  In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also
possess Heisenberg ($J$) and off-diagonal symmetric ($\Gamma$) couplings. We
investigate the quantum ($S = 1/2$) $K$-$J$-$\Gamma$ model on the honeycomb
lattice by a variational Monte Carlo (VMC) method. In addition to the "generic"
Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL)
phase, while the rest of the phase diagram contains different magnetically
ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which
in contrast to the KSL has a gapless spin response. In a magnetic field applied
normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral
spin-liquid phases, of which one is non-Abelian with Chern number $\nu = 5$ and
the other is Abelian with $\nu = 4$. These two phases could be distinguished by
their thermal Hall conductance.
