrals, and numerically integrated. To maximize the efficiency of our approach we introduce a generalized $k$-particle/hole propagator, which is used to analytically simplify the resultant integrals and reduce the dimensionality of the integration. We present results for the self-energy and spectral density to fifth order in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, for various values of the model asymmetry, and compare them to a numerical renormalization group calculation.

DOI
10.1103/physrevb.92.115131

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We study the renormalized perturbation theory of the single-impurity Anderson model, particularly the high-order terms in the expansion of the self-energy in powers of the renormalized coupling U\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}U\ifmmode~\else˜\fi​. Though the presence of counterterms in the renormalized theory may appear to complicate the diagrammatics, we show how these can be seamlessly accommodated by carrying out the calculation order-by-order in terms of skeleton diagrams. We describe how the diagrams pertinent to the renormalized self-energy and four vertex can be automatically generated, translated into integrals, and numerically integrated. To maximize the efficiency of our approach we introduce a generalized kkk-particle/hole propagator, which is used to analytically simplify the resultant integrals and reduce the dimensionality of the integration. We present results for the self-energy and spectral density to fifth order in U\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}U\ifmmode~\else˜\fi​, for various values of the model asymmetry, and compare them to a numerical renormalization group calculation.
The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.
We present a new method for calculating directly the one-particle self-energy of an impurity Anderson model with Wilson's numerical renormalization group method by writing this quantity as the ratio of two correlation functions. This way of calculating turns out to be considerably more reliable and accurate than that via the impurity Green's function alone. We give results for the self-energy for the case of a constant coupling between the impurity and the conduction band and the effective arising in the dynamical mean-field theory of the Hubbard model. The implications of the problem of the metal-insulator transition in the Hubbard model are also discussed.
The numerical renormalization group method is used to investigate zero-temperature phase transitions in quantum impurity systems, in particular in the particle–hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.
We consider single-particle spectra of a symmetric narrow-band Anderson impurity model, where the host bandwidth D is small compared to the hybridization strength ${\ensuremath{\Delta}}_{0}.$ Simple second-order perturbation theory (2PT) in U is found to produce a rich spectral structure that leads to rather good agreement with extant Lanczos results and offers a transparent picture of the underlying physics. It also leads naturally to two distinct regimes of spectral behavior, ${\ensuremath{\Delta}}_{0}Z/D\ensuremath{\gg}1$ and \ensuremath{\ll}1 (with Z the quasiparticle weight), whose existence and essential characteristics are discussed and shown to be independent of 2PT itself. The self-energy ${\ensuremath{\Sigma}}_{i}(\ensuremath{\omega})$ is also examined beyond the confines of PT. It is argued that on frequency scales of order $\ensuremath{\omega}\ensuremath{\sim}\sqrt{{\ensuremath{\Delta}}_{0}D},$ the self-energy in strong coupling is given precisely by the 2PT result, and we point out that the resultant poles in ${\ensuremath{\Sigma}}_{i}(\ensuremath{\omega})$ connect continuously to that characteristic of the atomic limit. This in turn offers a natural rationale for the known inability of the skeleton expansion to capture such behavior, and points to the intrinsic dangers of partial infinite-order summations that are based on PT in U.
I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.