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.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Integrable spin chains
- Posed by
- Balázs Pozsgay
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-06-08
- Model
- ChatGPT 5.4 Pro, ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Balázs Pozsgay
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 7 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
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.
What the AI did
The abstract says "research assistant". The Supplemental Material is far more specific, and it is what this classification rests on. The AI performed the algebraic computations for a classification of medium-range spin chains, producing the lists of integrable Hamiltonians in which this model first appeared; discovered the quadratic cross-relations between the generators of the two commuting halves; found the recipe for open boundary conditions, which is what makes a free-fermionic solution possible at all; and proved the paper's central theorem. In the author's words: "Theorem 5.1 is central in this work, and its proof is entirely the result of the AI."
Co-developed rather than discovered, for reasons the author supplies himself. Each step is a subproblem he formulated and the model solved, the overall strategy was his, and he ran a control: asked cold, with no context, whether it could construct such a model, the AI returned two general ideas and concrete models that all fell into the uninteresting family. It could not do this unprompted. His summary is that "both the author and the AI played an essential role".
Verification
An arXiv preprint, unrefereed, with no independent endorsement, and no mathematics was checked here.
The AI attribution was checked, and it is the reason this entry exists at all. The abstract's "research assistant" sits at or below the bottom of the contribution ladder, which would have put the paper out of scope; the Supplemental Material names four specific contributions and attributes the central theorem's proof outright. Reading it is what moved the classification, and it is linked below so a reader can do the same.
The author is unusually candid in two directions at once. He writes that the proof strategy is "relatively simple and relatively standard" and that "human researchers would have found this proof", and he separately reports a cold-start control in which the model failed to construct such a model without context. Both cut against his own result being read as more autonomous than it was.