A tight-binding study of the 1/5-depleted square lattice under magnetic flux reports that next-nearest-neighbor hopping distorts the Hofstadter butterfly and, the authors claim, creates nonzero total Chern numbers.
Topological insulators and superconductors
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Quantum Hall Effect and Chern Phases in the 1/5-Depleted Square Lattice
A tight-binding study of the 1/5-depleted square lattice under magnetic flux reports that next-nearest-neighbor hopping distorts the Hofstadter butterfly and, the authors claim, creates nonzero total Chern numbers.