Free fermions in disguise without exponential degeneracies: simple connections to the XY chain and the family of non-degenerate constructions

Free fermions in disguise (FFD) are spin chains whose exact spectra are free-fermionic even though no Jordan-Wigner transformation produces them; until 2026 every known example suffered degeneracies growing exponentially with system size. Pozsgay (arXiv:2606.09462) constructed the first FFD-type model free of exponential degeneracies by combining two commuting free-fermionic Hamiltonians whose compatibility is enforced by a global coupling parameter, realizing a continuous interpolation between the Jordan-Wigner-solvable XY chain and Fendley's original FFD model. The source paper leaves two questions explicitly open. First, is there a 'simple' connection — a finite-depth quantum circuit or a matrix product operator of finite bond dimension — relating the free-fermionic structure of the FFD-type model to that of the XY chain (which would suggest that even the original FFD model is obtainable from the XY chain by such means)? Second, which mechanisms beyond the commuting-pair construction can produce FFD-type models without exponential degeneracies, and which claw-free frustration graphs attaining the eigenmode-counting bound admit genuine spin-chain realizations? Both questions remain open.

Background

Exact solvability in one-dimensional quantum spin chains is organized around free fermions. The Jordan-Wigner (JW) transformation maps spin-1/2 operators to fermionic operators, and when the resulting Hamiltonian is quadratic (bilinear) in those fermions — as for the Ising and XY chains (the XY chain being the spin-1/2 chain with nearest-neighbour exchange terms built from the $X$ and $Y$ Pauli matrices, of which the transverse-field Ising chain is a special case) — the full spectrum follows from the eigenvalues of a matrix whose size grows only linearly with chain length.

In 2019 Fendley discovered a spin chain with three-site interactions whose spectrum is nevertheless free-fermionic, although no JW transformation, generalized or otherwise, produces it; he named the phenomenon "free fermions in disguise" (FFD). Elman, Chapman, and Flammia generalized the construction and proved rigorously that FFD models cannot be solved by generalized JW transformations, so the phenomenon lies genuinely outside the JW paradigm. Further families were found by Fendley and Pozsgay and by Fukai, Vona, and Pozsgay, all sharing one structural defect: degeneracies that grow exponentially with the volume and are homogeneous across the spectrum.

The solvability theory is graph-theoretic. A Pauli string (a product of Pauli matrices on distinct sites) squares to $\pm 1$ and either commutes or anticommutes with any other Pauli string; the frustration graph has one vertex per Hamiltonian term and an edge between two terms exactly when they anticommute. Graph-theoretic criteria for free-fermion solvability in terms of this graph were established independently by Chapman and Flammia, who reduced recognizing solvable models to recognizing line graphs (graphs whose vertices are the edges of another graph, two vertices adjacent when the corresponding edges share an endpoint), and by Ogura, Imamura, Kameyama, Minami, and Sato via simplicial homology. In the FFD literature the operative sufficient condition is that the frustration graph be claw-free (a claw being the induced star $K_{1,3}$) and free of even holes (a hole being an induced cycle of length at least four). The number of fermionic eigenmodes equals the independence number $S$ of the frustration graph, the size of a largest set of pairwise non-adjacent vertices. For a connected claw-free graph on $M$ vertices a classic theorem of Sumner and Las Vergnas gives $S \le \lceil M/2\rceil$. Meanwhile, the abstract algebra behind the terms — a graph-Clifford algebra, whose generators square to the identity and pairwise commute or anticommute — requires about $M/2 + O(1)$ qubits in its smallest faithful representation. Whenever the ratio $S/M$ is below $1/2$, the eigenmodes cannot resolve the Hilbert space and an extensive number of zero modes generates exponential degeneracy: for Fendley's original FFD model $S/M = 1/3$, and Vernier and Piroli showed that its Hilbert space factorizes exactly as $\mathcal{H} = \mathcal{H}_F \otimes \mathcal{H}_D$, with the "disguise" factor $\mathcal{H}_D$ carrying precisely this exponential degeneracy.

