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

TL;DR

This paper uncovers a novel twice hidden string order in the disordered phase of the spin-1/2 Kitaev-Gamma ladder, revealing complex phase structures and new phases relevant to Kitaev materials.

Contribution

It introduces a twice hidden string order parameter and characterizes new disordered phases in the $K ext{-} extGamma$ ladder model, advancing understanding of Kitaev material phases.

Findings

Identification of a distinct non-local string order parameter in the A$ extGamma$ phase.

Discovery of two new disordered phases in the $K extGamma$ ladder phase diagram.

Analysis of entanglement spectrum, edge states, and magnetic responses related to the phases.

Abstract

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.