Nonsymmorphic spin-space cubic groups and SU(2)${}_1$ conformal invariance in one-dimensional spin-1/2 models

Recently, extended gapless phases with emergent SU(2)${}_1$ conformal invariance occupying finite regions in the phase diagrams have been found in one-dimensional spin-1/2 models with nonsymmorphic $O_h$ symmetry groups. In this work, we investigate the question of whether the conditions for emergent SU(2)${}_1$ invariance can be loosened. We find that besides the nonsymmorphic $O_h$ group, the other four smaller nonsymmorphic cubic groups including $O$, $T_h$, $T_d$ and $T$ can also give rise to emergent SU(2)${}_1$ invariance. Minimal spin-1/2 models having these nonsymmorphic cubic groups as symmetry groups are constructed, and numerical evidences for the emergent SU(2)${}_1$ invariance are provided. Our work is useful for understanding gapless phases in one-dimensional spin systems with nonsymmorphic symmetries.