  We discuss the polarization amplitude of quantum spin systems in one
dimension. In particular, we closely investigate it in gapless phases of those
systems based on the two-dimensional conformal field theory. The polarization
amplitude is defined as the ground-state average of a twist operator which
induces a large gauge transformation attaching the unit amount of the U(1) flux
to the system. We show that the polarization amplitude under the periodic
boundary condition is sensitive to perturbations around the fixed point of the
renormalization-group flow rather than the fixed point itself even when the
perturbation is irrelevant. This dependence is encoded into the scaling law
with respect to the system size. In this paper, we show how and why the scaling
law of the polarization amplitude encodes the information of the
renormalization-group flow. In addition, we show that the polarization
amplitude under the antiperiodic boundary condition is determined fully by the
fixed point in contrast to that under the periodic one and that it visualizes
clearly the nontriviality of spin systems in the sense of the
Lieb-Schultz-Mattis theorem.
