that has a bilinear coupling with the electrical toroidal moment (see Eqs. (\[eq35a\]), (\[eq38\]), and (\[eq34\])). In the second scenario, the electrotoroidic system is also piezoelectric, and electrical toroidal moment and strain are coupled to each other (see Eqs. (\[eq35b\]), (\[eq39\]), and (\[eq34\])). An example of the latter can be found in Reference [@Prosandeev], where a pure gyrotropic phase transition leading to a piezoelectric, but non-polar, $P2_12_12_1$ state (that exhibits spontaneous electrical toroidal moments) was discovered in a perovskite film. Next, we describe the theoretical prediction of a material where the former scenario is realized.

Prediction and microscopic understanding of gyrotropy in electrotoroidic systems
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The system we have investigated numerically is a nanocomposite made of periodic squared arrays of BaTiO$_3$ nanowires embedded in a matrix formed by (Ba,Sr)TiO$_3$ solid solutions having a 85% Sr composition. The nanowires have a long axis oriented along the \[001\] pseudo-cubic direction (chosen to be the $z$-axis). They possess a squared cross-section of 4.8x4.8 nm$^2$ in the ($x$,$y$) plane, where the $x$- and $y$-axes are chosen along the pseudo-cubic \[100\] and \[010\] directions, respectively. The distance (along the $x$- or $y$-directions) between adjacent BaTiO$_3$ nanowires is 2.4 nm.

We choose this particular nanocomposite system 