The Dry Ten Martini Problem at Criticality

2026-10-16 · arXiv Quantum Physics

arXiv:2112.06869v3 Announce Type: replace-cross

Abstract: We solve the dry ten martini problem (lean-verified this time) by showing that every labeled almost Mathieu gap is open for $\lambda\neq 0$ 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.

原文链接：https://arxiv.org/abs/2112.06869

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