Identification of gapless phases by a twisting operator

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.