The classical-quantum disproportionation transition and magnetic ordering in RNiO$_3$ nickelates

The insulator-quasi-metal (bad metal) transition observed in Jahn-Teller (JT) magnets orthonickelates RNiO$_3$ (R = rare earth, or yttrium Y) is considered a canonical example of the Mott transition, traditionally described in the framework of Hubbard's $U-t$ model. However, in reality, the insulating phase of nickelates is the result of charge disproportionation (CD) with the formation of a system of spin-triplet ($S = 1$) electron [NiO$_6$]$^{10-}$ and spinless ($S = 0$) hole [NiO$_6$]$^{8-}$ centers, equivalent to a system of effective spin-triplet composite bosons moving in a nonmagnetic lattice. The effective CD-phase Hamiltonian takes into account local ($U$) and nonlocal ($V$) correlations, and the transfer of composite bosons ($t_b$). Within the framework of the effective field approximation, we have shown the existence of two types of CD phases: the high-temperature classical paramagnetic CO-phase of charge ordering of electron and hole centers, and the low-temperature magnetic quantum CDq phase with charge and spin density transfer between electron and hole centers, with ''uncertain valence'' [NiO$_{6}$]$^{(9\pm\delta)-}$ ($0 \le \delta \le 1$) and spin density $(1 \pm \delta)/2$ NiO$_6$-centers. In the classical CO phase, spin-triplet electron centers are surrounded by the nearest nonmagnetic hole centers, which ''turns off'' the strong superexchange interaction of the nearest neighbors. The magnetic ordering in the quantum CDq phase is determined by a strong traditional superexchange and an unusual bosonic double exchange mechanism.