Considerations on the quantum double-exchange Hamiltonian

Introduced by Zener [1] in the early 1950s the notion of double exchange together with mixed-valency manganites R1−xAxMnO3 (where R = La, Pr, Nd and A = Sr, Ca, Ba, Pb) attracted renewed attention when a colossal magnetoresistive effect was discovered in these compounds some years ago [2]. The magnetic and electronic properties of manganese oxides, to some extend, are believed to arise from the large Coulomb and Hund’s rule interaction of the manganese d shell electrons. Due to the almost octahedral coordination within the perovskite structure the d levels split into two subbands labeled according to their octahedral symmetry, eg and t2g. In the case of zero doping (x = 0) there are four electrons per Mn site which fill up the three t2g levels and one eg level, and by Hund’s rule, form a S = 2 spin state. Doping will remove the electron from the eg level, and by hopping via bridging oxygen sites these holes acquire mobility. However, this hopping acts in a background of local spins S = 3/2 formed by the t2g electrons and its amplitude depends on the overlap of the spin states at neighbouring sites (or, in a classical language, on their relative angle), it is largest if the total bond spin is maximal and vice versa [3]. Another ingredient, that is assumed to significantly influence the physical properties of manganites, is electronlattice interaction. Namely the twoeg orbitals, which are degenerate in a perfect cubic environment, will couple to lattice vibrations of the same symmetry, giving rise to a Jahn-Teller effect and polaronic behaviour in some regions of the phase diagram. It is this close interplay of three different subsystems (electrons in degenerate orbitals, background of localized spins, and lattice vibrations) that makes the physics of manganites both, rich and complicated. In the present work we concentrate on the double exchange (DE) part of the interactions and consider different possibilities for an approximate treatment of the exact DE Hamiltonian on a lattice in terms of effective electronic one- or two-band models. These can be used in a more elaborate modelling of the real materials (see our forthcoming work [4]). It turns out that quantum double exchange on a lattice is most suitably derived and described with the help of Schwinger bosons. We therefore include a detailed and pedagogic derivation of the quantum DE Hamiltonian using Schwinger bosons. Although this approach has been used before [5,6], we feel a comprehensive presentation of the subject is still missing. In two appendices we reexamine the derivation for two sites, and consider the semiclassical limit (S → ∞). In addition, by means of numerical experiments, we illustrate how this limit evolves from the quantum case.