  We study the Kitaev model with direction dependent interactions to
investigate how the flux configuration and/or the anisotropy in the exchanges
affect the Majorana excitations. Systematic numerical calculations demonstrate
how the anisotropy of the exchange couplings and flux configuration make the
Majorana excitation gapped. The induced gapped quantum spin liquid states are
distinct from the gapped one realized in the large anisotropic limit. The
nature of gapped states can be explained by the superlattice potential due to
flux configuration.
