}\delta _{mk} } \right) +\frac{1}{4} e_{ijm}\left( \frac{dQ_{kj}}{dE_i}-\frac{dQ_{ki}}{dE_j} \right) \right]\end{aligned}$$

Choosing a specific gauge [@gauge] and neglecting quadrupole moments (simulations reported below show that spontaneous and field-induced quadrupole moments can be neglected for the ferrotoroidics numerically studied in Section IV) lead to the reduction of Eq. (\[eq33before\]) to: $$\label{eq33}
g_{mk} =\frac{4\pi \omega }{c}\left[ \left( {\frac{dG_m }{dE_k }-\frac{dG_l }{dE_l
}\delta _{mk} } \right)  \right]$$

This formula nicely reveals that optical activity should happen when electrical toroidal moment [*linearly*]{} responds to an applied electric field.

Necessary conditions for gyrotropy in electrotoroidic systems
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According to Eq. (\[eq33\]), an electrotoroidic system possessing non-vanishing derivatives of its electrical toroidal moment with respect to the electric field automatically possesses natural optical activity. Let us now prove analytically that the occurrence of such non-vanishing derivatives requires additional symmetry breaking in electrotoroidic systems, namely that an electrical polarization or/and piezoelectricity should also exist, as well as couplings between electrical toroidal moment and electric polarization and/or strain.

For that, let us express the free energy of an electrotoroidic system that exhibits couplings between electrical toroidal moment **G**, polarization **P**, and strain $\eta$ as: $$\label{eq34}
F=F_0+\zeta_{ijkl}G_iG_j\eta_{kl}+\lambda_{ijkl}G_iG_jP_kP_l+q_{ijkl}P_iP_j\eta_{kl}-h_iG_i$$ where $h_i=( {\bm \nabla}\times \,{\bf E})_i$ is the field conjugate of $G_i$.

The equilibrium 