  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 (KG) model has been investigated by various numerical
techniques under a magnetic field, but definite conclusions about field-induced
spin liquids remain elusive and one reason may lie in the limited sizes of the
two-dimensional geometry it is possible to access numerically. We therefore
focus on the KG model defined on a two-leg ladder which is much more amenable
to a complete study, and determine the entire phase diagram in the presence of
a magnetic field along [111]-direction. Due to the competition between the
interactions and the field, an extremely rich phase diagram emerges with
fifteen 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 outline 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
insights into possible spin liquids in the two-dimensional limit.
