  Motivated by the recent experimental observation of the half-quantized
thermal Hall conductivity for the candidate material of the Kitaev spin liquid,
$\alpha$-RuCl$_3$[ Y. Kasahara et al., Nature 559, 227 (2018)], we investigate
effects of non-Kitaev exchange interactions, which exist in the real material,
on the gapped chiral spin liquid state realized in the case with an applied
magnetic field. It is found that off-diagonal exchange interactions enhance
significantly the mass gap of Majorana fermions. This result provides a
possible explanation for robust quantization of the thermal Hall conductivity
observed in the above-mentioned experiment. Furthermore, we demonstrate that if
the Kitaev spin liquid state and the zigzag antiferromagnetic order coexist
around the border of these two phases, off-diagonal exchange interactions can
induce Fermi surfaces of Majorana fermions, leading to a state similar to the
U(1) spin liquid.
