Electron Correlation and Jahn-Teller Interaction in Manganese Oxides

The interplay between the electron repulsion U and the Jahn-Teller electronphonon interation ELR is studied with a large d model for the ferromagnetic state of the manganese oxides. These two interactions collaborate to induce the local isospin (orbital) moments and reduce the bandwidth B. Especially the retardation effect of the Jahn-Teller phonon with the frequency Ω is effective to reduce B, but the strong Ω-dependence occurs even when the Coulombic interaction is dominating ( U >> ELR ) as long as ELR > Ω. The phonon spectrum consists of two components, i.e., the temperature independent sharp peak at ω = Ω̃ = Ω[(U + 4ELR)/U ] 1/2 and that corresponding to the Kondo peak. These results compared with the experiments suggest that Ω < ELR < U in the metallic manganese oxides. 74.25.Fy, 74.25.Ha, 74.72.-h, 75.20.Hr Typeset using REVTEX 1 Since the discovery of colossal magnetoresistance (CMR) the manganese oxides Re1−xAxMnO3 ( Re: rare earth metal ion, A: divalent metal ion ) have been attracting intensive interest recently [1]. One of the controversial issues is the role of orbital degeneracy of eg-electrons. In the conventional models of these compounds [2–5], the strong Hund’s coupling is considered to be of primary importance and the orbital degeneracy has been often neglected. Recently, however, Millis et al. [6,7] correctly pointed out that the Hund’s coupling alone is not enough to explain the CMR together with the insulating temperature dependence of the resistivity ρ(T ) above the ferromagnetic transition temperature Tc. The additional coupling they proposed to be important is the Jahn-Teller electron-phonon coupling, which lifts the double degeneracy of eg orbitals and gives rise to the polaronic effect. The small polaron formation above Tc leads to the insulating ρ(T ). In contrast with the insulating phase, the spin alignment below Tc will enlarge the band width B as B ∝ cos(θij/2) (θij is the angle between the two neighboring spins ~ Si and ~ Sj), and the Jahn-Teller interaction enters the weak coupling regime. This scenario seems to be consistent with the large isotope effect on Tc by replacing O 16 by O [8]. The strong correlation models of these compounds have been also studied by several authors especially for the undoped cases [9,10]. In these models the Jahn-Teller interaction is considered to be smaller than the Coulomb interactions, and the effective Hamiltonian for the spin-orbital coupled system and its phase diagram have been clarified. It is rather natural to assume the strong electron correlation because the strong Hund’s coupling originates from the strong electron-electron interactions ( ∼ 5eV ). However, it is not trivial if the electronelectron interaction continues to be strong even in the effective Hamiltonian describing the low energy physics after the screening by oxygen orbitals and conduction electrons, and it still remains the controversial issue whether the electron-electron interation and/or the Jahn-Teller coupling are in the strong coupling regime or not. Experimentally there are several anomalous features which cannot be explained by the weak Jahn-Teller coupling described above even in the low temperature ferromagnetic phase below Tc. 2 [a] In the neutron scattering experiment no temperature dependent phonon modes have been observed [11]. The recent Raman scattering experiment also shows that the JahnTeller phonons ( especially their frequencies ) are temperature independent and insensitive to the ferromagnetic transition at Tc [12]. [b] The photo-emission spectra show a small discontinuity at the Fermi edge even at T << Tc, which suggests some interactions still remain strong there [13]. [c] The optical conductivity σ(ω) at T << Tc is composed of two components, i.e., the narrow Drude peak (ω < 0.02eV) and the broad incoherent component extending up to ω ∼ 1eV [14]. The Drude weight is very small, which seems to be consistent with the photo-emission spectra. [d] The low temperature resistivity ρ(T ) can be fitted by ρ(T ) = ρ0 + AT , (1) where A is a large constant of the order of 500μΩcm/K [15], again suggesting the strong electron correlation. [e] Contrary to the case of resistivity [d], the coefficient of T -linear specific heat is very small with γ = 2mJ/K, which violates the Kadowaki-Woods law for these compounds [16]. Although [e] is difficult to reconcile with [a]-[d], we consider the latter as the evidences for the strong coupling even at T ≪ Tc. Because the spins are perfectly aligned at T ≪ Tc, the only remaining degrees of freedom are the orbital ones. In this paper, we study a large d model [17] for the ferromagnetic state including both the electron-electron interaction U and the Jahn-Teller coupling g. This is the generalization of Ref.[7] in two respects, (i) including the electron-electron interation, (ii) including the quantum fluctuations. Especially the latter is essential to describe the low temperature Fermi liquid state, which is described by the Kondo peak in the large d limit [17]. The strong electron-electron interaction with the reasonable magnitude of the Jahn-Teller coupling explains both the large isotope effect and [a]. Moreover, the features of strong correlation [b]-[d] are at least consistent with the large U picture although [e] still requires further studies, which we have not undertaken in this 3 paper. We start with the Hubbard-Holstein model for the ferromagnetic state. H = − ∑ i,j,α,α t ′ ij c † iαcjα′ + U ∑