Unveiling the phase diagram of a bond-alternating spin-$frac12$ $K$-$Gamma$ chain

The key to unraveling intriguing phenomena observed in various Kitaev materials lies in understanding the interplay of Kitaev ($K$) interaction and a symmetric off-diagonal $Gamma$ interaction. To provide insight into the challenging problems, we study the quantum phase diagram of a bond-alternating spin-$1/2$ $g_x$-$g_y$ $K$-$Gamma$ chain by density-matrix renormalization group method where $g_x$ and $g_y$ are the bond strengths of the odd and even bonds, respectively. The phase diagram is dominated by even-Haldane ($g_x > g_y$) and odd-Haldane ($g_x < g_y$) phases where the former is topologically trivial while the latter is a symmetry-protected topological phase. Near the antiferromagnetic Kitaev limit, there are two gapped $A_x$ and $A_y$ phases characterized by distinct nonlocal string correlators. In contrast, the isotropic ferromagnetic (FM) Kitaev point serves as a multicritical point where two topological phase transitions meet. The remaining part of the phase diagram contains three symmetry-breaking magnetic phases. One is a six-fold degenerate FM$_{U_6}$ phase where all the spins are parallel to one of the $pm hat{x}$, $pm hat{y}$, and $pm hat{z}$ axes in a six-site spin rotated basis, while the other two have more complex spin structures with all the three spin components being finite. Existence of a rank-2 spin-nematic ordering in the latter is also discussed.