Eq. (12) is the [*macroscopic*]{} magnetization, and as such, should not depend on the choice of a gauge. Note, however, that Eq. (22) is deduced from Eq. (21) via the annihilation of all the contributions stemming from ${\bf M}_0$. As a result, a specific choice of gauge was made in going from Eq.(21) to Eq.(22), namely the “P-only” gauge discussed in Ref. [@Hirst]. L.L. Hirst, [Reviews of Modern Physics]{}, **69**, 607 (1997). N. A. Spaldin, M. Fiebig, and M. Mostovoy, [ J. Phys. Cond. Mat.]{} **20**, 434203 (2008). S. Prosandeev, I. A. Kornev, L. Bellaiche, [ Phys. Rev. Lett.]{} [**107**]{}, 117602 (2011). L. Louis, I. Kornev, G. Geneste, B. Dkhil, and L. Bellaiche, [ J. Phys. Cond. Mat.]{} [**24**]{}, 402201 (2012).

J. Hlinka, T. Ostapchuk, D. Nuzhnyy, J. Petzelt, P. Kuzel, C. Kadlec, P. Vanek, I. Ponomareva, and L. Bellaiche, Phys. Rev. Lett. [**101**]{}, 167402 (2008).

Note that Eq. (\[eq34\]) involves the [*squares*]{} of the toroidal moment and of the polarization for the coupling interaction between these two physical quantities. As a result, one can easily understand that the presently studied nanocomposite has a ground state that is four times degenerated, due to the fact that the polarization and electrical 