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The integer quantum Hall (QH) effects characterized by topologicallyquantized and nondissipative transport are caused by an electrically insulatingincompressible phase that prevents backscattering between chiral metallicchannels. We probed the incompressible area susceptible to the breakdown oftopological protection using a scanning gate technique incorporatingnonequilibrium transport. The obtained pattern revealed the filling-factor($\nu$)-dependent evolution of the microscopic incompressible structureslocated along the edge and in the bulk region. We found that these specificstructures, respectively attributed to the incompressible edge strip and bulklocalization, show good agreement in terms of $\nu$-dependent evolution with acalculation of the equilibrium QH incompressible phases, indicating therobustness 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 effectbreakdown. These results demonstrate the ability of our method to image themicroscopic transport properties of a topological two-dimensional system.

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