Quasi-Periodic Topological Bulk-Bulk Localization

We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal-insulator transition. By constructing quasi-periodic transfer matrix equations from the limit of rational approximate projected Green's functions, we reduce results from SL(2,R) co-cycle theory (transfer matrix eigenvalue scaling) to consequences of translation invariant band theory. This reduction links the eigenfunction localization of the metal-insulator transition to the chiral edge modes of the Hofstadter Hamiltonian. Our analysis shows the localized phase roots in a topological "bulk-bulk" correspondence rather than self-duality, differentiating quasi-periodic localization from Anderson localization in disordered systems. These results and methods are widely relevant to systems beyond this paradigmatic model, including 2D cold atom realizations, and have direct application to Barry Simon's "Dry Ten Martini Problem" at criticality.