Spin Space Groups: Full Classification and Applications

Understand

In this work, we exhaust all the spin-space symmetries, which fully characterize collinear, non-collinear, commensurate, and incommensurate spiral magnetism, and investigate enriched features of electronic bands that respect these symmetries.

• We achieve this by systematically classifying the so-called spin space groups (SSGs) - joint symmetry groups of spatial and spin operations that leave the magnetic structure unchanged.
• Generally speaking, they are accurate (approximate) symmetries in systems where spin-orbit coupling (SOC) is negligible (finite but weaker than the interested energy scale); but we also show that specific SSGs could remain valid even in the presence of a strong SOC.
• By representing the SSGs as O($N$) representations, we - for the first time - obtain the complete classifications of 1421, 9542, and 56512 distinct SSGs for collinear ($N=1$), coplanar ($N=2$), and non-coplanar ($N=3$) magnetism, respectively.