Critical Integer Quantum Hall Topology in Maryland Model
One-dimensional tight binding models such as Aubry-Andre-Harper (AAH) model (with onsite cosine potential) and the integrable Maryland model (with onsite tangent potential) have been the subjects of extensive theoretical research in localization studies. AAH can be directly mapped onto the two-dimensional Hofstadter model, which manifests the integer quantum Hall topology on a lattice. However, no such connection has been made for the Maryland model (MM). In this talk, I will describe a generalized model that contains AAH and MM as the limiting cases with MM lying precisely at a topological quantum phase transition (TQPT) point. A remarkable feature of this critical point is that 1D MM retains well-‐defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the critical phenomena.
Recently, it was shown that 1D quasicrystal modeled by incommensurate AAH, Fibonacci, off-diagonal Harper are all topologically equivalent with 2D integer hall topology. This result was based on the identification of a special translational symmetry enforced by quasi-periodicity. MM belongs to the same symmetry class but it sits at a critical point of a topological phase transition. I will outline the consequences of including MM in this classification. I will explain how this quasiperiodic symmetry manifests in polarization, which is an experimentally relevant observable.