We derive and study a model of square planar, $d^8$ spin crossover materials that treats elastic, magnetic, and spin-orbit interactions on an equal footing. For 1D chains density matrix renormalization group calculations show that the competition between these interactions leads to six different phases. For weak spin-orbit coupling (SOC) and large antiferromagnetic interactions we find a symmetry-protected topological (SPT) Haldane phase. Moderate SOC introduces dynamical low-spin (LS) impurities that rapidly destroy the SPT state and drive the system into a topologically trivial high-spin (THS) phase. For strong SOC the $S^z=\pm 1$ HS states are high-energy excitations. Thus, the system can be understood as a transverse field Ising model with the SOC playing the role of the transverse field. Consistent with this, we find a quantum phase transition between the THS phase and a quantum disordered (QD) phase. However, if the magnetic coupling is nonzero or the HS and LS states of a single molecule are nondegenerate, the $Z_2$ (Ising) symmetry is broken and the phase transition becomes a crossover. Thus, the QD phase and the THS phases are adiabatically connected, as, equivalently, are the large-$D$ phase of the spin-1 Heisenberg model and the QD phase of the transverse field Ising model. We also find a ferroelastic LS phase, an antiferroelastic phase, with alternating HS and LS complexes, and a dimer phase, which results from the competition between antiferromagnetic and antiferroelastic interactions.

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• Density matrix renormalization group
• Exchange interaction
• Magnetic interactions
• Molecular magnetism
• Order parameters
• Phase diagrams
• Quantum phase transitions
• Spin-orbit coupling
• Symmetry protected topological states