From the Atomic Systems to the Extended Ones: the Hubbard Operators

It is sometimes convenient to emphasize the local aspects of a part of a crystalline system, and use the corresponding localized states to build a basis of the states of the whole system. In many cases it is only a subspace of these local states that is relevant, and the Hubbard operators provide a fairly simple way to write the corresponding projected Hamiltonian. Two examples of this type of treatment are presented int this work. The first is a Co2+ impurity in a MgO crystal interacting through a Jahn-Teller term with the crystal phonons, and it is shown how this interaction affects the electronic Raman scattering. The second is the Anderson lattice when the local electrons have an infinite Coulomb repulsion, and a diagrammatic expansion with cumulants is discussed. We propose a method to obtain approximate Green's functions for the Anderson lattice that employs the exact solutions of an atomic problem, and the corresponding spectral density of the local electrons is calculated.

Authorship

M. E. Foglio

Universidade EstadualUniversidade EstadualUniversidade Estadual, Instituto de Física `Gleb Wataghin', Campinas, São Paulo, BrazilUniversidade EstadualUniversidade EstadualUniversidade EstadualBrazilCampinas, São Paulo, BrazilUniversidade EstadualUniversidade EstadualUniversidade Estadual, Instituto de Física `Gleb Wataghin', Campinas, São Paulo, Brazil

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (10)
Tables (1)

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Figure 1.
Energy levels of the ground ⁴F term of a Co²⁺ split by a cubic field and by the spin orbit interaction. The levels are labeled by their symmetry properties.

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Figure 2.
One-phonon density of states in the MgO: (a) The curves computed by Peckham [14]; (b) the approximation used, where the dotted lines represent the acustic and optical phonon contributions.

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Figure 3.
Raman spectra of the MgO:Co²⁺ in the E_g configuration: (a) Electronic contribution as calculated by the present theory; (b) experimental curve obtained by Guha [15] at 18 K.

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Figure 4.
Typical cumulant diagrams for one-particle GF (a) The diagrams of the chain approximation (CHA) for the f-electron, represented by the filled square to the right, (b) As (a) but for the c-electrons , represented by an empty square. (c) A more complicated diagram, with cumulants of fourth and sixth order.

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Figure 5.
The spectral density of the f-electrons for a system with the following parameters: E_j,s = E_f = -0.5, m = 0., T = 0.001, a local hybridization V = 0.3 and an unperturbed band of electrons with a rectangular density of states of width p and centered at the origin, all in the same energy units. (a)The dotted curve shows the values obtained with the CHA. (b) The dashed curve gives the values obtained with the "multiple loop approximation'' (MLA), that corresponds to the family of diagrams shown in Fig. 6. No structure is observed close to w = 0., where the Kondo peak should be present.

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Figure 6.
The Multiple Loop Approximation (MLA). (a) The family of diagrams with only one fourth order cumulant. (b) Diagrams with infinite fourth order cumulants. (c) The diagrams that give the MLA correction to the GF in the CHA. They should be added to the diagrams in Fig. 4a.

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Figure 7.
The energies e_n,r = E_n,r-nm of the twelve eigenstates | n,rñ of the atomic limit are represented in this figure, and those corresponding to different occupations n = 0,1,2,3 are drawn in different columns. The index r that characterize the states is written above the corresponding levels, and the lines joining different levels are identified by numbers i, showing the possible transitions u_i that contribute to the GF. As with Eqs. (14 , 15), the frequencies have the chemical potential m subtracted, so that the Fermi surface corresponds to w = 0.

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Figure 8.
The spectral density r_f(w) of the f-electrons obtained with the AECA for several values of T, employing z = w+ih with h = 10^-4. The system has the following parameters: E_f = -0.5, m = 0., T = 0.001, a local hybridization V = 0.3 and a density of states of the unperturbed band electrons given by a rectangular density of states of width p centered at the origin, all in the same energy units. The effective cumulant was calculated employing a reduced hybridization V_a = D = V² and a frequency with a small imaginary part h_a = 0.003.

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Figure 9.
The spectral density of figure Fig. 8 in the neighborhood of w = 0. The integrated intensity of this structure decreases with T.

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Figure 10.
The spectral density of the f-electrons for the same system of Fig. 8 but with fixed T = 0.001 and for several values of the chemical potential m. It shows the crossover from the Kondo region to the intermediate valence region.

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Figure 1. Energy levels of the ground ⁴F term of a Co²⁺ split by a cubic field and by the spin orbit interaction. The levels are labeled by their symmetry properties.

Figure 2. One-phonon density of states in the MgO: (a) The curves computed by Peckham [14]; (b) the approximation used, where the dotted lines represent the acustic and optical phonon contributions.

Figure 3. Raman spectra of the MgO:Co²⁺ in the E_g configuration: (a) Electronic contribution as calculated by the present theory; (b) experimental curve obtained by Guha [15] at 18 K.

Figure 4. Typical cumulant diagrams for one-particle GF (a) The diagrams of the chain approximation (CHA) for the f-electron, represented by the filled square to the right, (b) As (a) but for the c-electrons , represented by an empty square. (c) A more complicated diagram, with cumulants of fourth and sixth order.

Figure 5. The spectral density of the f-electrons for a system with the following parameters: E_j,s = E_f = -0.5, m = 0., T = 0.001, a local hybridization V = 0.3 and an unperturbed band of electrons with a rectangular density of states of width p and centered at the origin, all in the same energy units. (a)The dotted curve shows the values obtained with the CHA. (b) The dashed curve gives the values obtained with the "multiple loop approximation'' (MLA), that corresponds to the family of diagrams shown in Fig. 6. No structure is observed close to w = 0., where the Kondo peak should be present.

Figure 6. The Multiple Loop Approximation (MLA). (a) The family of diagrams with only one fourth order cumulant. (b) Diagrams with infinite fourth order cumulants. (c) The diagrams that give the MLA correction to the GF in the CHA. They should be added to the diagrams in Fig. 4a.

Figure 7. The energies e_n,r = E_n,r-nm of the twelve eigenstates | n,rñ of the atomic limit are represented in this figure, and those corresponding to different occupations n = 0,1,2,3 are drawn in different columns. The index r that characterize the states is written above the corresponding levels, and the lines joining different levels are identified by numbers i, showing the possible transitions u_i that contribute to the GF. As with Eqs. (14 , 15), the frequencies have the chemical potential m subtracted, so that the Fermi surface corresponds to w = 0.

Figure 8. The spectral density r_f(w) of the f-electrons obtained with the AECA for several values of T, employing z = w+ih with h = 10^-4. The system has the following parameters: E_f = -0.5, m = 0., T = 0.001, a local hybridization V = 0.3 and a density of states of the unperturbed band electrons given by a rectangular density of states of width p centered at the origin, all in the same energy units. The effective cumulant was calculated employing a reduced hybridization V_a = D = V² and a frequency with a small imaginary part h_a = 0.003.

Figure 9. The spectral density of figure Fig. 8 in the neighborhood of w = 0. The integrated intensity of this structure decreases with T.

Figure 10. The spectral density of the f-electrons for the same system of Fig. 8 but with fixed T = 0.001 and for several values of the chemical potential m. It shows the crossover from the Kondo region to the intermediate valence region.

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From the Atomic Systems to the Extended Ones: the Hubbard Operators

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