Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry

TL;DR

This paper proves that quantum spin chains with half-odd-integer spins cannot have a unique gapped ground state if they are short-ranged, translation-invariant, and possess certain symmetries, extending Lieb-Schultz-Mattis theorems.

Contribution

It establishes new constraints on the ground states of quantum spin chains without continuous symmetry, using a novel approach based on matrix product states and operator algebra.

Findings

Half-odd-integer spin chains cannot have unique gapped ground states under specified symmetries.

The proof leverages the analogy between matrix product states and Cuntz algebra representations.

The result generalizes Lieb-Schultz-Mattis theorems to a broader class of quantum spin systems.

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 $\pi$ 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 $\pi({\mathcal A}_{R})''$ constructed from the ground state restricted to the right half-infinite chain.