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arXiv:2408.09684 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

A graph-theoretic framework for free-parafermion solvability

Ryan L. Mann, Samuel J. Elman, David R. Wood, Adrian Chapman

Published Apr 1, 2025• 5 citations
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Abstract

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.

DOI
10.1098/rspa.2024.0671

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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.
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.
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.
Exactly solvable models are essential in physics. For many-body spin- 1 / 2 systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions.
A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.
We consider quantum error-correcting subsystem codes whose gauge generators realize a translationinvariant, free-fermion-solvable spin model.In this setting, errors are suppressed by a Hamiltonian whose terms are the gauge generators of the code and whose exact spectrum and eigenstates can be found via a generalized Jordan-Wigner transformation.Such solutions are characterized by the frustration graph of the Hamiltonian: the graph whose vertices are Hamiltonian terms, which are neighboring if the terms anticommute.We provide methods for embedding a given frustration graph in the anticommutation relations of a spin model and present the first known example of an exactly solvable spin model with a two-dimensional free-fermion description and exact topological qubits.This model can be viewed as a free-fermionized version of the two-dimensional Bacon-Shor code.Using graph-theoretic tools to study the unit cell, we give an efficient algorithm for deciding if a given translation-invariant spin model is solvable, and explicitly construct the solution.Further, we examine the energetics of these exactly solvable models from the graph-theoretic perspective and show that the relevant gaps of the spin model correspond to known graph-theoretic quantities: the skew energy and the median eigenvalue of an oriented graph.Finally, we numerically search for models that have large spectral gaps above the ground-state spin configuration and thus exhibit particularly robust thermal suppression of errors.These results suggest that optimal models will have low dimensionality and odd coordination numbers, and that the primary limit to energetic error suppression is the skew energy difference between different symmetry sectors rather than single-particle excitations of the free fermions.
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.