Hidden Criticality in the Almost-Mathieu Operator via non-Hermitian Gauge Transformations and the Dry Ten Martini Problem

The almost Mathieu operator (AMO) famously hosts a metal-insulator phase transition at (the self-dual point). More recently, it was noticed that via a morita equivalence (the chiral gauge) the critical AMO has "hidden singularities" which segment it into two halves. In this work, we extend the chiral gauge to the non-critical AMO for which no singularities naturally occur, but then apply a bounded non-hermitian gauge transformation at the would-be singular bond to cut the non-critical operator into two half operators as well. In fact, we can apply this transformation to any number of bonds and embed the critical operator up to some bond in the non-critical AMO. We use this embedding to understand the resolvent set of the critical AMO, and coupled with projected green's function methods from Toeplitz operator theory, we bound the size of the labeled gaps of the critical AMO by the size of non-critical AMO gaps.