The Kondo effect in periodic narrow-band systems

The Kondo divergences owing to interaction of current carriers with local moments in highly correlated electron systems are considered within the Hubbard and s-d exchange models with infinitely strong onsite interaction, the many-electron Hubbard representation being used. The picture of density of states containing a peak at the Fermi level is obtained. Various forms of the self-consistent approximation are used. The problem of the violation of analytical properties of the Green’s function is discussed. Smearing of the “Kondo” peak owing to spin dynamics and finite temperatures is investigated. PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.), 71.27.+a Strongly correlated electron systems; heavy fermions, 71.28.+d Narrow-band systems; intermediate-valence solids The problem of strong correlations and magnetism in many-electron systems is one of the most important in the solid state theory. Since the Hubbard’s works of 60’s, a great progress has been achieved in understanding electronic structure of systems with strong on-site interaction. Last time, the role of the Kondo effect has been discussed within the large-d approach (d is space dimensionality) which reduces the initial periodic Hubbard model to an effective Anderson impurity model [1, 2]. Besides the Hubbard bands, an important role in the formation of density of states (DOS) picture belongs to a peak at the Fermi level, which was found for both half-filled and doped case (the latter case is considered in [2, 3]). It should be noted that this approach meets with a number of computational difficulties (e.g., consideration of finite temperatures is needed, and the low-temperature limit is non-trivial). The structure of the spectrum in large-d approaches is confirmed by the Monte-Carlo calculations. On the other hand, this feature was not reproduced by most preceding analytical approaches. In particular, the Hubbard-III approximation [4] does not take into account contributions of Fermi-like excitations in a proper way because of its single-site character. A detailed analysis of this approximation was performed in Refs. [5, 6] within the large-z (z is nearest-neighbor number) expansion. In the present paper we present a treatment that is based on the method of equations of motion for the many-electron Hubbard operators [7, 8] and is much more simple than the large-d approach. As the zero order this approach reduces to the simplest Hubbard-I approximation. General expressions for 1/zcorrections were obtained in Ref. [5]. Unfortunately, the terms with the one-particle occupation numbers (which just describe the Kondo effect) were neglected in Refs. [5, 6], and only a classical approximation was considered in Ref. [6]. We start from the s-d exchange model with the large s-d coupling parameter |I|, H = ∑ kσ tkc † kσckσ − I ∑ iσσ Siσσσ′c † iσciσ′ +Hd, (1) where tk is the band energy, Hd is the Heisenberg Hamiltonian of the localized-spin system, σ are the Pauli matrices. In the limit |I| → α∞ (here and hereafter α = sign I = ±), it is convenient to pass to the atomic representation of the Hubbard operators X i = |iβ〉〈iγ| where Hsd (the second term in (1)) takes the diagonal form [9]. For the electron concentration n < 1, after performing the procedure of projection onto the corresponding state space, the one-electron Fermi operators c†iσ are replaced by the many-electron operators g iσα. These are expressed in terms of the X-operators as g iσ+ = ∑ M {(S + σM + 1)/(2S + 1)}Xi(M + σ 2 ,+;M),