Hallmarks of electric and magnetic multipole orders in the band structure of variations of diamond and lonsdaleite

We have developed a comprehensive theoretical framework for electric and magnetic multipole order in compounds with crystal structures belonging to the diamond and lonsdaleite families. Both group-theoretical and microscopic tight-binding methods are employed to study the electronic band structure and identify key features enabled by symmetry-lowering electric and magnetic multipoles. Juxtaposing the effects of electric and magnetic multipoles of the same order, we establish dualities between crystal symmetries and their observable signatures. Our investigation points the way for how to systematically realize electric and magnetic multipole order under the umbrella of the diamond and lonsdaleite materials classes and it provides a rational structure for the range of currently known types of spin splittings (Rashba, Dresselhaus, Zeeman, AFM-induced, ...) that may arise due to these multipoles. We also clarify the conditions under which axial- and polar-toroidal order emerges within this material platform.