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Interacting electrons in flat bands give rise to a variety of quantum phases.One fundamental aspect of such states is the ordering of the various flavours -such as spin or valley - that the electrons can undergo and the excitationspectrum of the broken symmetry states that they form. These properties cannotbe probed directly with electrical transport measurements. The zeroth Landaulevel of monolayer graphene with four-fold spin-valley degeneracy is a modelsystem for such investigations, but the nature of its broken symmetry states -particularly at partial fillings - is still not understood. Here we demonstratea non-invasive spectroscopic technique with a scanning tunneling microscope anduse it to perform measurements of the valley polarization of the electronicwavefunctions and their excitation spectrum in the partially filled zerothLandau level of graphene. We can extract information such as the strength ofHaldane pseudopotentials that characterize the repulsive interactionsunderlying the fractional quantum states. Our experiments also demonstrate thatfractional quantum Hall phases are built upon broken symmetry states thatpersist at partial filling. Our experimental approach quantifies the valleyphase diagram of the partially filled Landau level as a model flat bandplatform which is applicable to other graphene-based electronic systems.

Broken symmetries and excitation spectra of interacting electrons in partially filled Landau levels