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arXiv:cond-mat/0406653 · Physical Review B

Renormalization of the periodic Anderson model: An alternative analytical approach to heavy-fermion behavior

A. Hübsch, K. W. Becker

Published Apr 20, 2005• 9 citations
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Abstract

In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy ${\ensuremath{\nu}}_{f}$ of the angular momentum. An expansion in $1∕{\ensuremath{\nu}}_{f}$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized $f$ level ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$, we obtain coupled equations for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$ and the averaged $f$ occupation $⟨{n}_{f}⟩$. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large ${\ensuremath{\nu}}_{f}$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small ${\ensuremath{\nu}}_{f}$.

DOI
10.1103/physrevb.71.155116

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In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy \ensuremathνf{\ensuremath{\nu}}_{f}\ensuremathνf​ of the angular momentum. An expansion in 1∕\ensuremathνf1∕{\ensuremath{\nu}}_{f}1∕\ensuremathνf​ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized fff level \ensuremathϵ\ifmmode~\else˜\fif{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremathϵ\ifmmode~\else˜\fi​f​, we obtain coupled equations for \ensuremathϵ\ifmmode~\else˜\fif{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremathϵ\ifmmode~\else˜\fi​f​ and the averaged fff occupation ⟨nf⟩⟨{n}_{f}⟩⟨nf​⟩. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large \ensuremathνf{\ensuremath{\nu}}_{f}\ensuremathνf​ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small \ensuremathνf{\ensuremath{\nu}}_{f}\ensuremathνf​.
Despite the advances in the development of numerical methods analytical approaches play a key role on the way towards a deeper understanding of strongly interacting systems. In this regards, renormalization schemes for Hamiltonians represent an important new direction in the field. Among these renormalization schemes the projector-based renormalization method (PRM) reviewed here might be the approach with the widest range of possible applications: As demonstrated in this review, continuous unitary transformations, perturbation theory, non-perturbative phenomena, and quantum-phase transitions can be understood within the same theoretical framework. This review starts from the definition of an effective Hamiltonian by means of projection operators that allows the evaluation within perturbation theory as well as the formulation of a renormalization scheme. The developed approach is then applied to three different many-particle systems: At first, we study the electron-phonon problem to discuss several modifications of the method and to demonstrate how phase transitions can be described within the PRM. Secondly, to show that non-perturbative phenomena are accessible by the PRM, the periodic Anderson model is investigated to describe heavy-fermion behavior. Finally, we discuss the quantum-phase transition in the one-dimensional Holstein model of spinless fermions where both metallic and insulating phase are described within the same theoretical framework.
This paper presents a renormalization approach to many-particle systems. By starting from a bare Hamiltonian H=H0+H1\mathcal{H}={\mathcal{H}}_{0}+{\mathcal{H}}_{1}H=H0​+H1​ with an unperturbed part H0{\mathcal{H}}_{0}H0​ and a perturbation H1,{\mathcal{H}}_{1},H1​, we define an effective Hamiltonian which has a band-diagonal shape with respect to the eigenbasis of H0.{\mathcal{H}}_{0}.H0​. This means that all transition matrix elements are suppressed which have energy differences larger than a given cutoff $\ensuremath{\lambda}$ that is smaller than the cutoff $\ensuremath{\Lambda}$ of the original Hamiltonian. This property resembles a recent flow equation approach on the basis of continuous unitary transformations. For demonstration of the method we discuss an exact solvable model, as well as the Anderson-lattice model where the well-known quasiparticle behavior of heavy fermions is derived.
We investigate an extended version of the periodic Anderson model (the so-called periodic Anderson-Hubbard model) with the aim to understand the role of interaction between conduction electrons in the formation of the heavy-fermion and mixed-valence states. Two methods are used: (i) variational calculation with the Gutzwiller wave function optimizing numerically the ground-state energy and (ii) exact diagonalization of the Hamiltonian for short chains. The fff-level occupancy and the renormalization factor of the quasiparticles are calculated as a function of the energy of the fff orbital for a wide range of the interaction parameters. The results obtained by the two methods are in reasonably good agreement for the periodic Anderson model. The agreement is maintained even when the interaction between band electrons, Ud{U}_{d}Ud​, is taken into account, except for the half-filled case. This discrepancy can be explained by the difference between the physics of the one- and higher-dimensional models. We find that this interaction shifts and widens the energy range of the bare fff level, where heavy-fermion behavior can be observed. For large-enough Ud{U}_{d}Ud​ this range may lie even above the bare conduction band. The Gutzwiller method indicates a robust transition from Kondo insulator to Mott insulator in the half-filled model, while Ud{U}_{d}Ud​ enhances the quasiparticle mass when the filling is close to half filling.
By means of a projector operator formalism we study the ground state properties of the Anderson Impurity Hamiltonian. The non-perturbative treatment of the model agrees with the previous one, obtained by Inagaki [Prog. Theor. Phys. 62, 1441 (1979)] by means of a perturbation expansion with respect the hybridization term. We go beyond the Inagaki's formalism to the next leading order. It provides a very accurate calculation of the energy spectrum in the total spin sector and, in particular, the ground state energy in the whole parameter space. For a one body spinless system, the dependence of the ground state energy as a function of the impurity level obtained by this procedure remarkably agrees with analytical results. For the many body case the occupancy of the impurity as a function of the parameters is studied and it agrees with the corresponding one obtained by using the Bethe ansatz and the Numerical Renormalization Group solution of the Hamiltonian. The magnetization and susceptibility of the impurity is analyzed by studying the response of the system to an external magnetic field, from which it is possible to extract the Kondo temperature. The dependence of the Kondo scale with the parameters of the model is in excellent agreement with well-known results.
The periodic Anderson model can be studied in the limit $U=\ensuremath{\infty}$ by employing the Hubbard X operators to project out the unwanted states. We had already studied this problem by employing the cumulant expansion with the hybridization as perturbation, but the probability conservation of the local states (completeness) is not usually satisfied when partial expansions like the ``chain approximation'' (CHA) are employed. To rectify this situation, we modify the CHA by employing a procedure that was used in the mean-field approximation of Coleman's slave-boson method. Our technique reproduces the features of that method in its region of validity, but avoids the unwanted phase transition that appears in the same method both when $\ensuremath{\mu}\ensuremath{\gg}{E}_{f}$ at low T and for all values of the parameters at intermediate temperatures. Our method also has a dynamic character that is absent from the mean-field slave-boson method.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.