Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities

We show that the ground-state expectation value of twisting operator is a topological order parameter for $\text{U}(1)$- and $\mathbb{Z}_{N}$-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems -- it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (non-local) order parameter must take values in $N$-th roots of unity, and its value can be changed by a generalized lattice translation acting as an $N$-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis ingappability for SU($N$) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the $N$-ality property to provide an efficient way to construct possible multi-critical phase transitions starting from a single Hamiltonian with a unique gapped ground state.