LAUSR

A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent $\mathrm{SU}{(2)}_{1}$ phase and a rank-1 spin ordered phase with ${O}_{h}\ensuremath{\rightarrow}{D}_{4}$ 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 rank-1 ordered phase with an ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}\ensuremath{\rightarrow}{D}_{3}$ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.