The Chern numbers for Hofstadter models with rational flux 2πp/q are partially determined by a Diophantine equation. A M od 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.1 The phase diagrams we consider should be distinguished from phase diagrams which describe the localization properties and the Liapunov exponent described e.g. in [12].
We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.
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