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Which frustration graphs give free fermions?: A converse to the even-hole, claw-free theorem, and exact free sectors of the periodic Fendley chain

Anonymous

Published Sep 27, 2026• 0 citations
Publisher

Abstract

Unrefereed mathematical candidate: a faithful-Pauli converse to the even-hole, claw-free global spectrum theorem; an exact central-sector criterion; explicit all-coupling periodic Fendley blocks at N=8,9; weak prescribed-mode heredity; and a repaired three-mode parity-completeness argument. Written proofs, exact computer-algebra certificates for 107 listed graphs, and numerical census evidence have separate assurance boundaries. Producer replay and a separately implemented model-review checker are not external human peer review, unaffiliated reproduction, formal verification or exhaustive novelty. The remaining 1621 numerical candidates are not all-coupling certified. Original prose/data CC0-1.0; original code MIT; third-party exclusions in LICENSES.md.

DOI
10.5281/zenodo.22994979

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arXiv:2305.15625•May 25, 2023•10 citations

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Free-fermion solvability in quantum spin systems is increasingly understood to be governed by a graph Clifford algebra defined from the frustration graph of the Hamiltonian. When the frustration graph belongs to certain classes, such as the even-hole-free and claw-free (ECF) class, the Hamiltonian is solvable by hidden free fermions: it admits a free-fermion solution although it does not reduce to a Majorana bilinear under the Jordan-Wigner transformation. However, unlike in the Jordan-Wigner case, where each mode is a linear combination of single Majorana fermions, the explicit operator structure of the hidden free-fermion modes---and that of the local conserved charges---has remained obscure. In this work, we derive a path-product expansion that expresses each free-fermion mode as a linear combination of products along induced paths in the extended frustration graph. The expansion follows from the Krylov generating function and yields the modes directly, without using the transfer matrix or nonlocal conserved charges; the resulting mode decomposition also computes infinite-temperature dynamical correlation functions. We further obtain explicit expressions for local conserved charges as linear combinations of path products along induced paths; these charges apply beyond the free-fermion (ECF) class to more general claw-free frustration graphs. We also identify a family of generalized conserved charges containing both the known nonlocal conserved charges and these local charges. For the homogeneous periodic Fendley model, the local conserved charges exhibit the same Catalan-tree pattern as those of the spin-$1/2$ XXX chain.

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A free fermions in disguise model with claws

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Recently, several spin chain models have been discovered that admit solutions in terms of “free fermions in disguise.” A graph-theoretical treatment of such models was also established, giving sufficient conditions for free fermionic solvability. These conditions involve a particular property of the so-called frustration graph of the Hamiltonian, namely that it must be claw-free. Additionally, one set of sufficient conditions also requires the absence of so-called even holes. In this paper, we present a model with disguised free fermions where the frustration graph contains both claws and even holes. Special relations between coupling constants ensure that the free fermionic property still holds. Notably, the central elements associated with the even holes can be removed by fixing the gauge, revealing our model to be an integrable deformation within the original algebra of free fermions in disguise. The transfer matrix of this model can be factorized in a special case, thereby proving the conjectured free fermionic nature of a special quantum circuit published recently by two of the present authors.

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Exactly solvable models are often identified from structural features of their interactions. For quantum spin Hamiltonians, these are captured by the frustration graph of anticommuting Pauli terms. The frustration graph characterizes broad families of spin models that map to free fermions for arbitrary, or generic, values of their coupling strengths. Nevertheless, there exist non-generically solvable models that admit free-fermion solutions only at finely tuned couplings. In this work, we characterize a family of such models in terms of graph structures. Given an arbitrary instance of Fendley's free-fermions-in-disguise (FFD) chain, we construct a set of operators, called Krylov currents, that are exactly quadratic in its effective fermionic modes and thus extend it to a linear family of free-fermion solvable Hamiltonians. As the Krylov currents are nonlinear in the FFD Hamiltonian terms, the elements of this generated family are non-generically solvable, and they accordingly fail the known conditions for generic solvability. We prove that every Krylov current admits an expansion in terms of induced paths of the frustration graph and that they generate the full special-orthogonal algebra of Gaussian transformations. Our results identify path support as a signature of non-generic solvability, and they provide a systematic route to finding exactly solvable many-body Hamiltonians.

