  The higher-spin Kitaev magnets, in which the Kitaev interaction and
off-diagonal exchange couplings are overwhelmingly large, have emerged as a
fertile avenue to explore exotic phases and unusual excitations. In this work,
we study the quantum phase diagram of the spin-1 Kitaev-$\Gamma$ model on the
honeycomb lattice using density-matrix renormalization group. It harbours six
distinct phases and the intriguing findings are three magnetically ordered
phases in which both time-reversal symmetry and lattice symmetry albeit of
different sort are broken spontaneously. The chiral spin state originates from
the order-by-disorder effect and exhibits an almost saturated scalar spin
chirality at the quantum level. Depending on the relative strength of the two
interactions, it also features columnar-like or plaquette-like dimer pattern as
a consequence of the translational symmetry breaking. In parallel, the nematic
ferromagnets are situated at ferromagnetic Kitaev side and possess small but
finite ferromagnetic ordering. The lattice-rotational symmetry breaking
enforces nonequivalent bond energy along one of the three bonds. Although the
intrinsic difference between the two nematic ferromagnets remains elusive, the
discontinuities in the von Neumann entropy, hexagonal plaquette operator, and
Wilson loop operator convincingly suggest that they are separated via a
first-order phase transition.
