Spin-nematic order in the spin-1/2 Kitaev-Gamma chain

A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one dimensional Kitaev-Gamma chain, an emergent SU(2)$_1$ phase and a rank-1 spin ordered phase with $O_h\rightarrow D_4$ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a higher-rank spin-nematic ordered phase, and a peculiar Kitaev phase. There is no numerical signature of spin orderings nor Luttinger liquid behaviours in the Kitaev phase. The transition between the spin-nematic and the "$O_h\rightarrow D_4$" phases and the nature of the Kitaev phase are worth further studies.