  Fractional excitations offer vast potential for both fundamental physics and
quantum technologies. However, their dynamics under the influence of gauge
fields pose a significant challenge to conventional models. Here, we
investigate the evolution of low-energy Majorana dispersions across various
crystalline phases of the \pi-flux in the Kitaev spin model on a honeycomb
lattice. We develop an effective tight-binding description for these low-energy
Majorana fermions, introducing a gauge potential through a superexchange-like
interaction that systematically eliminates the high-energy spectrum. We
identify conditions under which this superexchange interaction acts as a Z2
gauge field, governing the tight-binding hopping of Majorana Wannier orbitals.
Our study reveals an intriguing phase transition between two non-trivial
topological phases characterized by gapless flat-band (extensive) degeneracy.
To further explore flat band physics, we introduce a mean-field theory
describing a gauge-invariant Majorana density-wave order within these bands.
The resulting split Chern bands facilitate the partial filling of Chern bands,
effectively leading to fractional Chern states. Our work, encompassing both the
gauge-mediated tight-binding model and the mean-field theory, opens doors for
future exploration of $U(1)$, $SU(N)$ gauge-mediated tight-binding approach to
other fractional or entangled Wannier excitations.
