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arxiv: 2411.18685 · v5 · submitted 2024-11-27 · 🧮 math-ph · cond-mat.stat-mech· hep-th· math.MP· nlin.SI

All 4 x 4 solutions of the quantum Yang-Baxter equation

Marius de Leeuw , Vera Posch This is my paper

Pith reviewed 2026-05-23 16:27 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechhep-thmath.MPnlin.SI
keywords Yang-Baxter equationquantum R-matrixLax operatorsintegrable systems4x4 matricesnon-regular solutionsclassification
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The pith

The full set of 4x4 solutions to the quantum Yang-Baxter equation is now known after the non-regular cases are enumerated.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper finishes the classification of all 4 by 4 matrices satisfying the quantum Yang-Baxter equation by locating the non-regular solutions that remained after a recent listing of the regular ones. It presents several explicit new solutions and then examines the associated Lax operators. For regular solutions a direct correspondence exists between Lax operators and R-matrices that obey the Yang-Baxter equation. This link breaks for non-regular solutions, where some Lax operators generate regular R-matrices that instead satisfy only a modified Yang-Baxter equation.

Core claim

We complete the classification of 4 x 4 solutions of the Yang-Baxter equation. Regular solutions were recently classified and in this paper we find the remaining non-regular solutions. We present several new solutions, then consider regular and non-regular Lax operators and study their relation to the quantum Yang-Baxter equation. We show that for regular solutions there is a correspondence, which is lost in the non-regular case. In particular, we find non-regular Lax operators whose R-matrix from the fundamental commutation relations is regular but does not satisfy the Yang-Baxter equation. These R-matrices satisfy a modified Yang-Baxter equation instead.

What carries the argument

Non-regular 4x4 solutions to the quantum Yang-Baxter equation together with the regular/non-regular distinction for the associated Lax operators.

If this is right

  • Every 4x4 solution of the quantum Yang-Baxter equation is now accounted for.
  • Regular solutions maintain a direct link between their Lax operators and R-matrices obeying the Yang-Baxter equation.
  • Non-regular Lax operators can produce R-matrices that are regular yet obey only a modified Yang-Baxter equation.
  • The full list supplies all possible 4x4 R-matrices for constructing two-dimensional integrable quantum models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The completed list may be used as a lookup table when searching for new integrable spin chains or lattice models whose local operators are 4x4.
  • The modified Yang-Baxter equation satisfied by some non-regular cases could define a separate family of integrable systems worth separate classification.
  • The same separation into regular and non-regular cases might be attempted for 6x6 or larger matrices once their regular solutions are known.

Load-bearing premise

The recent classification of all regular 4x4 solutions is complete and contains no omissions or overlaps with the non-regular cases.

What would settle it

Discovery of any 4x4 matrix that satisfies the Yang-Baxter equation but appears in neither the prior regular list nor the new non-regular list provided here.

Figures

Figures reproduced from arXiv: 2411.18685 by Marius de Leeuw, Vera Posch.

Figure 1
Figure 1. Figure 1: Acting with different R-matrices will permute the Lax operators in a the triple product L1L2L3 by using the fundamental commutation relations. If we follow the arrows in the diagram, we see that R32R31R21R23R13R12 commutes with L1L2L3. For brevity, let us write A123 ≡ R32R31R21R23R13R12. (59) We have already shown that for w = 0 the R-matrix reduces to the Lax operator. We assume that the fundamental commu… view at source ↗
read the original abstract

In this paper, we complete the classification of 4 x 4 solutions of the Yang-Baxter equation. Regular solutions were recently classified and in this paper we find the remaining non-regular solutions. We present several new solutions, then consider regular and non-regular Lax operators and study their relation to the quantum Yang-Baxter equation. We show that for regular solutions there is a correspondence, which is lost in the non-regular case. In particular, we find non-regular Lax operators whose R-matrix from the fundamental commutation relations is regular but does not satisfy the Yang-Baxter equation. These R-matrices satisfy a modified Yang-Baxter equation instead.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to complete the classification of all 4×4 solutions of the quantum Yang-Baxter equation by enumerating the remaining non-regular solutions after a recent classification of the regular ones. It presents several new solutions, examines regular and non-regular Lax operators, and shows that the correspondence between Lax operators and the YBE holds for regular solutions but is lost for non-regular ones; in particular, it identifies non-regular Lax operators whose associated R-matrices are regular yet satisfy only a modified Yang-Baxter equation.

