The 134-type $A$-site ordered ${\mathrm{LaMn}}_3{\mathrm{Cr}}_4{\mathrm{O}}_{12}$, that belongs to the family of $A{A}_3^{\prime }{B}_4{\mathrm{O}}_{12}$ perovskites, with cubic structure and magnetically driven improper polarization is puzzling because the conventional mechanisms leading to the creation of polarization in type-II multiferroics do not apply. By symmetry analysis and first-principles simulations, we discover an original and previously overlooked energetic term that naturally explains the origin of improper polarization in ${\mathrm{LaMn}}_3{\mathrm{Cr}}_4{\mathrm{O}}_{12}$. In addition, we demonstrate the existence of three other energetic invariants in $A{A}_3^{\prime }{B}_4{\mathrm{O}}_{12}$ perovskites, for which other physical quantities (such as oxygen tiltings or shear strain) mediate the interaction between $G$-type antiferromagnetic vectors to create and even enhance improper polarization.

• Received 4 February 2020
• Accepted 11 June 2020

DOI:https://doi.org/10.1103/PhysRevB.101.214441