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A fixed-level certification dichotomy for an infinite family of frustration-free spin chains

Guillaume Lecomte

Preprint of the article "A fixed-level certification dichotomy for an infinite family of frustration-free spin chains" (22 pages, no figures). To every finite loopless directed graph T the paper attaches a rational, range-three, frustration-free spin-chain interaction: arcs of T carry hopping bands for domain walls, missing arcs act as hard local penalties. The main theorem is the exact three-term equivalence: T is acyclic <=> liminf_m gap_m(T) > 0 <=> delta_LTI(5) > 0, where gap_m is the spectral gap of the periodic chain on m sites and delta_LTI(5) is the value of the local translation-invariant (LTI) semidefinite certification hierarchy at level 5, its first sound level. The certification level is fixed while the size of the graph and the local Hilbert-space dimension are unbounded across the family. To the author's knowledge, this is the first fixed-level completeness result for a spectral-gap certification hierarchy over an infinite family of frustration-free chains: a single finite combinatorial invariant, acyclicity, decides both the thermodynamic spectral gap and the success of the certificate at one universal level. The acyclic direction is proved by an explicit rational certificate, a superpotential built from a topological height and three local corrections, giving delta_LTI(5) >= min{1/48, 1/sqrt(12H)} for a DAG of height H. The cyclic direction is made quantitative by a free-fermion analysis of the invariant winding sectors: directed cycles carry a diffusive lowest mode of energy of order 1/m^2, and the finite-volume gaps of the directed 2- and 3-cycles are computed exactly, 2(1-cos(pi/m)) and 2(1-cos(2pi/m)). The relationship with the PVBS chains of Bachmann and Nachtergaele and with earlier gapped-gapless classifications is discussed explicitly. All computer-assisted statements use exact symbolic arithmetic. The complete reproducibility package (symbolic pipeline, generated blocks with exact characteristic polynomials, Sturm data, LDL^T certificates, SHA-256 digests) is archived separately at https://doi.org/10.5281/zenodo.21790411. The manuscript has been submitted to Reports on Mathematical Physics.

•Aug 4, 2026•0 citations

No Szöllősi matrix lies in a quadruplet of mutually unbiased bases in dimension six

Anonymous

Unrefereed candidate excluding the specified closed two-circulant Szollosi family from MUB quadruplets in complex dimension six. The Fourier-family result re-proves prior work. The unrestricted dimension-six maximum remains conditional on MMVW Conjecture 2. The package separates an untrusted hull generator from an interval and graph checker; canonical half-open ownership repairs boundary coverage, and input-bound clean replay prevents stale cache acceptance. No end-to-end formal verification, unaffiliated reproduction or external human peer review is claimed. See the replay receipt for exact measured scope. Original prose and data CC0; original code MIT; third-party and review material retain their terms.

