Tensorial approach to altermagnetism

I present a tensorial approach to the description of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi mathvariant="bold">k</a:mi><a:mo>/</a:mo><a:mo>–</a:mo><a:mi mathvariant="bold">k</a:mi></a:mrow></a:math>-symmetric, time-reversal-odd splitting of electronic bands in magnetic materials, which can be of nonrelativistic origin and was recently given the name of “altermagnetism”. I demonstrate that tensors provide a general framework to discuss magnetic symmetry using both spin groups and magnetic point groups, which have often been contrasted in recent literature. I also provide a natural classification of altermagnets in terms of the lowest-order tensorial forms that are permitted in each of the 69 altermagnetic point groups. This approach clarifies the connection between altermagnetism and well-known bulk properties, establishing that the vast majority of altermagnetic materials must also be piezomagnetic and MOKE-active, and provides a rational criterion to search for potential altermagnets among known materials and to test them when the magnetic structure is unknown or ambiguous.