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arXiv:2610.03557

A graph-theoretic analysis of non-generic free-fermion solvability by Krylov decompositions

Jannis Ruh, Samuel J. Elman, Adrian Chapman

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

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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43 pages, 7 figures

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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.
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.
We present a graph-theoretic characterization of when a quantum spin model admits an exact solution via a mapping to free parafermions. Our characterization is based on the concept of a frustration graph, which represents the commutation relations between Weyl operators of a Hamiltonian. We show that a quantum spin system has an exact free-parafermion solution if its frustration graph is an oriented indifference graph. Furthermore, we show that if the frustration graph of a model can be dipath oriented via switching operations, then the model is integrable in the sense that there is a family of commuting independent set charges. Additionally, we establish an efficient algorithm for deciding whether this is possible. Our characterization extends that given for free-fermion solvability. Finally, we apply our results to solve three qudit spin models.
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.
Abstract Free fermions in disguise (FFD) Hamiltonians describe spin chains which can be mapped to free fermions, but not via a Jordan-Wigner transformation. Although the mapping gives access to the full Hamiltonian spectrum, the computation of spin correlation functions is generally hard. Indeed, the dictionary between states in the spin and free-fermion Hilbert spaces is highly non-trivial, due to the non-linear and non-local nature of the mapping, as well as the exponential degeneracy of the Hamiltonian eigenspaces. In this work, we provide a series of results characterizing the Hilbert space associated to FFD Hamiltonians. We focus on the original model introduced by Paul Fendley and show that the corresponding Hilbert space admits the exact factorization H = H F ⊗ H D , where H F hosts the fermionic operators, while H D accounts for the exponential degeneracy of the energy eigenspaces. By constructing a family of spin operators generating the operator algebra supported on H D , we further show that H D = H F ′ ⊗ H D ~ , where H F ′ hosts ancillary free fermions in disguise, while H D ~ is generated by the common eigenstates of an extensive set of commuting Pauli strings. Our construction allows us to fully resolve the exponential degeneracy of all Hamiltonian eigenspaces and is expected to have implications for the computation of spin correlation functions, both in and out of equilibrium.
Recently, a class of spin chains known as ``free fermions in disguise'' (FFD) has been discovered, which possess hidden free-fermion spectra even though they are not solvable via the standard Jordan-Wigner transformation. In this work, we extend this FFD framework to open quantum systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We establish a general class of exactly solvable open quantum systems within the FFD framework: if the Liouvillian frustration graph is claw-free and has a simplicial clique, the Liouvillian possesses a hidden free-fermion spectrum. In particular, the (even-hole, claw)-free condition automatically guarantees this, enabling exact computation of the Liouvillian gap and an infinite-temperature autocorrelation function. Our results provide the first realization of the FFD mechanism in open quantum systems.