Free fermions are integrable — they can be solved exactly because the particles don't interact. The Hubbard model is also integrable — it can be solved exactly even though the particles do interact. These seem like different miracles. One is trivial (no interaction), the other is deep (precise cancellations in the interaction allow exact solution).
Zhang (arXiv:2603.11172) connects them. A free fermionic system satisfies both the Yang-Baxter equation and Shastry's decorated star-triangle relation simultaneously. These two conditions together define “free” in a way that's more restrictive than merely being solvable but less restrictive than being noninteracting. The R-matrix — the object encoding how particles scatter — has difference form and conjugation symmetry.
The connection to interacting systems: some free fermionic R-matrices can be continuously deformed, via conjugation operators, into non-relativistic R-matrices describing interacting particles. The Hubbard model emerges this way. The interaction doesn't destroy integrability because it was introduced as a deformation of a structure that was already integrable. The integrability isn't preserved despite the interaction; it's preserved through the interaction, because the interaction is a conjugation of the free system.
Not all deformations work. The paper provides criteria for when a conjugation preserves integrability and gives an iterative procedure for constructing the R-matrix from the local Hamiltonian. The structural insight: the boundary between free and interacting integrable systems is not a wall but a gradient. Some interacting systems are “secretly free” in the sense that their integrability derives from a free system by continuous deformation. The Hubbard model's solvability isn't a coincidence — it's a perturbation of triviality.