Identification of gapless phases by a twisting operator

TL;DR

This paper establishes a necessary condition for gapped SO(3)-symmetric spin chains, linking the ground state to a spin singlet and the twisting operator's expectation value approaching unity, aiding in identifying gapless phases.

Contribution

It provides a rigorous theorem connecting the ground state properties and twisting operator behavior to the presence of gaps in SO(3)-symmetric spin chains, offering a new criterion for phase identification.

Findings

Ground states of gapped chains are spin singlets.

Twisting operator expectation approaches one in gapped phases.

Numerical verification across various models supports the theorem.

Abstract

We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases, and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.