In the presence of strong magnetic fields, two-dimensional electrons organize into highly degenerate energy bands known as Landau levels. These topological bands, along with the enhanced interactions within them, give rise to a diverse array of quantum phenomena collectively known as the quantum Hall effects. They not only lay the foundation for topological matter but also host a variety of exotic properties, many of which remain elusive due to limitations in measurement capabilities.

In this presentation, I will introduce our efforts to further the understanding of quantum Hall effects in graphene using a scanning tunneling microscope (STM), renowned for its capacity to capture high-resolution images and resolve intricate spectral features. Our experiments provided a direct visualization of Landau orbits, elucidated a decade long puzzle of symmetry breaking at charge neutral point, and resolved the topological excitation of quantum Hall ferromagnets. Moreover, we were able to access fractional quantum Hall effects with the STM for the first time. We take this opportunity to examine partially filled Landau levels from their excitation spectra and interaction with charge impurities. Looking ahead, STM holds significant potential in elucidating various topological and correlated quantum phases across a broad spectrum of van der Waals heterostructures.