  We introduce low-energy effective models for dense configurations of vortices
in the Kitaev honeycomb model. Specifically, we consider configurations of
vortices in which vortex-free plaquettes form triangular lattices against a
vortex-full background. Depending on the vortex density, these "dual"
configurations belong to either one of two families classified by translation
and inversion symmetry. As a function of a time-reversal symmetry breaking
term, one family exhibits gapped phases with even Chern numbers separated by
extended gapless phases, while the other exhibits gapped phases with even or
odd Chern numbers, separated by critical points. We construct an effective
model for each family, determine the parameters of these models by fitting the
integrated density of states, and reproduce energy spectra and Chern numbers of
the Kitaev honeycomb model. We also derive phase diagrams and determine these
models' validity.

Superconductivity;Quantum Physics