Quasiperiodicity induced eigenstate criticality and flat \mathbb{Z}_2 bands in a topological insulator

Flat topological bands have been sought after as a route to the fascinating phenomena of topology with strong correlations. The discovery of correlated insulator and superconductivity in various twisted multilayer graphene systems has shown that the superlattice pattern generated in van der Waals heterostructure, or more generally incommensurate effects, provides an ample playground for generating flat bands. We demonstrate that perturbing topological insulators with a quasiperiodic potential creates flat topological bands in a controllable fashion as well as induces non-trivial quantum phase transitions that are beyond the Landau-Ginzburg paradigm. Using a combination of analytic and numeric calculations, including a Convolutionary Neural Network classification of wavefunctions, we reveal the rich phase diagram induced by quasiperiodicity and topology. We show that quasiperiodicity can make a trivial insulator topological through a topological eigenstate phase transition that represent a unique universality class, and in the process creates essentially flat topological bands.