# Polarization amplitude near quantum critical points

## TL;DR

This paper investigates how the polarization amplitude in one-dimensional quantum spin systems near critical points reflects the renormalization-group flow and boundary conditions, revealing insights into the system's critical behavior and topological properties.

## Contribution

It demonstrates that the polarization amplitude's scaling law encodes RG flow information and distinguishes fixed points, especially under different boundary conditions, in quantum spin systems.

## Key findings

- Polarization amplitude under periodic boundary conditions is sensitive to RG flow perturbations.
- Under antiperiodic boundary conditions, the polarization amplitude is determined solely by the fixed point.
- The study visualizes the nontriviality of spin systems as per the Lieb-Schultz-Mattis theorem.

## Abstract

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.