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We discuss the thermodynamic and magnetic properties of a competinghigh-spin/low-spin two-orbital two-electron model on a square-lattice possiblyrelevant to the nickelates such as Ba$_2$NiO$_2$(AgSe)$_2$ (BNOAS). We focus onparameter regimes where a high-spin ($S=1$) and a low-spin ($S=0$) state areenergetically close to each other and discuss various exchange processes insuch a system. The model we study is a variant of, but different from theKondo-necklace model proposed for the system by Jin et al (H. S. Jin, W. E.Pickett and K. W. Lee, Phys. Rev. Research 2, 033197 (2020)). Although thereare similarities between the two models in terms of the ground-state phases andtheir symmetries, the detailed properties of the different phases and the phasetransitions are entirely different and should be easy to distinguish from anexperimental point of view.

Magnetism of competing high-spin/low-spin states in Ba$_2$NiO$_2$(AgSe)$_2$ and related two-orbital two-electron systems