Is the Heisenberg model in magnetism applicable only for direct exchange or for all the types of exchange interaction to dertermine the magnetic nature of any magnetic compound?
Heisenberg model of exchange interactions, usually written in form E = J S1.S2, applies quite well to situation when the two spins S1 and S2 are nearest neighbors. But even in perfect crystal lattice the more precise form of this interaction should be written as E = S1J S2, with J being a tensor rather than a scalar, with symmetry properties corresponding to those of host crystal lattice. Yet in conductive samples we have to take into account the exchange interactions "transmitted" over pretty large distances by itinerant charge carriers, i.e. electrons. This is called RKKY interaction. On the other hand, there are samples well described by still simplified Heisenberg model, taking into account only z-components of both spins (a.k.a. known as Ising model).
Unfortunately the Heisenberg model only has an analytical solution in one dimension using the Bethe ansatz. But numerically is a quite successful model for distinguishing, for instance, ferro to antiferromagnetic phases using statistics. But this model needs to assume a mean field approach on each spin besides their exchange coupling and implying the spins to be localized in the lattice (thus it is quite bad for metal phases or itinerant magnetism)
Thank you Mr Marek Wojciech Gutowski and Mr Daniel Baldomir for your answers. To be more expliscite, Can we apply the Heisenberg model to the superexchange or indirect exchange since it is bad to apply it for itinerant exchange.
The superexchange (or the double exchange), in general it is not possible to apply the Heisenberg model due to the hopping term. The usual is to employ a t-J (or a Hubbard) model directly. Only in the case that you have a a quite high Coulomb term U and half filling you can get an equivalent Heisenberg model from a t-J model where the exchange coupling constant J has the value
J=4t2/U
The Heisenberg Hamiltonian is usually restricted to localised spins interacting via short range magnetic couplings, it works well in insulating materials. It is true that in general it should not work in this form in itinerant systems...but in the case of localised spins interacting with itinerant carriers as for instance in manganites or diluted magnetic semi-conductors , the strategy consists in integrating out the carrier degrees of freedom, to derivate an effective Heisenberg model from which magnetisation, Tc, magnetic excitation spectrum can be calculated..see for instance the ref
Physical Review B 76 (2), 020401 (2007) and Europhysics Letters 92 (4), 47006 (2010) for instance for manganites and diluted magnetic semiconductors respectively.