I am looking for some understanding of which on-site symmetries in realistic crystalline materials (i.e. not just in random lattice models) can plausibly be expected to be realized and to induce symmetry protected/enriched topological phases (let me abbreviate "SPT" also for the "enhanced" case).
Specifically, I am wondering if finite cyclic groups $\mathbb{Z}/n$ for $n \geq 3$ can be expected, in principle, to show up as "protecting groups" of laboratory SPTs, at least in some approximation.
Even more specifically, I am wondering if it is plausible that $\mathbb{Z}/n$-SPTs may exist where the cyclic group acts as a subgroup $\mathbb{Z}/n \subset Spin(3) \simeq SU(2)$ of the group of on-site electron spin rotations.
(Feel free to read this as $\mathbb{Z}/n \subset SO(3)$ acting projectively.)
If not, is there some other generic way to (expect to) find on-site $\mathbb{Z}/n$-SPTs in realistic materials, for $n \geq 3$?
But if this is at all the case, how about the other finite subgroups of SU(2)/SO(3)? Is it plausible at all to have a real crystalline topological phase protected by any old finite subgroup of its on-site electron-spin-rotation group? In other words, can we expect an ADE-classification of types of on-site-SPTs, protected by finite subgroups of the on-site spin rotation?
(I am aware of most of the literature on on-site SPTs -- I think -- but most of it currently is concerned with more abstract questions of classifications, while tending to be a little shy about considering plausibly realistic examples of on-site SPT beyond the small $\mathbb{Z}/2$-symmetry and the full $SU(2)$ or $SO(3)$-symmetry of spin rotations.)
