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

Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis