Bond-Length-Driven Magnetic Transition in Quasi-One-Dimensional CrSb$X_3$ ($X$=S, Se)

Using {\it ab initio} calculations, we investigate the magnetic ground states of quasi-one-dimensional insulating CrSb$X_3$ ($X$ = S, Se) with infinite double-rutile chains. Within conventional band theory, without explicit Coulomb correlations ($U$), we obtain band gaps in close agreement with experiment. Remarkably, we find that the magnetic order is highly sensitive to the Cr-Cr bond length $d_{\rm Cr-Cr}$: increasing the bond length induces a transition from antiferromagnetic to ferromagnetic order at a critical distance $d^c_{\rm Cr-Cr} \approx 3.53 (\pm 0.05)$ \AA. Accordingly, CrSbS$_3$ lies near the transition boundary, whereas CrSbSe$_3$ is robustly ferromagnetic, in good agreement with experiment. Analysis of the exchange interactions reveals that the first-order phase transition is dominated by a sign reversal of the intrachain nearest-neighbor superexchange $J_1$ mediated by chalcogen ions, while the intrachain direct exchange $J_2$ remains ferromagnetic and changes only gradually. This behavior reflects an emergent Bethe-Slater-like behavior driven by competing exchange pathways in a quasi-1D transition-metal system, where the competition between $J_1$ and $J_2$ dictates the magnetic ground state. Besides, the electronic structures of the ground states of each compound are investigated.