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

TL;DR

This paper derives explicit upper and lower convex bounds on the partition function of the Hofstadter model for any rational flux and Bloch momenta, generalizing to asymmetric hopping and enabling bounds on derivatives.

Contribution

It provides the first explicit convex bounds on the Hofstadter model's partition function applicable to general flux and hopping configurations.

Findings

Bounds are valid for any rational flux and Bloch momenta.

Bounds extend to asymmetric and higher-order hopping models.

Allow derivation of bounds on derivatives of the partition function.

Abstract

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.