  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 excitation
spectrum of the broken symmetry states that they form. These properties cannot
be probed directly with electrical transport measurements. The zeroth Landau
level of monolayer graphene with four-fold spin-valley degeneracy is a model
system for such investigations, but the nature of its broken symmetry states -
particularly at partial fillings - is still not understood. Here we demonstrate
a non-invasive spectroscopic technique with a scanning tunneling microscope and
use it to perform measurements of the valley polarization of the electronic
wavefunctions and their excitation spectrum in the partially filled zeroth
Landau level of graphene. We can extract information such as the strength of
Haldane pseudopotentials that characterize the repulsive interactions
underlying the fractional quantum states. Our experiments also demonstrate that
fractional quantum Hall phases are built upon broken symmetry states that
persist at partial filling. Our experimental approach quantifies the valley
phase diagram of the partially filled Landau level as a model flat band
platform which is applicable to other graphene-based electronic systems.