Significance. A complete classification of 4×4 QYBE solutions would be a useful reference for integrable systems and quantum algebra. The explicit new families and the observation that regularity of the R-matrix does not guarantee the YBE (but only a modified version) in the non-regular case provide concrete distinctions that could guide further constructions of integrable models.

major comments (2)
  1. [Introduction] Introduction and classification section: the central claim that the work finds 'all' 4×4 solutions requires that the cited prior classification of regular solutions is exhaustive and that the regularity condition partitions the full solution set without gaps or overlaps. The manuscript supplies no independent algebraic argument, computational cross-check over the 256-dimensional space of 4×4 matrices, or explicit verification that every solution is captured by one of the two categories.
  2. [Lax operators section] Lax-operator analysis (near the discussion of fundamental commutation relations): the examples of non-regular Lax operators producing regular R-matrices that fail the YBE are load-bearing for the distinction drawn between regular and non-regular cases. The precise form of the 'modified Yang-Baxter equation' satisfied by these R-matrices is not stated as an equation, nor is it verified component-wise for the listed matrices.
minor comments (2)
  1. Explicit component-wise expressions or a table listing the new non-regular solutions would improve readability and allow direct comparison with the prior regular list.
  2. The abstract states that 'several new solutions' are presented; a count or enumeration of how many distinct families (up to equivalence) are found would clarify the scope of the new results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Introduction] Introduction and classification section: the central claim that the work finds 'all' 4×4 solutions requires that the cited prior classification of regular solutions is exhaustive and that the regularity condition partitions the full solution set without gaps or overlaps. The manuscript supplies no independent algebraic argument, computational cross-check over the 256-dimensional space of 4×4 matrices, or explicit verification that every solution is captured by one of the two categories.

    Authors: Our claim to have completed the classification rests on combining the enumeration of non-regular solutions presented here with the recently published classification of all regular solutions. The regularity condition is an algebraic property defined in the literature that partitions the set of solutions into two disjoint classes by construction. We do not re-derive or independently verify the completeness of the regular classification within this manuscript, as our contribution focuses on the remaining non-regular cases. A full computational search over the 256-dimensional space of 4×4 matrices would require solving a system of 64 independent cubic equations and lies outside the scope of the present work. revision: no

  2. Referee: [Lax operators section] Lax-operator analysis (near the discussion of fundamental commutation relations): the examples of non-regular Lax operators producing regular R-matrices that fail the YBE are load-bearing for the distinction drawn between regular and non-regular cases. The precise form of the 'modified Yang-Baxter equation' satisfied by these R-matrices is not stated as an equation, nor is it verified component-wise for the listed matrices.

    Authors: We agree that the modified Yang-Baxter equation should be stated explicitly and that a component-wise verification would strengthen the presentation. In the revised manuscript we will insert the explicit form of the modified equation and supply a component-wise check for the representative examples discussed in that section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct enumeration of non-regular solutions.

full rationale

The paper completes the 4x4 YBE classification by enumerating non-regular solutions after citing a recent external classification of regular solutions. No equations or steps in the provided abstract or description reduce by construction to inputs, fitted parameters, or self-citations. The new families, Lax operator relations, and modified YBE observations are obtained independently via search or algebraic methods rather than tautological renaming or self-definition. The partition into regular/non-regular is taken as given from prior work and does not create a load-bearing loop within this manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard linear algebra over the complex numbers and the conventional definition of the Yang-Baxter equation; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of matrix multiplication and the definition of the Yang-Baxter equation hold over the complex numbers.
    Implicit background for any algebraic classification of matrix solutions.

pith-pipeline@v0.9.0 · 5644 in / 1177 out tokens · 21247 ms · 2026-05-23T16:27:20.815217+00:00 · methodology

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Forward citations

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  2. The quantum group structure of long-range integrable deformations

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Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 2 Pith papers · 4 internal anchors

  1. [1]

    THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL

    L. A. Takhtajan and L. D. Faddeev, “THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL” , Russian Mathematical Surveys 34, 11 (1979) , https://dx.doi.org/10.1070/RM1979v034n05ABEH003909

    work page doi:10.1070/rm1979v034n05abeh003909 1979

  2. [2]

    Some Exact Results for the Many-Body Problem in one Dimensi on with Repulsive Delta-Function Interaction

    C. N. Yang, “Some Exact Results for the Many-Body Problem in one Dimensi on with Repulsive Delta-Function Interaction ”, Phys. Rev. Lett. 19, 1312 (1967) , https://link.aps.org/doi/10.1103/PhysRevLett.19.1312

    work page doi:10.1103/physrevlett.19.1312 1967

  3. [3]

