Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder

Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma ($\Gamma$) interaction, causes magnetic frustration and its interplay with the Kitaev ($K$) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined $K\Gamma$ model, quasi-one dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic $\Gamma$ limit (denoted by A$\Gamma$ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the A$\Gamma$ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of $K\Gamma$ ladder is also reported.