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In two dimensions, topological phases of free Majorana fermions coupled to a$\mathbb{Z}_2$ gauge field are known to be classified according to the Chernnumber $\nu \in \mathbb{Z}$. Its value mod 16 specifies the type of anyonicexcitations. In this paper, we investigate triangular vortex configurations(and their dual) in the Kitaev honeycomb model and show that fourteen of thesesixteen phases can be obtained by adding a time-reversal symmetry-breakingterm. Missing phases are $\nu=\pm 7$. More generally, we prove that anyperiodic vortex configuration with an odd number of vortices per geometric unitcell can only host even Chern numbers whereas odd Chern numbers can be found inother cases.

Parity of Chern numbers in the Kitaev honeycomb model and the sixteenfold way