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A Graph-Theoretic Framework for Free-Parafermion Solvability

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

TL;DR

The paper introduces a graph-theoretic framework for free parafermion solvability in qudit spin systems, tying exact solvability to the structure of the frustration graph $G$. A key result is that an oriented indifference graph $G$ yields an exact free-parafermion solution, with single-particle energies determined by the roots of the independence polynomial via $Z_G(-\varepsilon_k^{-d})=0$, and parafermionic modes $\psi_{p,k}$ reconstructing the Hamiltonian as a sum over parafermionic bundles. If $G$ is dipath orientable (via switching), the model becomes integrable with a family of commuting independent-set charges, and the authors provide a polynomial-time algorithm to test this condition. The framework is demonstrated on three qudit spin models, extending free-fermion methods to parafermions and offering practical tools for discovering new solvable systems. This work broadens the reach of exact solvability techniques to higher-dimensional and more complex quantum spin models, with potential implications for topological phases and quantum computation.

Abstract

We present a graph-theoretic characterisation of when a quantum spin model admits an exact solution via a mapping to free parafermions. Our characterisation 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. Further, 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 characterisation extends that given for free-fermion solvability. Finally, we apply our results to solve three qudit spin models.

A Graph-Theoretic Framework for Free-Parafermion Solvability

TL;DR

The paper introduces a graph-theoretic framework for free parafermion solvability in qudit spin systems, tying exact solvability to the structure of the frustration graph . A key result is that an oriented indifference graph yields an exact free-parafermion solution, with single-particle energies determined by the roots of the independence polynomial via , and parafermionic modes reconstructing the Hamiltonian as a sum over parafermionic bundles. If is dipath orientable (via switching), the model becomes integrable with a family of commuting independent-set charges, and the authors provide a polynomial-time algorithm to test this condition. The framework is demonstrated on three qudit spin models, extending free-fermion methods to parafermions and offering practical tools for discovering new solvable systems. This work broadens the reach of exact solvability techniques to higher-dimensional and more complex quantum spin models, with potential implications for topological phases and quantum computation.

Abstract

We present a graph-theoretic characterisation of when a quantum spin model admits an exact solution via a mapping to free parafermions. Our characterisation 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. Further, 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 characterisation extends that given for free-fermion solvability. Finally, we apply our results to solve three qudit spin models.
Paper Structure (14 sections, 22 theorems, 93 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Qudit Models
  5. Parafermions
  6. Free-Parafermion Solvability and Integrability
  7. Dipath Orientation via Switching Operations
  8. Applications
  9. Conclusion & Outlook
  10. Proof of Lemma \ref{['lemma:IndependencePolynomialTransferMatrices']}
  11. Proof of Lemma \ref{['lemma:HamiltonianParafermionicModeCommutator']}
  12. Proof of Lemma \ref{['lemma:ParafermionicModeCommutator']}
  13. Proof of Lemma \ref{['lemma:ProjectorOperatorRelations']}
  14. Proof of Lemma \ref{['lemma:HamiltonianProjectorOperatorRelation']}

Key Result

Proposition 1

Let $G$ be a graph and let $v$ be a vertex in $G$. The transfer operator $T_G(x)$ and independence polynomial $Z_G(x)$ satisfy the following recursion relations.

Figures (2)

  • Figure 1: The frustration graph for the one-dimensional model introduced in Ref. alcaraz2020integrable with $n=14$ and $p=2$.
  • Figure 2: The frustration graph for the one-dimensional model with $n=2$.

Theorems & Definitions (48)

  • Definition 1: Independence polynomial
  • Definition 2: Frustration graph
  • Definition 3: Independent set charges
  • Definition 4: Transfer operator
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • ...and 38 more