The source broke this barrier. Pozsgay takes a double tensor product of two Ising-type algebras with generators $B_j$ and $\tilde B_j$ ($j = 1,\dots,M$; each squares to a coupling-dependent constant, and only neighboring generators anticommute), and builds two Hamiltonians $H_1 = \sum_j B_j + \sum_j A_j$ and $H_2 = \sum_j \tilde B_j + \sum_j \tilde A_j$ from the linear generators and the three-generator terms $A_j = i\lambda B_j B_{j+1}\tilde B_j$ and $\tilde A_j = -i\lambda \tilde B_j \tilde B_{j+1} B_{j+1}$. Each of $H_1, H_2$ separately has a frustration graph that is claw-free and free of even holes and hence is free-fermionic, but supplies only half the eigenmodes needed; the combined generator set does not satisfy these graph conditions, so for generic couplings integrability would be destroyed. The crucial ingredient is the global-$\lambda$ condition — all cubic coupling constants collapse to one parameter — under which quadratic relations among the generators guarantee $[H_1, H_2] = 0$ and the compatibility of the two free-fermionic structures. Together they supply $2\lfloor (M+1)/2 \rfloor$ eigenmodes, matching the $M + O(1)$ qubits of the minimal representation up to $O(1)$, so no exponential degeneracy occurs for generic couplings. In a second representation the combination $H_1 - H_2$ is a chain of two-site $Y_j X_{j+1}$-type terms (a U(1)-invariant XY chain with a Dzyaloshinskii–Moriya interaction, i.e. an antisymmetric spin-spin coupling) plus three-site corrections, and it interpolates continuously between the XY chain and Fendley's FFD model; along this line the degeneracy is constant, except at the FFD endpoint. The author reads this as evidence that FFD-type and JW-solvable models "are not truly separate families".

Two notions from tensor-network theory frame the first open question. A finite-depth quantum circuit is a unitary built from local gates arranged in a number of layers that does not grow with system size; a matrix product operator (MPO) is a one-dimensional tensor network specified by local tensors and an auxiliary bond dimension, and a bond dimension bounded by a fixed constant makes the operator "simple" in the same sense. The source paper itself shows that the model's transfer matrix (the generating function of the commuting conserved charges) admits an MPO of bond dimension 3 and its fermionic operators an MPO of bond dimension 4 — so what is at issue is not MPO-representability of the operators, but whether some circuit or MPO intertwines the free-fermionic structures of the two model families.

Problem Statement

The two questions below are posed together in the source paper and share one scientific objective: locating hidden free-fermion (FFD-type) solvability relative to Jordan-Wigner solvability — structurally through question (A) and in scope through question (B).

(A) The simple-connection question. The model of arXiv:2606.09462 is a perturbation of a JW-solvable XY chain, its free-fermionic operators varying continuously with the interpolation parameter $\lambda$. Does there exist a "simple" connection between the two structures: a finite-depth unitary quantum circuit, or a matrix product operator of finite bond dimension independent of chain length, that relates the free-fermionic structure (mode algebra, transfer matrices, or Hamiltonian family) of the FFD-type interpolation model to that of the Jordan-Wigner-solvable XY chain? If such a circuit or MPO exists, it would suggest that even Fendley's original FFD model — the $\lambda \to \infty$ endpoint — can be obtained from the XY chain by such simple means. An answer is either an explicit construction together with a proof of the claimed structural property at all system sizes, or a rigorous proof that no such connection can exist; the source states that currently it is not known whether such a connection can or cannot exist.

(B) The mechanism-family question. It is not yet understood whether mechanisms other than the commuting-pair/global-$\lambda$ construction of the source can produce FFD-type models free of exponential degeneracies; the author reports having discovered one further model with a related but slightly different mechanism, but the full family of possible constructions has not been characterized. Determine which mechanisms (equivalently, which graph-Clifford-algebraic or frustration-graph data) yield non-degenerate FFD-type models — in particular, which connected claw-free frustration graphs attaining the independence-number bound $\lceil M/2 \rceil$ admit spin-chain realizations whose free-fermion solvability is not equivalent, under simple unitary transformations, to Jordan-Wigner solvability. An answer may consist of genuinely new explicit constructions (with proofs of free-fermionic solvability and of the eigenmode count excluding exponential degeneracy) together with a characterization of the family, or of a classification theorem showing the commuting-pair mechanism is essentially the only one.

