Heart of Entanglement: Chiral, Nematic, and Incommensurate Phases in the Kitaev-Gamma Ladder in a Field

The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid-state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive. One reason for this may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically, and missed incommensurately ordered states may be interpreted as a spin liquid. Here we focus on the Kitaev-Gamma ladder model as a guide to the phase space of disordered states which could potentially become a spin liquid in the two-dimensional limit. We determine the entire phase diagram in the presence of a magnetic field along the [111] direction. Because of the competition between the interactions and the field, an extremely rich phase diagram emerges with 15 distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin-nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outlines a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a C3 symmetric cluster of the two-dimensional honeycomb lattice, and offer insight into possible spin liquids in the two-dimensional limit.