Paolo Radaelli
In the past two decades, there has been a resurgence of interest in compounds having electronic bands with lifted spin degeneracy, partly motivated by the requirement of new materials for spintronics. In addition to spin polarisation in ferromagnets, it is well known that spin degeneracy can be lifted even in non-magnetic materials by the famous Rashba- Dresselhaus (R-D) effect, which requires spin-orbit coupling (SOC) and is therefore largest in the presence of heavy elements. The R-D effect also requires the absence of inversion symmetry, due either to the bulk crystal structure being acentric or to symmetry breaking at interfaces. More recently, several groups came to the rather surprising realisation that spin degeneracy can also be lifted in some fully compensated antiferromagnets (AFM), including collinear AFM. In some cases, DFT calculations demonstrated that this splitting persists even when spin-orbit coupling (SOC) is removed, due to the interaction between electron spins and the ‘effective Zeeman field’ (largely of magnetic exchange origin) produced by ordered magnetic moments. In the specific context of collinear antiferromagnets, this phenomenon has been named ‘altermagnetism’, though this type of spin splitting is also found in non-collinear systems. Whereas altermagnetic textures are k/ − k-symmetric and time-reversal-odd, certain non-collinear magnetic structures were shown to display k/ − k-anti-symmetric, time-reversal-even textures – a phenomenon now known as anti-altermagnetism.
I will present a general approach to construct k/ − k-symmetric, time-reversal-odd spin textures based on a multipolar tensor expansion, which complies with the full magnetic symmetry of the crystal [1,2]. I will further demonstrate that these spin textures decompose in an ‘altermagnetic-like’ (SOC-independent) component, which is invariant by rotation in spin space and is described by the spin-group or colour-group formalisms, and a SOC-dependent component that depends on spin orientation, and is described by the more familiar magnetic point groups, which also determine macroscopic properties. I will also outline an extension of this procedure to deal with “anti-altermagnetic”, k/ − k-antisymmetric, time-reversal-even spin textures (including but not limited to so-called p-wave textures). I will present a clear explanation of the role of magnetic point groups, spin groups and colour groups in the description and classification of these textures.
References†
[1] Radaelli, P. G. Tensorial approach to altermagnetism. Phys. Rev. B 110, 214428 (2024).
[2] P.G. Radaelli & G. Gurung “Color symmetry and altermagneticlike spin textures in noncollinear antiferromagnets”, Phys. Rev. B 112, 014431 (2025)