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Quantum and thermal effects in the double exchange ferromagnet

The physics of the ferromagnetic phase of the ``double exchange'' model has been widely discussed in the context of the CMR manganites. Usually, the double exchange ferromagnet is treated is classically, by mapping it onto an effective Heisenberg model. However this mapping does not permit a correct treatment of quantum or thermal fluctuation effects, and the results obtained lack many of the interesting features seen in experiments on the manganites. Here we outline a new analytic approach to systematically evaluating quantum and thermal corrections to the magnetic and electronic properties of the double exchange ferromagnet.

Ferromagnetic manganites and double exchange Systems in which charge carriers move in a background of local magnetic moments are ubiquitous among doped d-electron oxides, the two most widely studied examples being the colossal magnetoresistance (CMR) manganites and the high temperature superconducting (HTc) cuprates.
It was first suggested by Zener [1] that ferromagnetism in the manganites originated from the kinetic energy of the doped holes, which couple to the local moments of the Mn atoms through a strong Hund's first rule coupling. This idea was later formalised in terms of the strong coupling limit of a ferromagnetic Kondo lattice Hamiltonian [2,3] Here, t is the hopping integral for eg electrons transfered between neighbouring Mn atoms, whose half filled t2g levels are modelled as a spin S = 3/2 local moment, and J ≫ t > 0 is the Hund's rule coupling between itinerant and localised electrons. The orbital degeneracy of the eg levels (which are Jahn-Teller active) has been neglected, as have next nearest neighbour hoppings and all Heisenberg super-exchange processes. This minimal model -a single band of tight binding electrons coupled to a background of local moments is for historical reasons known as the "double exchange" (DE) model, and we will refer to its Ferromagnetic ground state as the "double exchange ferromagnet" (DEFM). The model Equation (1) has been widely studied for fifty years, and we shall not pretend here to offer a review of either its diverse physical properties, or of the many ingenious approaches which have been used to study them. Rather, we present a concise introduction to a simple and controlled way of performing a spin wave expansion for such a Hamiltonian, and a brief overview of the effect of quantum and thermal fluctuations on the DEFM. We believe that the methods developed can easily be transferred to other magnetic d-electron oxides.

Spin wave Expansion
In the strong Hund's rule coupling limit t/J → 0, so far as magnetic properties are concerned, it makes little sense to distinguish between localised and itinerant electrons. For example, oxidised Mn atoms in the canonical CMR manganite La1−xCaxMnO3 can exist either as Mn 4+ of Mn 3+ ions, which are respectively spin S=3/2 and spin S=2 local moments. Where the Hund's rule coupling is so strong, it is therefore convenient to work in the local basis of eigenstates of total (i.e. "itinerant" eg plus "localised" t2g electron) spin T = S + s.
Taking this as a starting point, we have derived a controlled large S expansion of Equation (1) which can be applied to any magnetically ordered state. Our original approach [5,6,7,8] was to generalise the Holstein-Primakoff (HP) transformation [4] by allowing the length of the fluctuating spin T to be an operator quantity, and introducing local "up"(f ) and local "down" (p) electron operators such that T = S + (f † f − p † p)/2. A more general derivation of the associated spin and charge algebra based on the Schwinger Boson representation of Penc and Lacaze [9] will be presented elsewhere [10].
Where the operator f † acts on an empty site, it creates a low energy state with total spin T = S + 1/2, while the the operator p † creates a high energy state with total spin T = S − 1/2. All information about low energy spin excitations is encoded in the generalised Holstein-Primakoff bosonsã, so in the limit t/J → 0, we can project out the high energy, low spin states associated with p electrons altogether, and obtain a very simple representation of Equation (1) : where ǫ k is the (tight binding) electron dispersion and v 13 24 the vertex for interaction between band electron (f ) and spin wave modes (ã). For compactness, a further four-boson/twofermion vertex has been dropped. The generalisation of this Hamiltonian to bilayer systems is straightforward [11].
From this starting point it is possible to use a conventional diagrammatic perturbation theory to evaluate spin-wave and electron self energy corrections. The leading zero temperature semi-classical process is the one loop diagram of Figure (1a), which generates a dispersion for the spin waves, and so stabilises the FM ground state.
Quantum and Thermal Effects At a semi-classical level (i.e. considering only the O(1/S) one loop diagrams of Figure (1a and 1b)) the DEFM can be mapped onto an independent set of spin excitations, described by a nearest-neighbour Heisenberg model (see e. g. [3,7]), and a band of tight binding electrons with temperature dependant bandwidth [12]. Once the leading quantum effects are taken into account (the remaining O(1/S 2 ) diagrams of Figure (1)), this mapping breaks down. The spin waves become damped (see Figure (2)), even at zero temperature, and both quantum and thermal fluctuations dynamically generate new non-nearest neighbour couplings between spins, which progressively modify the form of spin wave dispersion (see Figure (3)). Thermal corrections to the overall scale of the spin wave dispersion in the DEFM are broadly similar to those in the Heisenberg FM, but are considerably enhanced relative to the classically equivalent Heisenberg model (see Figure (4)).
At zero temperature, in the limit t/J → 0, the low energy f electron states remain undamped. However at finite temperature they interact through the transverse spin susceptibility, and have a finite lifetime which is controlled by the phase space for spin fluctuations [12]. Conclusions We have shown that it is possible to derive a controlled large S expansion for the Kondo lattice which, when applied to the DEFM in the limit t/J → 0, reduces the model to a set of composite spin modes interacting with spinless electrons. At a semi-classical level spin and charge modes do not interact, but at O(1/S 2 ) interactions are restored and the physics of the system substantially modified. At a qualitative level, the departures from Heisenberg model behaviour caused by quantum and thermal fluctuation effects in the DEFM resemble those seen in inelastic neutron scattering experiments on the CMR manganites. However there remains some doubt that a minimal model of the form Equation (1) is sufficient to fit experimental data for the manganites quantitatively [11].