Spin symmetries are crystalline symmetries such as discrete rotations, mirrors and translations where the transformations acting on spin space are not necessarily the same as those acting on real space. These generalize standard magnetic symmetries where the spin space and real space transformations are locked to one another. Spin symmetries are most evidently relevant to condensed matter physics when there is a separation of scales between the effects of time reversal symmetry breaking at zero spin-orbit coupling and the spin-orbit coupling/crystal field scales.

In this talk, I plan to discuss systems where spin symmetries play an essential role in determining the physics. These include magnetic excitations in collinear Heisenberg magnets including experimentally relevant cases. Group theory helps to clarify the nature of the resulting magnons. I also discuss cases of anisotropic exchange where there are residual spin symmetries that can then survive far from the zero spin-orbit coupling limit. I plan to spend a significant amount of time explaining Landau theory adapted to spin groups and, in particular, their application to altermagnets. From this perspective, altermagnets are compensated magnets where the staggered magnetization (or its generalizations) transforms under a nontrivial irreducible representation of the point group of the lattice. This leads to the conclusion that altermagnets have completely constrained secondary multipolar order parameters that are time reversal odd, spin symmetric and spatially anisotropic. Landau theory provides a powerful tool to explore the phenomenology of these systems.

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