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Classical spin order near antiferromagnetic Kitaev point 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 rank-1 ordered phase with an $O_h\rightarrow D_3$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the $O_h\rightarrow D_3$ 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 $O_h\rightarrow D_{3d}\rightarrow D_3$ 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 $O_h\rightarrow D_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the $O_h\rightarrow D_3$ 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.