Diagnosing energy gap in quantum spin liquids via polarization amplitude

Identifying whether a many-body ground state is gapped or gapless is a fundamental yet challenging problem, especially in quantum spin liquids. In this work, we develop a gap-diagnostic scheme based on the polarization amplitude defined via a twist operator, evaluated within the infinite density-matrix renormalization group (iDMRG) framework. As a benchmark, analysis of the spin-$1/2$ XXZ chain demonstrates that the polarization amplitude clearly distinguishes the gapless Tomonaga-Luttinger liquid from the gapped N\'eel phase. We then extend this framework to infinite cylinders of the spin-$1/2$ XY-$J_\chi$ model on the square lattice. We find that the polarization amplitude sharply detects the transition between the gapless XY phase and the gapped chiral spin liquid phase. These results show that polarization amplitudes provide a strong energy-gap diagnostic in two-dimensional frustrated quantum magnets, including quantum spin liquids.