  Exactly solvable lattice models for spins and non-interacting fermions
provide fascinating examples of topological phases, some of them exhibiting the
localized Majorana fermions that feature in proposals for topological quantum
computing. The Chern invariant $\nu$ is one important characterization of such
phases. Here we look at the square-octagon variant of Kitaev's honeycomb model.
It maps to spinful paired fermions and enjoys a rich phase diagram featuring
distinct abelian and nonabelian phases with $\nu= 0,\pm1,\pm2,\pm3$ and $
\pm4$. The $\nu=\pm1 $ and $\nu=\pm3$ phases all support localized Majorana
modes and are examples of Ising and $SU(2)_2$ anyon theories respectively.
