An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. Yet the standard Jordan–Wigner transformation captures only a narrow subclass of models solvable via a mapping to free fermions. In this talk, I will present a unified graph-theoretical framework that goes beyond Jordan-Wigner, and extends the known class of free-fermion models. Our approach is based on the frustration graph of the Hamiltonian (the complement of the commutation graph), and provides a sufficient condition for generic free fermion solvability. In particular, we show how structural properties of the frustration graph determine solvability: line graphs reproduce Jordan–Wigner–type mappings, while simplicial claw-free graphs identify a broader class of “free fermions in disguise” (a superset of the Jordan-Wigner-type free fermions). I will also outline recent generalisations to qudit and parafermionic systems, described by oriented frustration graphs, as well as links to operator symmetries and contextual subspace reductions.

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