Symmetry replication and toroidic effects in the multiferroic pyroxene NaFeSi2O6

The magnetoelectric and toroidic effects occurring in NaFeSi2O6 are analyzed theoretically. The symmetry-breaking mechanism giving rise to the incommensurate antiferromagnetic-ferroelectric phase observed below 6 K is shown to be induced by replication of a single transition order parameter. It implies an effective-continuous symmetry of the system identified as the phason rotation of the incommensurate order parameter. The magnetic-field induced toroidal moment is expressed in terms of the macroscopic and spin variables. It shows that the toroidal susceptibility components vary critically as the electric polarization, denoting the inherently magnetoelectric nature of the scaling properties of the ferrotoroidic state. The toroidal moment arising under applied field in antiferromagnetic-ferroelectric structures is induced simultaneously with a spin wave which exhibits the same symmetry properties but spans different degrees of freedom. The difference between microscopic spins degrees of freedom and the electromagnetic macroscopic magnetization and toroidal fields is emphasized.