  Topological phases of matter remain a focus of interest due to their unique
properties -- fractionalisation, ground state degeneracy, and exotic
excitations. While some of these properties can occur in systems of free
fermions, their emergence is generally associated with interactions between
particles. Here we quantify the role of interactions in general classes of
topological states of matter in all spatial dimensions, including parafermion
chains and string-net models. Using the interaction distance [Nat. Commun. 8,
14926 (2017)], we measure the distinguishability of states of these models from
those of free fermions. We find that certain topological states can be exactly
described by free fermions, while others saturate the maximum possible
interaction distance. Our work opens the door to understanding the complexity
of topological models and to applying new types of fermionisation procedures to
describe their low-energy physics.
