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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.