Non-magnetic insulating phase induced by Jahn-Teller effect in RNiO$_3$

We propose a three-dimensional multi-orbital tight-binding model for rare-earth nickelates RNiO$_3$ that treats charge, spin, orbital, and lattice degrees of freedom on equal footing. All model parameters, including the on-site interactions $U$ and $J$ and the electron-phonon (el-ph) coupling to the breathing mode, are extracted from hybrid-functional DFT calculations for the small-bandwidth nickelate LuNiO$_3$. The model describes three competing insulating phases governed by the interplay of $U{-}3J$ and el-ph coupling to the breathing and Jahn--Teller (JT) modes. For large $U{-}3J$, the insulating state is stabilized by local JT distortions on high-spin Ni$^{3+}$ sites. For smaller $U{-}3J$, the system undergoes charge disproportionation, $2\mathrm{Ni}^{3+}\rightarrow\mathrm{Ni}^{2+}+\mathrm{Ni}^{4+}$, resulting in the spin-polarized charge-ordered state observed experimentally below the N\'eel temperature in small-bandwidth RNiO$_3$. When the JT energy on the Ni$^{2+}$ site exceeds Hund's exchange $3J$, a distinct charge- and orbital-ordered insulating phase emerges in which the two $e_g$-electrons occupy the same orbital with opposite spin. The stability of this phase is further confirmed by self-consistent calculations within the full three-dimensional tight-binding model. This newly predicted metastable state, characterized by JT distortions in a nonmagnetic charge-ordered RNiO$_3$ phase, shows that the onset of magnetic order is not required for the metal-insulator transition in RNiO$_3$.