Wannier band transitions in disordered $\pi$-flux ladders

TL;DR

This paper investigates the robustness of Wannier band topology in disordered $ ext{pi}$-flux ladders, revealing a new disorder-induced transition and connecting Wannier and energy band topologies even with disorder present.

Contribution

It introduces a real-space RG method to analyze disorder effects on Wannier topological phases and establishes a link between Wannier and energy band topologies under disorder.

Findings

Wannier topology remains robust against disorder.

Disorder induces a transition between non-trivial and trivial Wannier phases.

Connection established between Wannier topology and energy band topology in disordered systems.

Abstract

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 work, we focus on the simplest example of a Wannier topological insulator: a mirror-symmetric $\pi$-flux ladder in 1D. We find that the Wannier topology is robust to disorder, and derive a real-space renormalization group procedure to understand a new type of strong disorder-induced transition between non-trivial and trivial Wannier topological phases. We also establish a connection between the Wannier topology of the ladder and the energy band topology of a related 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.