# Probing breakdown of topological protection: Filling-factor-dependent evolution of robust quantum Hall incompressible phases

## TL;DR

This study uses a scanning gate technique to visualize how quantum Hall incompressible phases evolve with filling factor and break down under high current, revealing microscopic transport properties of topological 2D systems.

## Contribution

It introduces a method to image the microscopic evolution and breakdown of quantum Hall incompressible phases under nonequilibrium conditions.

## Key findings

- Incompressible edge and bulk structures show filling-factor-dependent evolution.
- These structures align with equilibrium quantum Hall phase calculations.
- High current destroys the filling-factor dependence of incompressible patterns.

## Abstract

The integer quantum Hall (QH) effects characterized by topologically quantized and nondissipative transport are caused by an electrically insulating incompressible phase that prevents backscattering between chiral metallic channels. We probed the incompressible area susceptible to the breakdown of topological protection using a scanning gate technique incorporating nonequilibrium transport. The obtained pattern revealed the filling-factor ($\nu$)-dependent evolution of the microscopic incompressible structures located along the edge and in the bulk region. We found that these specific structures, respectively attributed to the incompressible edge strip and bulk localization, show good agreement in terms of $\nu$-dependent evolution with a calculation of the equilibrium QH incompressible phases, indicating the robustness of the QH incompressible phases under the nonequilibrium condition. Further, we found that the $\nu$ dependency of the incompressible patterns is, in turn, destroyed by a large imposed current during the deep QH effect breakdown. These results demonstrate the ability of our method to image the microscopic transport properties of a topological two-dimensional system.