The Dry Ten Martini Problem at Criticality

Abstract

We solve the dry ten martini problem (lean-verified this time) by showing that every labeled almost Mathieu gap is open for $\lambda \ne 0$ and every irrational frequency. In particular, we prove that the maximum logarithmic height of each labeled gap is non-decreasing in the coupling $\lambda$ and extend an initial open seed gap for each gap label $n$ from $\lambda \in (0,{\lambda }_n)$ to all $\lambda >0$. The non-decreasing property follows from restoring time-reversal symmetry to the almost Mathieu operator $H$ by pairing it with its magnetic inverse $\bar{H}$. This allows us to express the logarithmic coupling derivative of the maximal gap height as the expectation value of a non-negative operator, yielding the required monotonicity.

arXiv comment12 + 11 pages, 2 figures; This version improves on the presentation of v2 and provides a lean-verification of the theorem