References

1. B. Pozsgay, "Free fermions in disguise without exponential degeneracies", arXiv:2606.09462 (2026). DOI: 10.48550/arXiv.2606.09462. https://arxiv.org/abs/2606.09462
2. P. Fendley, "Free fermions in disguise", J. Phys. A: Math. Theor. 52, 335002 (2019). DOI: 10.1088/1751-8121/ab305d. arXiv:1901.08078. https://arxiv.org/abs/1901.08078
3. S. J. Elman, A. Chapman, S. T. Flammia, "Free fermions behind the disguise", Commun. Math. Phys. 388, 969 (2021). DOI: 10.1007/s00220-021-04220-w. arXiv:2012.07857. https://arxiv.org/abs/2012.07857
4. P. Fendley, B. Pozsgay, "Free fermions beyond Jordan and Wigner", SciPost Phys. 16, 102 (2024). DOI: 10.21468/SciPostPhys.16.4.102. arXiv:2310.19897. https://arxiv.org/abs/2310.19897
5. A. Chapman, S. T. Flammia, "Characterization of solvable spin models via graph invariants", Quantum 4, 278 (2020). DOI: 10.22331/q-2020-06-04-278. arXiv:2003.05465. https://arxiv.org/abs/2003.05465
6. M. Ogura, Y. Imamura, N. Kameyama, K. Minami, M. Sato, "Geometric criterion for solvability of lattice spin systems", Phys. Rev. B 102, 245118 (2020). DOI: 10.1103/PhysRevB.102.245118. arXiv:2003.13264. https://arxiv.org/abs/2003.13264
7. K. Fukai, B. Pozsgay, I. Vona, "Solving models with generalized free fermions I: Algebras and eigenstates", Phys. Rev. E 114, 024124 (2026). DOI: 10.1103/nly2-3zq3. arXiv:2602.03431. https://arxiv.org/abs/2602.03431
8. K. Fukai, B. Pozsgay, I. Vona, "Solving models with generalized free fermions II: Path-product expansion and conserved charges", arXiv:2605.31453 (2026). DOI: 10.48550/arXiv.2605.31453. https://arxiv.org/abs/2605.31453
9. K. Fukai, I. Vona, B. Pozsgay, "A free fermions in disguise model with claws", SciPost Phys. 21, 011 (2026). DOI: 10.21468/SciPostPhys.21.1.011. arXiv:2508.05789. https://arxiv.org/abs/2508.05789
10. E. Vernier, L. Piroli, "The Hilbert-space structure of free fermions in disguise", J. Stat. Mech. (2026) 013101. DOI: 10.1088/1742-5468/ae29f5. arXiv:2507.15959. https://arxiv.org/abs/2507.15959
11. K. Fukai, B. Pozsgay, "Quantum circuits with free fermions in disguise", J. Phys. A: Math. Theor. 58, 175202 (2025). DOI: 10.1088/1751-8121/adcd18. arXiv:2402.02984. https://arxiv.org/abs/2402.02984
12. D. Szász-Schagrin, D. Cristani, L. Piroli, E. Vernier, "Construction and simulability of quantum circuits with free fermions in disguise", Quantum Sci. Technol. 11, 015044 (2026). DOI: 10.1088/2058-9565/ae390d. arXiv:2509.22585. https://arxiv.org/abs/2509.22585
13. D. Szász-Schagrin, P. Bayona-Pena, L. Piroli, E. Vernier, "Correlation and entanglement dynamics of free fermions in disguise", arXiv:2607.02359 (2026). DOI: 10.48550/arXiv.2607.02359. https://arxiv.org/abs/2607.02359
14. J. Ruh, S. J. Elman, "Expanding the class of free fermions via twin-collapse methods", arXiv:2509.09092 (2025). DOI: 10.48550/arXiv.2509.09092. https://arxiv.org/abs/2509.09092
15. A. Sinha, S. Maity, P. Padmanabhan, V. Korepin, "Hidden Ising models from the generalized Yang-Baxter equation", arXiv:2605.30007 (2026). DOI: 10.48550/arXiv.2605.30007. https://arxiv.org/abs/2605.30007
16. B. Pozsgay, R. Sharipov, A. Tiutiakina, I. Vona, "Adiabatic gauge potential and integrability breaking with free fermions in disguise", SciPost Phys. 17, 075 (2024). DOI: 10.21468/SciPostPhys.17.3.075. arXiv:2402.12979. https://arxiv.org/abs/2402.12979