Simultaneous nanoscale imaging of local conductivity and chemical potential in a quantum Hall isospin ferromagnet

Quantum Hall isospin ferromagnetism in multilayer graphene offers a versatile playground for exploring flat band correlated physics, driven by the intricate coupling of spin, valley, orbital, and layer degrees of freedom. However, a nanoscale probe capable of simultaneously mapping local conductivity and chemical potential in these exotic phases has yet to be realized. Here, we introduce scanning conductivity and chemical potential microscopy (SCCM), a technique integrating scanning microwave impedance microscopy and Kelvin probe force microscopy. We demonstrate SCCM by probing the quantum Hall states and many-body Landau level energy spectrum in bilayer graphene. Applied to marginally twisted double bilayer graphene, SCCM then reveals a cascade of quantum Hall isospin ferromagnetic states with unexpected re-emergence behaviors. Significantly, experimental many-body Landau level energy spectrum further uncovers the intricate connections of these complex phenomena to inter-subband Landau level crossings and Landau level single-particle wavefunctions. These insights enable the construction of a comprehensive quantum Hall phase diagram. Our results demonstrate SCCM's capability in decoding complex quantum phenomena, establishing it as a versatile nanoscale probe for electron correlation and topology.