Spin-cavity interactions in relativistic Jahn-Teller systems under strong light-matter coupling

We extend our recent work on the cavity-modified spin Zeeman effect of an effective spin-1/2-system[J. Chem. Phys. 163, 174307 (2025)] to a relativistic Jahn-Teller scenario under strong light-matter coupling. Here, the effective spin-1/2-system is realized via a single electron or a single hole in a doubly-degenerate molecular orbital system of trigonal symmetric transition metal complexes. Both single-particle and single-hole systems are subject to both vibronic and spin-orbit coupling (SOC) augmented by interactions with a quantized cavity field via the cavity Zeeman interaction. Methodologically, we combine the relativistic $E\times e$-Jahn-Teller model with a recently introduced effective Hamiltonian formalism based on quasi-degenerate perturbation theory, which treats the cavity-spin interaction in leading order beyond the dipole approximation. We derive analytic expressions for Kramers pair energies in weak and strong SOC regimes as well as related cavity-modified effective electronic g-factors. We find cavity-induced modifications of the electronic g-factor to become relevant in the weak SOC regime for both single-particle and single-hole systems while being effectively quenched under strong SOC. Alternating signs of the cavity-Zeeman correction render single-particle and single-hole scenarios distinct in their response to the cavity field from a g-factor perspective.