1D Localization-Delocalization Physics and Toroidal Representations of Transfer Matrices

Wavefunction localization is a characteristic phenomenon occurring in disordered and quasiperiodic systems as well as with edges states in topological phases. We study the quasiperiodic Aubry-Andre-Harper (AAH) model, known to exhibit a unique localization-delocalization transition in one dimensions, defying standard Anderson localization. Generalizations of the AAH model include next-nearest neighbor (NNN) hopping, or additional incommensurate on-site terms and have so far been studied numerically. For such extended models the appearance of a mobility edge i.e. an energy cut-off dictating which wavefunctions undergo the localization-delocalization transition is expected. To study properties of these models, we employ transfer matrices which are known to characterize localization physics through Lyapunov exponents. We use the symplectic nature of transfer matrices to represent them as points on a torus. Related wavefunctions then form toroidal curves. We obtain distinct toroidal curves for localized, delocalized and critical wavefunctions, thus demonstrating a geometrical characterization of localization physics. Applying the transfer matrix method to the NNN AAH model, we formulate a geometrical picture that captures the emergence of the mobility edge in a visually striking way.