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

Understanding and identifying various phases and transitions among them is an essential topic in condensed matter physics, where non-perturbative results are powerful. As an important topic, Lieb-Schultz-Mattis (LSM) theorem gives general but strict constraints on quantum phases about ingappability --- a translation-symmetric lattice system cannot have a unique gapped ground state under some symmetry condition¹. In this work, we derive a stronger and rigorous statement than LSM theorem in the sense that the LSM ingappability is a lemma following our result. We prove that, in U(1)*Z_N-symmetric symmetry-protected topological phase (SPT) in one dimension², the ground-state expectation value of LSM twisting operator is a topological-phase order parameter --- it is quantized in the thermodynamic limit and cannot be changed under adiabatic deformation within one SPT phase. This (non-local) order parameter is valued in N-th unit root and will be changed by a lattice translation, which generates LSM ingappability for SU(N) spins³ if we further impose translation symmetry. Furthermore, our rigorous result for SPT phases can also produce a large number of LSM ingappabilities where the translation symmetry is even replaced by antiunitary symmetry, e.g., magnetic translations.

[1] E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961).

[2] F. Pollmann, E. Berg, A.M. Turner, and M. Oshikawa, Phys. Rev. B 85, 075125 (2012).

[3] I. Affleck, and E. Lieb, Lett. Math. Phys. 12, 57 (1986).