Unusual excitations and double-peak specific heat in a bond-alternating spin-$1$ $K$-$\mathrm{\Gamma }$ chain

One-dimensional gapped phases that avoid any symmetry breaking have drawn enduring attention. In this paper, we study such phases in a bond-alternating spin-1 $K$-$\mathrm{\Gamma }$ chain built of a Kitaev ($K$) interaction and an off-diagonal $\mathrm{\Gamma }$ term. In the case of isotropic bond strength, a Haldane phase, which resembles the ground state of a spin-$1$ Heisenberg chain, is identified in a wide region. A gapped Kitaev phase situated at dominant ferromagnetic and antiferromagnetic Kitaev limits is also found. The Kitaev phase has extremely short-range spin correlations and is characterized by finite ${\mathbb{Z}}_2$-valued quantities on bonds. Its lowest entanglement spectrum is unique, in contrast to the Haldane phase whose entanglement spectrum is doubly degenerate. In addition, the Kitaev phase shows a double-peak structure in the specific heat at two different temperatures. In the pure Kitaev limit, the two peaks are representative of the development of short-range spin correlation at $T_h\simeq 0.5680$ and the freezing of ${\mathbb{Z}}_2$ quantities at $T_l\simeq 0.0562$, respectively. By considering bond anisotropy, regions of Haldane phase and Kitaev phase are enlarged, accompanied by the emergence of dimerized phases and three distinct magnetically ordered states.