Symmetry replication and toroidic effects in the multiferroic pyroxene ${\text{NaFeSi}}_2{\text{O}}_6$

The magnetoelectric and toroidic effects occurring in ${\text{NaFeSi}}_2{\text{O}}_6$ 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.

Figure 1

Symmetry replication mechanism. (a) In the parent high-symmetry phase, the vector order parameter vanishes, and the continuous symmetry group is $\infty .m$ $\left(C_{\infty \text{v}}\right)$. When only one vector order parameter (b) or two collinear order parameters (c) are involved, a single ordered phase is stabilized which possesses the symmetry $m$ generated by the reflexion $\sigma$. By contrast, two dephased noncollinear order parameters (d) stabilize an additional vector state having the lowest-symmetry group 1, reduced to the identity.

Figure 2

Position of the Fe atoms in the conventional monoclinic unit-cell of the paramagnetic structure of NFS. Atoms 1, 2, 3 and 4 are located, respectively, at $\left(0,1-\delta ,1/4\right)$, $\left(0,\delta ,3/4\right)$, (1/2,1/2-$\left(1/2,1/2-\delta ,1/4\right)$, and $\left(1/2,1/2+\delta ,3/4\right)$.

Figure 3

Phase diagrams associated with the free energy $F_1$ [Eq. (3)] for (a) $a_3<0$ and (b) $a_3>0$. All curves correspond to second-order transitions. Phases II and III are respectively observed in NFS below $T_N$ and above a threshold magnetic field $B_{th}$.

Figure 4

Connection between the magnetic point groups of phases I-to-V, obtained by minimization of ${\text{F}}_1$, and equilibrium conditions fulfilled by the coupled order-parameter for each phase.

Figure 5

Phase diagram associated with the free energy $F_2$ [Eq. (4)]. The curves correspond to second-order transitions. Phase ${\text{II}}^{\prime }$ is observed below $T_F$ in NFS.

Figure 6

Field dependence of the magetotoroidal susceptibility. (a) When the ${\text{II}}^{\prime }\text{-III}$ transition is first order. (b) When there is a sequence, ${\text{II}}^{\prime }\text{-II-III}$, of second-order transitions.

Figure 7

Calculated orientations of the Fe spins in the monoclinic unit cells of phases II, ${\text{II}}^{\prime }$, and III at zero field.

Figure 8

Orientations of the Fe spins giving rise to a nonzero toroidal moment, deduced from Eqs. (17, 21): (a) when no additional restriction is put on the symmetry; (b) when the symmetry groups keeps a twofold axis parallel to $\vec{b}$.