The Dry Ten Martini Problem at Criticality

has presented a rich playground for deep mathematical questions about linear operators. While many question regarding this deceptively simple linear operator have been answered [1–8], the critical point, V = t, is difficult to access via existing analytic methods. Of particular interest is the stability of the model’s topological properties, namely whether or not the spectral gaps labeled by the gap-labeling theorem [9] form open sets in the almost-Mathieu spectrum. This was formally posed as the “dry ten martini problem”: Dry Ten Martini Problem. Consider an energy in the spectrum of the critical almost-Mathieu operator E ∈ Σ, satisfying Ĥ |ψ〉 = E |ψ〉 with Ĥ as in Eq. (1) and V = 1. If the integrated density of states, N(E) = mα + n with m,n ∈ Z and Θ = 2πα ∈, then E belongs to the boundary of a component of R− Σ. Remark: This is equivalent to saying the compliment of the spectrum, R − Σ, is composed of open sets for gaps labeled by the gap-labeling theorem. This problem has been tackled for quasi-periodic models without self-duality [10] and proven for the absolutely continuous and pure-point like regimes of the spectrum [11, 12]. The self-dual version was an open problem [10, 11, 13] until recently (during the writing of this manuscript). Ref. [2] proved the purely the critical almost-Mathieu operator is purely singularly continuous and thus has no eigenvalues. This directly implies all spectral gaps inherited from rational approximates remain open in the irrational limit. We present here an alternative proof for the above statement for the critical almost-Mathieu operator in Section IV. Unlike, Ref. [2] and others, which approach the problem in the irrational limit, we take advantage of the recently formulated rational transfer matrix approximates [14, 15] to construct the irrational almost-Mathieu transfer matrix from a continued fraction sequence of rational approximates. These approximates are taken in the higher dimensional parent Hamiltonian representation for the almost-Mathieu operator: