Reinvestigating Electronic-Structural Interplays in LaMnO$_3$ from Canonical Jahn-Teller Distortion Notations

LaMnO3 is considered as a prototypical Jahn-Teller perovskite compound, exhibiting a metal to insulator transition at $T_{JT} = 750K$ related to the joint appearance of an electronic orbital ordering and a large lattice Jahn-Teller distortion. From first-principles, we revisit the behavior of LaMnO$_3$ and show that is not only prone to orbital ordering but also to charge ordering. Both charge and orbital orderings appear to be enabled by rotations of the oxygen octahedra and the subtle competition between them is monitored by a large tetragonal compressive strain, that is itself a Jahn-Teller active distortion. Equally, the competition of ferromagnetic and antiferromagnetic orders is slave of the same tetragonal strain. Our results further indicate that the metal to insulator transition can be thought as a Peierls transition. It also questions the applicability of the Kugel-Khomskii model and the cooperative Jahn-Teller Effect to LaMnO$_3$. As a basis to our discussion, we make the inventory of - and introduce canonical notations for - lattice distortions in perovskites deforming the oxygen octhedra and are connected to charge and orbital orderings.