(\[eq35a\]) and (\[eq38\]) for the studied case, that is: $$\label{eqnew}
g_{11} =-\frac{4\pi \omega }{c}\frac{dG_3 }{dE_3
}=
-\frac{4\pi \omega }{c} \chi^{(G)}_{3n}\frac{\partial h_n}{\partial
P_l}\chi^{(P)}_{l3}=-\frac{4\pi \omega }{c} (\lambda_{n3l3} +\lambda _{n33l}+\lambda_{3nl3} +\lambda _{3n3l}) \chi^{(G)}_{3n} G_3 P_3
 \chi^{(P)}_{l3}$$

The usual temperature dependencies of the order parameter and its conjugate field imply that $G_3$ and $P_3$ should be proportional to $\sqrt{(T_G-T)}$ and $\sqrt{(T_C-T)}$, respectively, while their responses, $ \chi^{(G)}_{3n}$ and $\chi^{(P)}_{l3}$, should be proportional to $1/(T_G-T)$ and $1/(T_C-T)$, respectively. This explains why the behavior of $g_{11}$ as a function of $T$ is well described by $A/\sqrt{(T_C-T)(T_G-T)}$.

Summary
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In summary, we propose the existence of a new class of spontaneously optically active materials, via the use of different techniques (namely, analytical derivations, phenomenologies and first-principles-based simulations). These materials are electrotoroidics for which the electric toroidal moment changes linearly under an applied electric field. Such linear change is demonstrated to occur if at least one of the following two conditions is satisfied: (1) the electric toroidal moment is coupled to a spontaneous electrical polarization; or (2) the electric toroidal moment is coupled to strain and the whole system is piezoelectric. We also report a realization of case (1), and further show that applying an electric field in 