Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry
Yoshiko Ogata, Hal Tasaki
TL;DR
For 1D quantum spin chains with half-odd-integer spin $S$, the authors prove Lieb–Schultz–Mattis-type no-go theorems ruling out a unique gapped ground state under translation invariance combined with either time-reversal symmetry or ${\mathbb Z}_2\times{\mathbb Z}_2$ symmetry. They develop a rigorous framework that ties matrix product state intuition to representations of the Cuntz algebra on the right-half-chain via the split property, producing MPS-like intertwiners $s_\mu$ and analyzing projective symmetry representations. The two main results are established by constructing symmetry-implementing antilinear or linear automorphisms $\hat{\Xi}$ and $\hat{R}_j$ on the right-half-chain algebra and showing internal consistency forces a contradiction when $2S$ is odd. This work generalizes LSM-type conclusions beyond continuous symmetries and provides a robust method applicable to broader symmetry classes and decay conditions, with implications for symmetry-protected topological phases in one dimension.
Abstract
We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses time-reversal symmetry or ${\mathbb Z}_2 \times {\mathbb Z}_2$ symmetry (i.e., the symmetry with respect to the $π$ rotations of spins about the three orthogonal axes). The proof is based on the deep analogy between the matrix product state formulation and the representation of the Cuntz algebra in the von Neumann algebra $π({\mathcal A}_{R})''$ constructed from the ground state restricted to the right half-infinite chain.