  We study the effects of the flux configurations on the emergent Majorana
fermions in the $S=1/2$ Kitaev model on a honeycomb lattice, where quantum
spins are fractionalized into itinerant Majorana fermions and localized fluxes.
A quantum spin liquid appears as the ground state of the Kitaev model in the
flux-free sector, which has intensively been investigated so far. In this flux
sector, the Majorana fermion system has linear dispersions and shows power law
behavior in the Majorana correlations. On the other hand, periodically-arranged
flux configurations yield low-energy excitations in the Majorana fermion
system, which are distinctly different from those in the flux-free state. We
find that one of the periodically arranged flux states results in the gapped
Majorana dispersion and the exponential decay in the Majorana correlations. The
Kitaev system with another flux configuration exhibits a semi-Dirac like
dispersion, leading to the power law decay with a smaller power than that in
the flux-free sector along symmetry axes. We also examine the effect of the
randomness in the flux configurations and clarify that the Majorana density of
states is filled by increasing the flux density, and power-law decay in the
Majorana correlations remains. The present results could be important to
control the motion of Majorana fermions, which carries the spin excitations, in
the Kitaev candidate materials.
