Symmetries and anomalies of Kitaev spin-S models: Identifying symmetry-enforced exotic quantum matter

We analyze the internal symmetries and their anomalies in the Kitaev spin- S S models. Importantly, these models have a lattice version of a Z₂ ℤ2 1-form symmetry, denoted by Z₂^1 ℤ21. There is also an ordinary 0-form Z₂^ (x) ×Z₂^ (y) ×Z₂T ℤ2 (x) ×ℤ2 (y) ×ℤ2T symmetry, where Z₂^ (x) ×Z₂^ (y) ℤ2 (x) ×ℤ2 (y) are π spin rotations around two orthogonal axes, and Z₂T ℤ2T is the time reversal symmetry. The anomalies associated with the full Z₂^ (x) ×Z₂^ (y) ×Z₂T×Z₂^1 ℤ2 (x) ×ℤ2 (y) ×ℤ2T×ℤ21 symmetry are classified by Z₂^17 ℤ217. We find that for S∈Z S∈ℤ the model is anomaly-free, while for S∈Z+12 S∈ℤ+12 there is an anomaly purely associated with the 1-form symmetry, but there is no anomaly purely associated with the ordinary symmetry or mixed anomaly between the 0-form and 1-form symmetries. The consequences of these symmetries and anomalies apply to not only the Kitaev spin- S S models, but also any of their perturbed versions, assuming that the perturbations are local and respect the symmetries. If these local perturbations are weak, generically these consequences still apply even if the perturbations break the 1-form symmetry. A notable consequence is that there should generically be a deconfined fermionic excitation carrying no fractional quantum number under the Z₂^ (x) ×Z₂^ (y) ×Z₂T ℤ2 (x) ×ℤ2 (y) ×ℤ2T symmetry if S∈Z+12 S∈ℤ+12, which implies symmetry-enforced exotic quantum matter. We also discuss the consequences for S∈Z S∈ℤ.