Interplay between trimerization, ferroelectric, and magnetic order in the hexagonal manganites

The hexagonal manganites h-RMnO$_3$ with R = Sc, Y, In, Dy -- Lu are a model system for multiferroics with pronounced magnetoelectric effects in which the magnetic and ferroelectric order emerge independently (so-called split-order-parameter or type-I multiferroics). In spite of many years of intense investigations the system never seizes to surprise us with novel, unexpected manifestations of its five-fold long-range order (antiferrodistortive, ferroelectric, antiferromagnetic Mn order, and rare-earth order on the 2a and 4b sites). Here I will discuss several such examples: (i) ``Incompatible'' magnetic order of the Mn and rare-earth sublattices according to different symmetry representations in combination with triggered ordering at the 4b site. (ii) The {it absence} of ferroelectric order in h-InMnO$_3$ down to low temperatures in spite of its apparent similarities to, in particular, h-YMnO$_3$. (iii) Observation of a direct and rigid coupling of the ferroelectric to the antiferrodistortive order in annealing experiments at 1300 K. In summary, all these experiments allow us to present a comprehensive model for the microscopic origin of the ferroelectric and the multiple magnetic (re-) ordering in this important group of compounds.