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arXiv:2603.11172

Integrable Free and Interacting Fermions

Zhao Zhang

nlin.SIcond-mat.stat-mechhep-thmath-phquant-phSubmitted Mar 11, 2026• Announced Mar 13, 2026• 0 citations

v1 · Mar 11, 2026 v2 · Jun 10, 2026 v3 · Oct 29, 2026

Abstract

Integrability conditions on local Hamiltonians for one-dimensional quantum systems to be free and interacting fermions are introduced. The definition of free fermion is the simultaneous satisfaction of the Yang-Baxter equation and Shastry's decorated star-triangle relation by the $R$-matrix, which is more general than the previous `free-fermion algebra' by Maassarani and more special than free fermions as in the context of exactly solvable quantum models or integrable classical two-dimensional vertex models dual to quantum spin chains. Free fermionic $R$-matrices are of the difference form and have a conjugation symmetry. These free Hamiltonians may sometimes be deformed by the conjugation operator to describe an integrable interacting system with non-relativistic $R$-matrices, as are the cases of the Hubbard model and the XY model in a longitudinal field. A further criterion is obtain on precisely when such deformations remain integrable. A practical procedure is proposed to iteratively solve the free fermionic $R$-matrices from local Hamiltonians, which can be used to construct non-relativistic $R$-matrices if the conditions are met.

Comments: 23 pages, 5 figures

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Integrability conditions on local Hamiltonians for one-dimensional quantum systems to be free and interacting fermions are introduced. The definition of free fermion is the simultaneous satisfaction of the Yang-Baxter equation and Shastry's decorated star-triangle relation by the

R

-matrix, which is more general than the previous `free-fermion algebra' by Maassarani and more special than free fermions as in the context of exactly solvable quantum models or integrable classical two-dimensional vertex models dual to quantum spin chains. Free fermionic

R

-matrices are of the difference form and have a conjugation symmetry. These free Hamiltonians may sometimes be deformed by the conjugation operator to describe an integrable interacting system with non-relativistic

R

-matrices, as are the cases of the Hubbard model and the XY model in a longitudinal field. A further criterion is obtain on precisely when such deformations remain integrable. A practical procedure is proposed to iteratively solve the free fermionic

R

-matrices from local Hamiltonians, which can be used to construct non-relativistic

R

-matrices if the conditions are met.

A bootstrap program is presented for algebraically solving the univariate R R -matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the R R -matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively in a self consistent way using a lemma due to Kennedy, which reconstructs the R R -matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However, at least for the most common examples, they always turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are indeed implied by the Reshetikhin condition.

A new type of Yang-Baxter equation (YBE) for

R

-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the

Z_N

symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for

N>2

, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated

R

-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter

R

-matrix---realized as an intertwiner of cyclic representations of

U_q(\mathfrak{sl}_2)

at a root of unity---and Shastry's two-parameter

R

-operator obtained from the decorated YBE.

We propose a new type of the Yang-Baxter equation (YBE) and the decorated Yang-Baxter equation (DYBE). Those relations for the fermionic R-operator were introduced recently as a tool to treat the integrability of the fermion models. Using the YBE and the DYBE for the XX fermion model, we construct the fermionic R-operator for the one-dimensional (1D) Hubbard model. It gives another proof of the integrability of the 1D Hubbard model. Furthermore a new approach to the SO(4) symmetry of the 1D Hubbard model is discussed.

Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.