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arXiv:cond-mat/0011216 · Physical review. B, Condensed matter

Spectral properties of a narrow-band Anderson model

Steffen Schäfer, David E. Logan

Published Jan 9, 2001• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.63.045122

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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 \ensuremathΔ0.{\ensuremath{\Delta}}_{0}.\ensuremathΔ0​. 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, \ensuremathΔ0Z/D\ensuremath≫1{\ensuremath{\Delta}}_{0}Z/D\ensuremath{\gg}1\ensuremathΔ0​Z/D\ensuremath≫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 \ensuremathΣi(\ensuremathω){\ensuremath{\Sigma}}_{i}(\ensuremath{\omega})\ensuremathΣi​(\ensuremathω) is also examined beyond the confines of PT. It is argued that on frequency scales of order \ensuremathω\ensuremath∼\ensuremathΔ0D,\ensuremath{\omega}\ensuremath{\sim}\sqrt{{\ensuremath{\Delta}}_{0}D},\ensuremathω\ensuremath∼\ensuremathΔ0​D​, the self-energy in strong coupling is given precisely by the 2PT result, and we point out that the resultant poles in \ensuremathΣi(\ensuremathω){\ensuremath{\Sigma}}_{i}(\ensuremath{\omega})\ensuremathΣi​(\ensuremathω) 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.
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.
The symmetric Anderson impurity model with a hybridization vanishing at the Fermi level, Δ I ∝|ω| r , is studied via the numerical renormalization group (NRG) at T = 0; and detailed comparison made with predictions arising from the local moment approach (LMA), a recently developed many-body theory which is found to provide a remarkably successful description of the problem. Results for the `normal' ( r = 0) impurity model are obtained as a specific case, and likewise compared. Particular emphasis is given both to single-particle excitation dynamics, and to the transition between the strong-coupling (SC) and local moment (LM) phases of the model. Scaling characteristics and asymptotic behaviour 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 recently established result, the modified spectral functions ℱ(ω)∝|ω| r D (ω) are found to contain a generalized Kondo resonance that is ubiquitously pinned at the Fermi level; and which exhibits a characteristic low-energy Kondo scale, ω K ( r ), that narrows progressively upon approach to the SC→LM transition, where it vanishes. Universal scaling of the spectra as the transition is approached thus results. The scaling spectrum characteristic of the normal Anderson model is recovered as a particular case, that exemplifies behaviour characteristic of the SC phase generally, and which is captured quantitatively by the LMA. In all cases the r -dependent scaling spectra are found to possess characteristic low-energy asymptotics, but to be dominated by generalized Doniach-Sunjic tails, in agreement with LMA predictions.
In an attempt to obtain a better description than the local-density approximation (LDA) for the electronic structure of realistic narrow-band systems, we have studied a perturbation-theory method for the self-energy and have tested it on the electronic properties of the periodic one-dimensional Anderson model. For a number of models we have calculated a wide range of properties such as quasiparticle energies, spectral densities, reduced hybridizations, and various magnetic correlation functions. For two different models we were able to compare with Monte Carlo results and found good agreement. In general, we have obtained a description of narrow-band systems that is more reasonable than the one given by the LDA.
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 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, for various values of the model asymmetry, and compare them to a numerical renormalization group calculation.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc{U}_{c}Uc​ and exhibits strong indications of a lower critical occupation number nc{n}_{c}nc​. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.