Symmetry constraints on the electrical polarization in multiferroic materials

The symmetry conditions for the development of a macroscopic electrical polarization as a secondary order parameter to a magnetic ordering transition and the constraints on the direction of the polarization vector are determined by a nonconventional application of the theory of irreducible corepresentations. In our approach, which is suitable for both magnetic and structural modulations, antiunitary operators are employed to describe symmetry operations that exchange the propagation vector $\mathbf{k}$ with $-\mathbf{k}$, rather than operations combined with time reversal as in classical corepresentation analysis. Unlike the conventional irreducible representations, corepresentations can capture the full symmetry properties of the system even if the propagation vector is in the interior of the Brillouin zone. It is shown that ferroelectricity can develop even for a completely collinear structure, and that helical and cycloidal magnetic structures are not always polar. In some cases, symmetry allows the development of polarization parallel to the magnetic propagation vector. Our analysis also highlights the unique importance of magnetic commensurability, enabling one to derive the different symmetry properties of equivalent commensurate and incommensurate phases even for a completely generic propagation vector.