Fig. 1: Competing interactions in Pr⁴⁺ oxides.

a Hierarchy of single-ion energy scales for Pr⁴⁺ ions in octahedral oxygen environments starting from a ²F, (S = 1/2, L = 3) free-ion state. For spin-orbit coupling (SOC) as the dominant energy scale (J_eff = 1/2 limit, left, brown), the low-symmetry crystal field (CF) lifts the degeneracy of the ground-state ²F_5/2 and excited ²F_7/2 multiplets into seven Kramers doublets (KDs, with selected radial squared wave-functions represented). For approximate O_h symmetry, the Γ₇ doublet, in the \(\left|J,{m}_{J}\right\rangle\) basis, is given by \({\Gamma }_{7}^{\pm }=\sin \theta \left|\frac{5}{2},\pm \frac{5}{2}\right\rangle+\cos \theta \left|\frac{5}{2},\mp \frac{3}{2}\right\rangle\), where \({\sin }^{2}\theta \, \approx \, 1/6\) (See SI). For CF as the dominant energy scale (S_eff = 1/2 limit, right, blue), SOC and distortion from O_h symmetry lift the ²A_2u ground state and the triply degenerate ²T_2u and ²T_1u excited states into seven KDs. The ground-state doublet is given in the \(\left|{m}_{l},{m}_{s}\right\rangle\) basis by \(\left|\pm \right\rangle=A\,\left|\mp 3,\pm \frac{1}{2}\right\rangle -\,B\,\left|\mp 2,\mp \frac{1}{2}\right\rangle \,+\,C\,\left|\pm 1,\pm \frac{1}{2}\right\rangle -\,D\,\left|\pm 2,\mp \frac{1}{2}\right\rangle\), where \({({A}^{2}/{B}^{2})}^{{\Gamma }_{7}} \, \approx \, 2.6\) and \({({C}^{2}/{D}^{2})}^{{\Gamma }_{7}} \, \approx \, 0.33\) for the \(\left|{\Gamma }_{7}\right\rangle\) doublet⁶² (See SI). In the intermediate limit, the competition between Δ_CF and ζ_SOC scales in Pr⁴⁺ yields seven KD (indicated by the mix of brown/blue lines) with magnetic properties that are distinct from the J_eff = 1/2 and S_eff = 1/2 limits, such as large magnetic super-exchange achievable by tuning the ligand field. b–d Crystal structure, magnetic lattice dimensionality, and Pr⁴⁺ coordination environment for the oxides studied in this work: Na₂PrO₃ (2-Pr), Sr₂PrO₄ (1-Pr), and Li₈PrO₆ (0-Pr), respectively.