Structurally triggered orbital and charge orderings in TlMnO$_3$ and related compounds

Rare earth perovskites ($R^{3+}$M$^{3+}$O$_3$), with $e_g^1$ electronic occupation of the M $d$ states, display different types of metal-insulator transition. For manganites (M=Mn), metal-insulator transition is usually induced by the Jahn-Teller ($JT$) distortions, which stabilize orbital orderings (OO) at Mn sites. Among them, LaMnO$_3$ shows a $C$ type OO and crystallizes with $Pbnm$ structure. Whereas, TlMnO$_3$ shows a very distinct $G$ type OO with an unusual $P\overline{1}$ structure. Employing first principles calculations, and symmetry mode analysis we rationalize structural and electronic origin of $G$-type OO in TlMnO$_3$. Going further, we consider nickelates (M=Ni), where metal-insulator transition is driven by a breathing distortion, which stabilizes the charge ordering (CO) at Ni sites. Interestingly, different $JT$ and breathing distortions are very similar MO$_6$ octahedral distortions and stem from high frequency phonon modes of ideal $Pm\overline3m$ structure. Our comparative study reveals that following a common triggering mechanism these modes appear in their respective ground states.