Anomaly matching and symmetry-protected critical phases in SU(N) spin systems in 1+1 dimensions

We study (1+1)-dimensional SU(N) spin systems in the presence of the global SU(N) rotation and lattice translation symmetries. By matching the mixed anomaly of the PSU(N)×Z in the continuum limit, we identify a topological index for spin model evaluated as the total number of Young-tableaux boxes of spins per unit cell modulo N, which characterizes the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be regarded as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase (symmetry-protected critical phase) -- only a class of SU(N) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our prediction agrees with several examples known in previous studies of SU(N) models.