Anomalies and unnatural stability of multi-component Luttinger liquids in ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ spin chains

We study translationally invariant spin chains where each unit cell contains an $n$-state projective representation of a ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ internal symmetry, generalizing the spin-1/2 XYZ chain. Such spin chains possess a generalized Lieb-Schulz-Mattis (LSM) constraint, and we demonstrate that certain $(n-1)$-component Luttinger liquids possess the correct anomalies to satisfy these LSM constraints. For $n=3$, using both numerical and analytical approaches, we find that such spin chains with nearest-neighbor interactions appear to be gapless for a wide range of microscopic parameters and described by a two-component conformally invariant Luttinger liquid. This implies the emergence of $n-1$ conserved $U(1)$ charges from only discrete microscopic symmetries. Remarkably, the system remains gapless for an unnaturally large parameter regime despite the apparent existence of symmetry-allowed relevant operators in the field theory. This suggests that either these spin chains have hidden conserved quantities not previously identified, or the parameters of the field theory are simply unnatural due to frustration effects of the lattice Hamiltonian. We argue that similar features are expected to occur in: (1) ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ symmetric chains for $n$ odd, and (2) ${\mathbb{S}}_n\times {\mathbb{Z}}_n$ symmetric chains for all $n>2$. Finally, we suggest the possibility of a lower bound growing with $n$ on the minimum central charge of field theories that possess such LSM anomalies.