Localization as a consequence of quasiperiodic bulk-bulk correspondence
Dan S. Borgnia, Robert-Jan Slager
Abstract
We report on a direct connection between band theory, quasiperiodic topology, and the almost-Mathieu (Aubry-Andr\'e) metal insulator transition (MIT). By constructing the transfer matrix equations of one-dimensional (1D) quasiperiodic operators from rational approximate projected Green's functions, we relate the quasiperiodic Lyapunov exponents to the chiral edge modes of rational-flux Hofstadter Hamiltonians. We thereby show that the insulating phase is rooted in a topological ``bulk-bulk'' correspondence, a bulk-boundary correspondence between the 1D Aubry-Andr\'e system (boundary) and its two-dimensional (2D) parent Hamiltonian (bulk). We extend this connection to random disorder via a Fourier expansion in quasiperiodic modes, demonstrating our results are widely applicable to systems beyond this paradigmatic model. The uncorrelated disorder limit is characterized by the breakdown of bulk-boundary driven quasiperiodic localization.