  The topological non-triviality of insulating phases of matter is by now well
understood through topological K-theory where the indices of the Dirac
operators are assembled into topological classes. We consider in the context of
the Kitaev chain a notion of a generalized Dirac operator where the associated
Clifford algebra is centrally extended. We demonstrate that the central
extension is achieved via taking rational operator powers of Pauli matrices
that appear in the corresponding BdG Hamiltonian. Doing so introduces a
pseudo-metallic component to the topological phase diagram within which the
winding number is valued in $\mathbb Q$. We find that this phase hosts a mode
that remains extended in the presence of weak disorder, motivating a
topological interpretation of a non-integral winding number. We remark that
this is in correspondence with Ref.[J. Differential Geom. 74(2):265-292]
demonstrating that projective Dirac operators defined in the absence of
spin$^\mathbb C$ structure have rational indices.
