| Abstract: |
| The almost Mathieu operator (AMO) famously hosts a 1D metal-insulator phase transition. Away from the transition, the AMO spectrum has been shown to form a Cantor set for all irrational frequencies (Ten Martini Problem) with all labeled gaps open (Dry Ten Martini Problem). This work closes problem for the remaining $\lambda =1$ critical point at Diophantine frequencies, showing the critical AMO Cantor set spectrum attains all labeled gaps. Our proof depends on two key ingredients:
(1) Sub/Supercritical Embedding -- Hidden singularities of critical AMO in the chiral gauge and a finite similarity transform permit us to embed finite-volume restrictions of the critical AMO into the non-critical operator.
(2) Bulk-Edge Correspondence -- The closing of labeled gaps at $\lambda =1$ forces small rank-one errors via bulk-edge correspondence and permit the gluing of the finite-volume restrictions back together.
Together, these imply that if a labeled gap closes, the infinite critical operator may be glued together
and fully embedded in any non-critical operator (at re-scaled energy) for which the non-critical AMO resolvent is lower bounded by a diverging critical AMO resolvent. This is too strong a condition and violates the AMO gap continuity, arriving at a contradiction. |
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