Quasiperiodicity in one dimensional models
Most materials studied in physics have a periodic structure i.e. there is a pattern of, for instance, atoms that repeats in some regular interval. How are periodic systems having irrational period different than those with a rational one? Such systems are called quasiperiodic or referred to as quasicrystals.
Collaborators and I studied the famous Aubry-Andre-Harper (AAH) model, a one-dimensional tight-binding model that is equivalent to the Hofstadter problem at its critical point and exhibits a localization-delocalization (metal-insulator) transition despite being one-dimensional (in contrast to well-known works on Anderson localization). Generalizations of the AAH model can also feature mobility edges in which case only the states of the system having certain energies undergo to localization-delocalization transition.
Working with collaborators, I co-developed a transfer matrix method that allowed us to obtain information about wavefunctions of the system by studying curves on a torus. Toroidal curves for localized, delocalized and critical wavefunctions of the AAH and AAH-like models are distinct, demonstrating a geometrical characterization of localization physics. Applying the transfer matrix method to AAH-like models, we formulated a geometrical picture that also captures the emergence of the mobility edge. Consequently, this work connects with ongoing experimental ultracold atomic studies of a previously proposed model having exact mobility edges.
F. A. An, K. Padavic, et al, Observation of interaction-shifted mobility edges in generalized Aubry-Andre lattices, in preparation