Ferroelectricity in the multiferroic hexagonal manganites

Since their discovery in 1963 the hexagonal manganites have consolidated their role as exotic ferroelectrics with astonishing functionalities. Their introduction as room-temperature device ferroelectrics was followed by observations of giant flexoelectricity, multiferroicity with magnetoelectric domain and domain-wall coupling, protected vortex domain structures, topological domain-scaling behaviour and domain walls with tunable conductance and magnetism. Even after half a century, however, the emergence of the ferroelectric state has remained the subject of fierce debate. We resolve the interplay of electric polarization, topological trimerization and temperature by direct access to the polarization for temperatures up to 1,400 K. Nonlinear optical experiments and piezoresponse force microscopy, complemented by Monte Carlo simulations, reveal a single phase transition with ferroelectricity determined by topology rather than electrostatics. Fundamental properties of the hexagonal manganites, including an explanation for the two-phase-transition controversy as a finite-size scaling effect, are derived from this and highlight why improper ferroelectrics are an inherent source of novel functionalities.