Upper and Lower Bounds on the Partition Function of the Hofstadter Model

Using unitary equivalence of magnetic translation operators, explicit upper and lower convex bounds on the partition function of the Hofstadter model are given for any rational ``flux" and any value of Bloch momenta. These bounds (i) generalize straightforwardly to the case of a general asymmetric hopping and to the case of hopping of the form $t_{jn}(S_j^n+S_j^{-n})$ with $n$ arbitrary integer larger than or equal $2$, and (ii) allow to derive bounds on the derivatives of the partition function.