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Free fermion systems enjoy a privileged place in physics. With theirsimple structure they can explain a variety of effects, ranging frominsulating and metallic behaviours to superconductivity and the integerquantum Hall effect. Interactions, e.g. in the form of Coulombrepulsion, can dramatically alter this picture by giving rise toemerging physics that may not resemble free fermions. Examples of suchphenomena include high-temperature superconductivity, fractional quantumHall effect, Kondo effect and quantum spin liquids. The non-perturbativebehaviour of such systems remains a major obstacle to their theoreticalunderstanding that could unlock further technological applications.Here, we present a pedagogical review of “interaction distance"[Nat. Commun. 8, 14926 (2017)] – a systematicmethod that quantifies the effect interactions can have on the energyspectrum and on the quantum correlations of generic many-body systems.In particular, the interaction distance is a diagnostic tool thatidentifies the emergent physics of interacting systems. We illustratethis method on the simple example of a two-site Fermi-Hubbard model. 

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 possess 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 fourfold 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 tunnelling 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 the 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. 

Scanning tunneling spectroscopy is used to image valley ordering in graphene in the presence of a magnetic field.