Free Fermions in Disguise without Exponential Degeneracies

A number of spin chains are solvable by hidden free-fermionic structures that go beyond the Jordan-Wigner transformation, the family known as "free fermions in disguise". Every example in the literature shared an awkward feature: degeneracies growing exponentially with the volume, and homogeneous across the spectrum, so every energy level carried the same degeneracy. The question is whether that feature is forced. Can a model in this family have a spectrum free of exponential degeneracies, or does the hidden structure always impose them? This exhibits one. The model is a particular perturbation of two Ising chains, and can equally be read as an interpolation between a Jordan-Wigner solvable chain and Fendley's original FFD model. For generic coupling constants its spectrum has no exponential degeneracies.

An existence question settled by exhibiting an object, not a general theorem: one model in the family has no exponential degeneracies for generic couplings, and nothing here says which others do. The route is worth recording because it is not the one anyone was looking down. The author had tried and failed to find such a model directly. It surfaced instead from an unrelated classification of medium-range spin chains, where the AI's computations produced a list of integrable Hamiltonians and one of them combined two terms of a standard XY chain with two of Fendley's FFD model, a combination nobody had considered. The author noticed it in the list; the rest followed from asking the model a sequence of increasingly specific questions.