Phase Diagrams of Kitaev Models for Arbitrary Magnetic-Field Orientations

Abstract: The Kitaev model is an exactly solvable quantum spin model within the language of the constrained real fermions. In spite of numerous studies along special magnetic-field orientations, there is a limited amount of knowledge on the complete field-angle characterization, which can provide valuable information on the existence of fractionalized excitations. For this purpose, we first extend previous studies on the field-angle response of the ferromagnetic Kitaev model to its antiferromagnetic version. Yet, the realistic description of the candidate Kitaev materials, within the edge-sharing octahedra paradigm, require additional coupling terms, including a large off-diagonal term $\Gamma$ along with possible anisotropic corrections $\Gamma_p$. It is therefore not sufficient to depend on the topological properties of the bare Kitaev model as the only source for the observed thermal Hall conductivity signals and an understanding of these extended Kitaev models with a complete field response is demanded. Starting from the zero-field phase diagram of realistic K-$\Gamma$-$\Gamma_p$ models, we identify antiferromagnetic zig-zag and (partially) polarized phases as well as two unusual Kitaev(-$\Gamma$) spin-liquid phases. The magnetic field response of these phases for arbitrary field orientations provides a remarkably rich phase diagram. A partially polarized phase is revealed between two ordered phases with a suppressed magnetization, finite fractionalization and finite Chern number. This phase is characterized as an extended Kitaev-$\Gamma$ spin-liquid. To comply our findings with experiments, we reproduce the asymmetry in the extent of the intermediate phases specifically for the two different field directions $\theta = \pm 60^o$.