Rational control of magnonic and electronic band splittings

We provide a theoretical demonstration of controllable nonrelativistic spin splitting in both electronic and magnonic bands via targeted structural distortions tied to specific phonon modes. Using ${\mathrm{MnF}}_{2}$ as a model system, we identify a $d$-wave magnon band splitting between magnon modes of specific handedness, directly correlated with the nonrelativistic spin splitting observed in the electronic structure. Crucially, we show that structural distortions associated with the ${\mathrm{A}}_{2u}$ and ${\mathrm{A}}_{1g}$ phonon modes (8.52 and 9.74 THz) modulate these splittings without altering the antiferromagnetic order. The effect originates from changes in the nonmagnetic ligand environment, highlighting the key role of lattice degrees of freedom in governing spin dynamics. Our findings establish a novel route for structure-mediated control of spin splitting, opening possibilities for tunable magnonic and spintronic functionalities in antiferromagnetic materials.