•Sep 30, 2026•0 citations
Unrefereed mathematical candidate: a faithful-Pauli converse to the even-hole, claw-free global spectrum theorem; an exact central-sector criterion; explicit all-coupling periodic Fendley blocks at N=8,9; weak prescribed-mode heredity; and a repaired three-mode parity-completeness argument. Written proofs, exact computer-algebra certificates for 107 listed graphs, and numerical census evidence have separate assurance boundaries. Producer replay and a separately implemented model-review checker are not external human peer review, unaffiliated reproduction, formal verification or exhaustive novelty. The remaining 1621 numerical candidates are not all-coupling certified. Original prose/data CC0-1.0; original code MIT; third-party exclusions in LICENSES.md.
We show that a quantum spin system has an exact description by non-interacting fermions if its frustration graph is claw-free and contains a simplicial clique. The frustration graph of a spin model captures the pairwise anticommutation relations between Pauli terms of its Hamiltonian in a given basis. This result captures a vast family of known free-fermion solutions. In previous work, it was shown that a free-fermion solution exists if the frustration graph is either a line graph, or (even-hole, claw)-free. The former case generalizes the celebrated Jordan-Wigner transformation and includes the exact solution to the Kitaev honeycomb model. The latter case generalizes a non-local solution to the four-fermion model given by Fendley. Our characterization unifies these two approaches, extending generalized Jordan-Wigner solutions to the non-local setting and generalizing the four-fermion solution to models of arbitrary spatial dimension. Our key technical insight is the identification of a class of cycle symmetries for all models with claw-free frustration graphs. We prove that these symmetries commute, and this allows us to apply Fendley's solution method to each symmetric subspace independently. Finally, we give a physical description of the fermion modes in terms of operators generated by repeated commutation with the Hamiltonian. This connects our framework to the developing body of work on operator Krylov subspaces. Our results deepen the connection between many-body physics and the mathematical theory of claw-free graphs.
Free-fermion solvability in quantum spin systems is increasingly understood to be governed by a graph Clifford algebra defined from the frustration graph of the Hamiltonian. When the frustration graph belongs to certain classes, such as the even-hole-free and claw-free (ECF) class, the Hamiltonian is solvable by hidden free fermions: it admits a free-fermion solution although it does not reduce to a Majorana bilinear under the Jordan-Wigner transformation. However, unlike in the Jordan-Wigner case, where each mode is a linear combination of single Majorana fermions, the explicit operator structure of the hidden free-fermion modes---and that of the local conserved charges---has remained obscure. In this work, we derive a path-product expansion that expresses each free-fermion mode as a linear combination of products along induced paths in the extended frustration graph. The expansion follows from the Krylov generating function and yields the modes directly, without using the transfer matrix or nonlocal conserved charges; the resulting mode decomposition also computes infinite-temperature dynamical correlation functions. We further obtain explicit expressions for local conserved charges as linear combinations of path products along induced paths; these charges apply beyond the free-fermion (ECF) class to more general claw-free frustration graphs. We also identify a family of generalized conserved charges containing both the known nonlocal conserved charges and these local charges. For the homogeneous periodic Fendley model, the local conserved charges exhibit the same Catalan-tree pattern as those of the spin-1/21/21/2 XXX chain.
Recently, several spin chain models have been discovered that admit solutions in terms of “free fermions in disguise.” A graph-theoretical treatment of such models was also established, giving sufficient conditions for free fermionic solvability. These conditions involve a particular property of the so-called frustration graph of the Hamiltonian, namely that it must be claw-free. Additionally, one set of sufficient conditions also requires the absence of so-called even holes. In this paper, we present a model with disguised free fermions where the frustration graph contains both claws and even holes. Special relations between coupling constants ensure that the free fermionic property still holds. Notably, the central elements associated with the even holes can be removed by fixing the gauge, revealing our model to be an integrable deformation within the original algebra of free fermions in disguise. The transfer matrix of this model can be factorized in a special case, thereby proving the conjectured free fermionic nature of a special quantum circuit published recently by two of the present authors.
Exactly solvable models are often identified from structural features of their interactions. For quantum spin Hamiltonians, these are captured by the frustration graph of anticommuting Pauli terms. The frustration graph characterizes broad families of spin models that map to free fermions for arbitrary, or generic, values of their coupling strengths. Nevertheless, there exist non-generically solvable models that admit free-fermion solutions only at finely tuned couplings. In this work, we characterize a family of such models in terms of graph structures. Given an arbitrary instance of Fendley's free-fermions-in-disguise (FFD) chain, we construct a set of operators, called Krylov currents, that are exactly quadratic in its effective fermionic modes and thus extend it to a linear family of free-fermion solvable Hamiltonians. As the Krylov currents are nonlinear in the FFD Hamiltonian terms, the elements of this generated family are non-generically solvable, and they accordingly fail the known conditions for generic solvability. We prove that every Krylov current admits an expansion in terms of induced paths of the frustration graph and that they generate the full special-orthogonal algebra of Gaussian transformations. Our results identify path support as a signature of non-generic solvability, and they provide a systematic route to finding exactly solvable many-body Hamiltonians.
Preprint of the article "A fixed-level certification dichotomy for an infinite family of frustration-free spin chains" (22 pages, no figures). To every finite loopless directed graph T the paper attaches a rational, range-three, frustration-free spin-chain interaction: arcs of T carry hopping bands for domain walls, missing arcs act as hard local penalties. The main theorem is the exact three-term equivalence: T is acyclic <=> liminf_m gap_m(T) > 0 <=> delta_LTI(5) > 0, where gap_m is the spectral gap of the periodic chain on m sites and delta_LTI(5) is the value of the local translation-invariant (LTI) semidefinite certification hierarchy at level 5, its first sound level. The certification level is fixed while the size of the graph and the local Hilbert-space dimension are unbounded across the family. To the author's knowledge, this is the first fixed-level completeness result for a spectral-gap certification hierarchy over an infinite family of frustration-free chains: a single finite combinatorial invariant, acyclicity, decides both the thermodynamic spectral gap and the success of the certificate at one universal level. The acyclic direction is proved by an explicit rational certificate, a superpotential built from a topological height and three local corrections, giving delta_LTI(5) >= min{1/48, 1/sqrt(12H)} for a DAG of height H. The cyclic direction is made quantitative by a free-fermion analysis of the invariant winding sectors: directed cycles carry a diffusive lowest mode of energy of order 1/m^2, and the finite-volume gaps of the directed 2- and 3-cycles are computed exactly, 2(1-cos(pi/m)) and 2(1-cos(2pi/m)). The relationship with the PVBS chains of Bachmann and Nachtergaele and with earlier gapped-gapless classifications is discussed explicitly. All computer-assisted statements use exact symbolic arithmetic. The complete reproducibility package (symbolic pipeline, generated blocks with exact characteristic polynomials, Sturm data, LDL^T certificates, SHA-256 digests) is archived separately at https://doi.org/10.5281/zenodo.21790411. The manuscript has been submitted to Reports on Mathematical Physics.
Unrefereed candidate excluding the specified closed two-circulant Szollosi family from MUB quadruplets in complex dimension six. The Fourier-family result re-proves prior work. The unrestricted dimension-six maximum remains conditional on MMVW Conjecture 2. The package separates an untrusted hull generator from an interval and graph checker; canonical half-open ownership repairs boundary coverage, and input-bound clean replay prevents stale cache acceptance. No end-to-end formal verification, unaffiliated reproduction or external human peer review is claimed. See the replay receipt for exact measured scope. Original prose and data CC0; original code MIT; third-party and review material retain their terms.