  We study quantum phase transitions in the honeycomb Kitaev model under a
magnetic field, focusing on the topological nature of Majorana fermion
excitations. We find a gapless phase between the low-field gapless quantum spin
liquid and the high-field gapped forced-ferromagnetic state for the
antiferromagnetic Kitaev model in the [001] field by using the Majorana
mean-field theory, in conjunction with the exact diagonalization and the
spin-wave theory supporting the validity of this approach. The transition
between the two gapless phases is driven by a topological change of the
Majorana spectrum --- line node formation interconnecting two Majorana cones.
The peculiar change of the Majorana band topology is rationalized by a sign
change of the effective Kitaev coupling by the magnetic field, which does not
occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away
from [001], the two gapless phases become gapped and topologically nontrivial,
characterized by nonzero Chern numbers with different signs. The sign change of
the Chern number leads to a reversal of the thermal edge current in the
half-quantized thermal Hall effect.
