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 global SU(N) rotation and lattice translation symmetries. Knowing the mixed anomaly of the two symmetries at low energy, we identify, by the anomaly matching argument, a topological index for the spin model-the total number of Young-tableau boxes of spins per unit cell modulo N-characterizing the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases. As an example of the latter case, we show that 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 results agree with several examples known in previous studies of SU(N) models, and predict a general constraint on the structure factor which is measurable in experiments.