  Topological phases supported by quasi-periodic spin-chain models and their
bulk-boundary principles are investigated by numerical and K-theoretic methods.
We show that, for both the un-correlated and correlated phases, the operator
algebras that generate the Hamiltonians are non-commutative tori, hence the
quasi-periodic chains display physics akin to the quantum Hall effect in two
and higher dimensions. The robust topological edge modes are found to be
strongly shaped by the interaction and, generically, they have hybrid
edge-localized and chain-delocalized structures. Our findings lay the
foundations for topological spin pumping using the phason of a quasi-periodic
pattern as an adiabatic parameter, where selectively chosen quantized bits of
magnetization can be transferred from one edge of the chain to the other.
