Wannier band transitions in disordered pi-flux ladders

Boundary obstructed topological insulators are an unusual class of higher-order topological insulators with topological characteristics determined by the so-called Wannier bands. Boundary obstructed phases can harbor hinge/corner modes, but these modes can often be destabilized by a phase transition on the boundary instead of the bulk. While there has been much work on the stability of topological insulators in the presence disorder, the topology of a disordered Wannier band, and disorder-induced Wannier transitions have not been extensively studied. In this talk, we explore the effect of disorder on the simplest example of a Wannier topological insulator: a mirror-symmetric pi-flux ladder in 1D. We demonstrate that the Wannier topology is robust to disorder, and establish a connection between the Wannier topology and the energy band topology of a system with a physical boundary cut, something which has generally been conjectured for clean models, but has not been studied in the presence of disorder. These results suggest that Wannier topology can be a powerful tool for studying the boundaries of disordered BOTIs.