Kondo Hamiltonian

The behaviour pertinent to the opposite limit of the well localized f states is found in most rare-earth compounds. The 4f states are situated more than 5 eV below EF in most of them. The strength of the interaction of f and conduction-band electrons is considerable (= 0.1 eV), but contributes only indirectly to the magnetic coupling of f-moments via polarization of conduction electrons (RKKY), the 4f-moment magnitude remaining preserved. For f states closer to EF, as is the case of y-Ce or some Ce compounds (a situation comparable To some actinide compounds), the interactions between the f- and conduction-band states becomes stronger. The Kondo Hamiltonian can be written as... 

Hk Kondo Hamiltonian RKKY Ruderman-Kittel-Kasuya-... 

For the non-symmetrical Anderson model with C/ + 2ef 0, one has in the Coqblin Schrieffer and Kondo Hamiltonians in addition to the exchange a potential scattering term. The Kondo resonance is no longer at the Fermi energy ep = 0 but is shifted. This shift leads to a smaller resistivity p(T), the maximum of which, however, is still at T = 0. The potential scattering has a dramatic effect on the thermoelectric power, which vanishes in the symmetric case C/ + 2ef = 0, but has a huge peak near 7k for C/ 4- 2ef 0. 

When T>Tk Kondo s derivation of the logarithmic decrease in resistivity started with the Hamiltonian (cf. (11))... 

The Kondo-lattice Hamiltonian conserves total spin and being an interacting model is nontrivial to solve. However, as with the conjugated systems, it is possible to solve finite Kondo chains efficiently by employing the VB method. The VB... 

The exact diagonalization method has been widely exploited in the study of polyenes as well as small conjugated molecules. It has also been employed in studying spin systems and systems with interacting fermions and spins such as Kondo lattices. These studies have been mainly confined to low- dimensions. The exact diagonalization techniques also allow bench-marking various approximate many-body techniques for model quantum cell Hamiltonians. 

As described by Liu (1961) the nature of the exchange interaction changes when the energy required to promote an electron from the conduction band into the 4f band is small. In this case the admixture interaction between conduction and 4f electrons must be taken into account. This interaction leads to "an effective attractive sp-f exchange interaction of the Schrieffer-Wolff kind though the form of the Hamiltonian is still given by eq. (19). Such admixture interactions are applicable to Ce and Yb alloys (see Coqblin and Blandin, 1968) and lead to a Kondo effect. 

It is suggestive that the narrow Kondo resonance states of individual 4f impurities will form heavy quasiparticle bands in a periodic lattice of 4f ions. A satisfactory microscopic theory of heavy-band formation has yet to be developed. The Hamiltonian of eq. (107) can be generalized to the lattice by introducing a Bose field h,- at every lattice site. However, in this model it is no longer practicable to restrict to physical states with = 1 at every site. The most successful approach so far consists in a mean-field approximation for the Bose field (Coleman 1985, 1987, Newns and Read 1987) that is valid for large N and r < It can be applied both for the impurity and the lattice model. It starts from the observation that in the limit with QJN= fixed, the rescaled... 

In the traditional view, the interaction which is believed to be responsible for the Kondo effect proceeds via the conduction electron-impurity spin exchange interaction. The hamiltonian for this interaction (the so-called s-d or s-f hamiltonian) is... 

In the event that the amount of hybridization is too strong for the exchange hamiltonian to be an adequate starting point for a theoretical description of the Kondo-like anomalies in the physical properties, the characteristic temperature is often identified with a spin fluctuation temperature T,f which has the obvious definition Tsf = h/keTsf. In this view, Tjt, rather than Tk, is a boundary which separates high temperature (compared to Tsf) magnetic behavior from low temperature nonmagnetic behavior. 

For —Sf A, U A and U + 2ef = 0 the Anderson Hamiltonian, eq.(1), can be transformed into the Coqblin-Schrieffer (CS) Hamiltonian. In this Kondo limit charge fluctuations are completely suppressed and the model describes an effective 4f-electron spin j which interacts via exchange with the conduction electrons... 

Allen and Martin (1982) and Lavagna et al. (1982, 1983) proposed that the y-a transition in Ce was related to the Kondo effect. Since Kondo systems have densities of states with two peaks, one very close to the Fermi level and one below, this offered a possible explanation of the two peaks in the photoelectron spectra, assuming that the photoabsorption process would transfer the structure in the density of states to the emitted spectrum. Detailed calculations of a photoelectron spectrum were not, however, carried out. The Kondo (or Abrikosov-Suhl) resonance in the density of states can be obtained from the Anderson Hamiltonian in the limit U CO (Lacroix 1981). For a Fermi level in the center of a valence band of width 2D and a constant density of states coupled by a constant matrix element V to an Nf-fold degenerate localized level at energy Sf below two peaks can arise in the density of states of the coupled system at low temperatures if is not too small. As... 

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