if there are other necessary conditions, in addition to the existence of an electrical toroidal moment, for such materials to be optically active.

In this work, we carry out analytical derivations, original phenomenologies and first-principles-based computations that successfully address all the aforementioned important issues. In particular, we find that electrotoroidic materials do possess spontaneous optical activity, but only if their electric toroidal moment changes [*linearly*]{} under an applied electric field. This linear dependence is further proved to occur if the electrotoroidic materials also possess a spontaneous electrical polarization that is coupled to the electric toroidal moment, or if they are also piezoelectric with the strain affecting the value of the electric toroidal moment. We also find that, in the former case, the applied electric field further allows the control of the sign of the optical activity. Our atomistic approach also reveals the evolution of the microstructure leading to the occurrence of field-switchable gyrotropy, and shows that the optical rotatory strength can be significant in some electrotoroidic systems.

Relation between gyrotropy and electrical toroidal moment in electrotoroidic systems
====================================================================================

Let us first recall that the gyrotropy tensor elements, $g_{ml}$, are defined via [@Landau]: $$\label{eq74}
g_{mk}=\frac{\omega }{2c}e_{ijm}\gamma _{ijk}$$ where $e_{ijm}$ is the 