Anderson lattice

Fig. 21. Real part of the conductivity of YbFe4St>i2- The symbols on die left axis represent dc values at different temperatures. Below T (fv 50 K), a narrow peak at zero frequency and a gap-like feature at 18 meV gradually develop. Inset Renormalized band structure calculated from die Anderson lattice Hamiltonian. % and f denote bands of free carriers and localized electrons, respectively. At low temperatures a direct gap A opens. The Fermi level, Ep is near die top of die lower band,, resulting in hole-like character and enhanced effective mass of die quasiparticles (Dordevic et al., 2001).

To relate the above results to the physical systems, we now give the comparison of the calculated band structure in the Anderson lattice-like limit of the Emery model with the ARPES data on LSCO [6]. The regime of the bare parameters is Ap[Pg.142]

The observed evolution of the shape of the band-structure upon doping satisfies the Luttinger sum rule [9], It should be noted that only in the Anderson lattice-like limit of the Emery model it is possible to obtain the observed evolution of the FS upon doping. In all other cases, the oxygen symmetry of the FS can be attributed to the (non-renormalized) oxygen band and therefore the strong doping dependence of the band structure cannot be expected. 

A theoretical description of electron-phonon coupling effects in unstable-moment compounds demands a thorough understanding of their electronic excitation spectrum. It has already been qualitatively described in the introduction. In this section we give a firmer theoretical foundation to these concepts for the 4f impurity case as well as the periodic lattice of 4f ions in a metalUc compound. For the Ce or Yb systems the physics for these two cases is adequately described by the (degenerate) Anderson and Anderson lattice Hamiltonians respectively. For a single impurity it reads n = flfj... 

A different approach was used by Razafimandimby et al. (1984), d Am-brumenil and Fidde (1985) and Fulde et al. (1988) for the Kondo lattice to calculate quasiparticle bands. They start from the observation, Nozieres (1974), that for r a Fermi liquid description can be used for the scattering by Kondo ions. Its phase shift is assumed to have a resonant behaviour around the Fermi energy, with T defining the energy scale. A periodic lattice of resonant scattering centers then leads to narrow quasipartiele bands, whieh have been caleulated within the KKR formalism. In the simplest approximation they are equivalent to those obtained from the mean-field approximation of the Anderson lattice. The method of Razafimandimby et al. has been used for a realistic... 

Fig. 27. Resistivity p(T) (a), thermoelectric power S(T) and Lorenz number L T) (b), calculated with LNCA techniques for a sixfold degenerate Anderson lattice model in the Kondo regime (Cox and Grewe 1988). Impurity results, scaled with concentration are shown for comparison. The resistivity results exhibit the logarithmic increase with decreasing temperature and the coherence-derived decrease below T = to the residual value due to impurities, which is quadratic in the Fermi liquid regime T < T. S(T ) is positive definite for the simple model situation chosen.

The extension of the work described here to the Anderson lattice is an interesting but difficult problem. It would also be interesting to include more terms in the Anderson impurity model, describing for instance multiplet effects. Auger decay and the direct f-d Coulomb interaction. The determination of the parameters in the model from first principles is an important problem. Much progress has been made by Herbst et al. (1978) and by Herbst and Wilkins (1979) for 17, Ui and % and there are promising attempts to estimate from band structure calculations. 

Below we briefly review a few of the developments after this chapter had been submitted (February, 1986). There has been a substantial amount of work on the extension of the impurity model to a lattice model. Rice and Ueda (1985) and Fazekas and Brandow (1987) used a generalized Gutzwiller ansatz for the spin-degenarate Anderson lattice. Saso and Seino (1986) and Blankenbecler et al. (1987) used the Monte Carlo technique for studying a finite one-dimensional chain. Millis and Lee (1987) used the slave boson formalism for the large degeneracy Anderson lattice. A renormalization group calculation has also been performed for the two-impurity model by Jones and Varma (1987). 

In spite of the fact that the fits in figs. 25 and 26 are very good, it leaves one rather unsatisfied because the degree of valence mixing for SmBs (0.3 and 0.7) is not what the more recent numbers yield (0.4 and 0.6). However, the model used for the computation of the susceptibility corresponds in a simple way to the alloy analog of the Anderson lattice model, which has been shown to give good results, e.g. by... 

Fig. 35. Predictions of a slave-boson treatment of a Kondo insulator for the specific heat Cp (solid line) and temperature derivative of the f-occupation number dUf/dT (solid circles). As opposed to the Anderson impurity model, where dnf/dT peaks at a temperature which is 2-3 times larger than the temperature where Cp is maximum, for this Anderson Lattice calculation both quantities peak at roughly the same temperature. From Riseborough (1992).

Although there are few treatments of the Anderson Lattice that simultaneously compute Cp, X over a broad temperature range, two recent examples suggest that for the... 

Thalmeier (1988) has calculated the adiabatic bulk modulus of the Anderson Lattice by including a strain dependence of Lkf Lkf —> Kkf + (dVy Vyds)s. Using a slave-boson method for e = 0 and second order perturbation theory with respect to dVkf(V)/ds, he finds... 

In the first method, the Landau scattering amplitudes are calculated up to order l/Af in a slave-boson scheme. The 1/N terms represent hybridization (charge) fluctuations. From this, it was found for the Anderson lattice by Lavagna et al. (1987) that a weak instability occurred in the / = 2 (d wave) channel. Soon after, though, Zhang... 

At this stage, it should be remarked that, as to be discussed in the spin-fluctuation section, inclusion of 1/iV terms in the Anderson lattice theories leads to a qualitatively diflferent picture. Moreover, Millis (1987) has shown that the normal state T lnT correction to the specific heat in these theories are two orders of magnitude too small to explain the experimental data in UPtj. In the Hubbard model approach, Rasul (1991) has shown that order l/N corrections actually lead to a repulsive interaction, so conclusions at the 1/iV level must be regarded with some suspicion. 

Encouraged by the correct description of the thermodynamic properties, of the electron-hole symmetric Anderson impurity model (Horvatic and Zlatic 1985, Okada et al. 1987), by straightforward perturbation theory in Us, several groups have examined the spectral density of the Anderson lattice model using second-order perturbation theory... 

Zlatic et al. 1986). Clearly, the peak at the Fermi level sharpens as the ground state of the system approaches a magnetic instability. To date, there has been little work directed at performing RPA in a self-consistent manner, for the Anderson lattice. 

Sheng and Cooper (1995) took an alternative approach to the heavy fermion materials. The model which they use is based on the Anderson lattice, however, the on-site couplings play a crucial role. The most novel aspect of their model is the detailed recognition of how the non-spherical crystalline environment causes hybridization between the on site f electrons and ligand electrons centered off site. Their approach to the model is to first diagonalize the local parts of the Hamiltonian. For Ce based heavy fermion systems this is done in the space of local two particle states, supplemented with the trivial vacuum state 10). The periodic nature of the lattice is then re-introduced by treating the localized two electron states as forming composite particles. 

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