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

Abstract

We analyze the internal symmetries and their anomalies in the Kitaev spin-SS models. Importantly, these models have a lattice version of a \mathbb{Z}_2ℤ2 1-form symmetry, denoted by \mathbb{Z}_2^{[1]}ℤ2[1]. There is also an ordinary 0-form \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^Tℤ2(x)×ℤ2(y)×ℤ2T symmetry, where \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}ℤ2(x)×ℤ2(y) are \piπ spin rotations around two orthogonal axes, and \mathbb{Z}_2^Tℤ2T is the time reversal symmetry. The anomalies associated with the full \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^T×\mathbb{Z}_2^{[1]}ℤ2(x)×ℤ2(y)×ℤ2T×ℤ2[1] symmetry are classified by \mathbb{Z}_2^{17}ℤ217. We find that for S∈\mathbb{Z}S∈ℤ the model is anomaly-free, while for S∈\mathbb{Z}+\frac{1}{2}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-SS 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 \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^Tℤ2(x)×ℤ2(y)×ℤ2T symmetry if S∈\mathbb{Z}+\frac{1}{2}S∈ℤ+12, which implies symmetry-enforced exotic quantum matter. We also discuss the consequences for S∈\mathbb{Z}S∈ℤ.