Collinear Magnetic Multipole Hamiltonians: Unveiling Additional Splitting in Altermagnets and Beyond

The flourishing exploration into new antiferromagnetic materials with surprising spin splitting has introduced the potential to design spintronic devices with advantageous spin transport properties. Anisotropies in hopping integrals between distinct orbital states are established mechanisms for non-relativistic spin splitting (NRSS) in electronic bands. However, when it comes to magnon dispersion, many studies on altermagnets often overlook additional symmetry breaking in the exchange between orbitally-projected spin density matrices. This study investigates orbitally-decomposed exchange in bismuth ferrite (BFO), where the antiferrodistortive (AFD) mode provides a toggle for enabling these additional NRSS effects, teasing apart detailed contributions from varying orbital and multipolar interactions.

Using a first-principles framework based on the magnetic force theorem, we show that changes in crystal environments induce broken symmetries in the exchange interaction between multipolar orders, with continuous dependence on the AFD mode in BFO. Spin wave excitations from this Hamiltonian enhance our understanding of how multipolar effects manifest in magnon dispersion and pinpoint order parameters. Our orbitally resolved methodology aims to connect this theory to experimentally observable signatures of magnetic multipolar orders, with potential implications for the design of advanced spintronic devices.