Exactly solvable lattice models for spins or hopping
fermions provide fascinating examples of topological phases.
Some of them support localized Majorana fermions, which
feature in topologically protected quantum computing. The
Chern invariant $\nu$ is one important characterization of
such phases. Systems with arbitrarily large Chern numbers
are known, but systems supporting Majorana fermions have
mainly provided ground states with $\nu=0,\pm1$ although
symmetry arguments in some cases allow for any integer
$\nu$. With the rich variety of phases exhibited by
spin-triplet p-wave fermions in mind, we look at the
square-octagon variant of Kitaev's honeycomb model. It maps
to spinful paired fermions and indeed 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. We show that transitions between topological
phases are accompanied by stepwise transfer of Chern number
between the four bands and then finally describe the edge
spectra at topological domain walls, highlighting the one
between distinct $\nu=0$ phases.

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