  The aim of this work is to unravel a basic microscopic picture behind complex
magnetic properties of hexagonal manganites. For these purposes, we consider
two characteristic compounds: YMnO3 and LuMnO3, which form different magnetic
structures in the ground state. First, we establish an electronic low-energy
model, which describes the behavior of the Mn 3d bands of YMnO3 and LuMnO3, and
derive parameters of this model from the first-principles calculations. From
the solution of this model, we conclude that, despite strong frustration
effects in the hexagonal lattice, the relativistic spin-orbit interactions lift
the degeneracy of the magnetic ground state so that the experimentally observed
magnetic structures are successfully reproduced by the low-energy model. Then,
we analyze this result in terms of interatomic magnetic interactions, which
were computed using different approximations (starting from the model
Hamiltonian as well as directly from the first-principles electronic structure
calculations in the local-spin-density approximation). We argue that the main
reason why YMnO3 and LuMnO3 tend to form different magnetic structures is
related to the behavior of the single-ion anisotropy, which reflects the
directional dependence of the lattice distortion: namely, the expansion and
contraction of the Mn-trimers, which take place in YMnO3 and LuMnO3,
respectively. On the other hand, the magnetic coupling between the haxagonal
planes is controlled by the next-nearest-neighbor interactions, which are less
sensitive to the direction of the trimerization. Finally, using the Berry-phase
formalism, we evaluate the magnetic-state dependence of the ferroelectric
polarization, and discuss potential applications of the latter in
magnetoelectric switching phenomena.
