Perturbation Expansion for the Anderson Hamiltonian. III

Abstract

Thermodynamical quantities are expanded in perturbation series with respect to the Coulomb repulsion U for the Anderson Hamiltonian with electron-hole symmetry and 'their general terms of perturbation are investigated. By these processes the mutual. relations among these quantities are discussed. It is confirmed that the thermodynamical quantities such as the specific heat. and the scattering t-matrix are expressed in terms of two quantities; the even and odd parts of the susceptibility as far as low-lying excitations are concerned. These results are entirely consistent with the Nozieres phenomenological Fermi liquid theory based on the s-d exchange model. . §I. Introduction In the :first paper1h*> of' this series, we presented a perturbation theoretical approach to the Anderson Hamiltonian2> with electron-hole symmetry and showed that each term of the perturbation expansion for the thermodynamical quantities, in particular for the free energy, can be expressed by an imaginary time integral of the product of two antisymmetric determinants constructed by unperturbed local d-electron temperature Green's functions. In the second paper by Yam ada, B) each term in the perturbation expansions for the specific heat, the susceptibility, the resistivity and the density-of~state s for the localized d-state has been calculated up to fourth order. In the course of calculation it . has been found that some general relations hold between thermodynamical quantities, in particular, the T-linear specific heat· is proportional to the even part of the susceptibility. This finding immediately leads us to the result that the ratio of the susceptibility to the coefficient of the T-linear specific heat becomes twice as large in the s-d limit as the value in the case of no correlation. This paper deals with further development along this line: Here, discussion is mainly concentrated on the. general properties of the perturbation series for thermodynamical quantities in the Anderson model. The basic Hamiltonian in this paper is the Anderson Hamiltonian with electron-hole symmetry (Ed= -tU), which is divided into two parts, the unperturbed part H 0 and the perturbation H' as follows: