A study of the ambiguity in the solutions to the Diophantine equation for Chern numbers

The Chern numbers for Hofstadter models with rational flux 2 pi p/q are partially determined by a Diophantine equation. A mod q ambiguity remains. The resolution of this ambiguity is only known for the rectangular lattice with nearest neighbors hopping where it has the form of a 'window condition'. We study a Hofstadter butterfly on the triangular lattice for which the resolution of ambiguity is open. In the model many pairs (p, q) satisfy a window condition which is shifted relative to the window of the square model. However, we also find pairs (p, q) where the Chern numbers do not belong to any contiguous window. This shows that the rectangular model and the one we study on the triangular lattice are not adiabatically connected: many gaps must close. Our results suggest the conjecture that the mod q ambiguity in the Diophantine equation generically reduces to a sign ambiguity.