The Dry Ten Martini Problem at Criticality

Abstract:We prove (actually this time) that every allowed gap label of the almost Mathieu operator is realized by an open gap at critical coupling for every irrational frequency. The main step is to show that the maximum Lyapunov exponent in each labeled gap is nondecreasing as the subcritical coupling increases. In the Landau representation, an antiunitary conjugation relates commuting left and right magnetic actions that agree on the rooted resolvent. A boundary balance on the lattice diagonals yields a self-adjoint compression and expresses $\lambda h_n'(\lambda)$ as a nonnegative weighted norm ratio of Green-function entries. Together with noncritical gap opening and upper semicontinuity of the Lyapunov exponent, this establishes critical gap opening.

13 + 2 pages, 1 figure; This version repairs the original proof via substantially different methods. A former co-author R. J. S. on the original manuscript did not contribute to the updated version. All work on the new proof was done by the corresponding author and the new co-author M. F

Mathematical Physics (math-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Quantum Physics (quant-ph)

arXiv:2112.06869 [math-ph]

(or arXiv:2112.06869v2 [math-ph] for this version)