    Partition function of the Eight-Vertex lattice model

    R. J. Baxter, “Partition function of the Eight-Vertex lattice model” , Annals of Physics 70, 193 (1972) , https://www.sciencedirect.com/science/article/pii/0003491672903351

    work page arXiv 1972

  4. [4]

    Yang-Baxter Equations

    J. H. H. Perk and H. Au-Yang, “Yang-Baxter Equations”, math-ph/0606053, https://arxiv.org/abs/math-ph/0606053

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Yang-Baxter Equation in Integrable Systems

    M. Jimbo, “Yang-Baxter Equation in Integrable Systems” , World Scientific Publishing Company Pte Limited (1990)

    work page 1990

  6. [6]

    Baxter’s equations and algebraic geometry

    I. M. Krichever, “Baxter’s equations and algebraic geometry” , Functional Analysis and Its Applications 15, 92 (1981) , https://api.semanticscholar.org/CorpusID:121076970

    work page 1981

  7. [7]

    Suzuki ,\ http://www.sciencedirect.com/science/article/pii/037596019090962N journal journal Physics Letters A \ volume 146 ,\ pages 319 ( year 1990 ) NoStop

    J. Hietarinta, “All solutions to the constant quantum Yang-Baxter equatio n in two dimensions” , Physics Letters A 165, 245 (1992) , https://www.sciencedirect.com/science/article/pii/037596019290044M

    work page arXiv 1992

  8. [8]

    Hietarinta’s classification of 4 × 4 constant Yang-Baxter operators using algebraic approach

    S. Maity, V. K. Singh, P. Padmanabhan and V. Korepin, “Hietarinta’s classification of 4 × 4 constant Yang-Baxter operators using algebraic approach” , arxiv:2409.05375

    work page arXiv

  9. [9]

    Hopf algebras and the quantum Yang-Baxter equation

    V. G. Drinfeld, “Hopf algebras and the quantum Yang-Baxter equation” , Sov. Math. Dokl. 32, 254 (1985)

    work page 1985

  10. [10]

    A q difference analog of U(g) and the Yang-Baxter equation

    M. Jimbo, “A q difference analog of U(g) and the Yang-Baxter equation” , Lett. Math. Phys. 10, 63 (1985)

    work page 1985

  11. [11]

    Solving and classifying the solutions of the Yang-Baxter e quation through a differential approach. Two-state systems

    R. S. Vieira, “Solving and classifying the solutions of the Yang-Baxter e quation through a differential approach. Two-state systems” , Journal of High Energy Physics 2018, R. S. Vieira (2018) , http://dx.doi.org/10.1007/JHEP10(2018)110

    work page doi:10.1007/jhep10(2018)110 2018

  12. [12]

    Classifying integrable spin-1/2 chains with nearest neig hbour interactions

    M. de Leeuw, A. Pribytok and P. Ryan, “Classifying integrable spin-1/2 chains with nearest neig hbour interactions”, Journal of Physics A: Mathematical and Theoretical 52, 5052 01 (2019) , http://dx.doi.org/10.1088/1751-8121/ab529f

    work page doi:10.1088/1751-8121/ab529f 2019

  13. [13]

    Classifying Nearest-Neighbor Interactions and Deformations of AdS

    M. de Leeuw, C. Paletta, A. Pribytok, A. L. Retore and P. R yan, “Classifying Nearest-Neighbor Interactions and Deformations of AdS” , Phys. Rev. Lett. 125, 031604 (2020) , arxiv:2003.04332

    work page arXiv 2020

  14. [14]

    Yang-Baxter and the Boost: splitting the difference

    M. de Leeuw, C. Paletta, A. Pribytok, A. L. Retore and P. R yan, “Yang-Baxter and the Boost: splitting the difference”, SciPost Physics 11, P. Ryan (2021) , http://dx.doi.org/10.21468/SciPostPhys.11.3.069

    work page doi:10.21468/scipostphys.11.3.069 2021

  15. [15]

    All regular 4 × 4 solutions of the Yang-Baxter equation

    L. Corcoran and M. de Leeuw, “All regular 4 × 4 solutions of the Yang-Baxter equation” , arxiv:2306.10423

    work page arXiv

  16. [16]

    New spectral-parameter dependent solutions of the Yang-Baxter equation

    A. S. Garkun, S. K. Barik, A. K. Fedorov and V. Gritsev, “New spectral-parameter dependent solutions of the Yang-Baxter equation” , arxiv:2401.12710

    work page arXiv

  17. [17]

    Quantization of Lie Groups and Lie Algebras

    L. D. Faddeev, N. Y. Reshetikhin and L. A. Takhtajan, “Quantization of Lie Groups and Lie Algebras” , Alg. Anal. 1, 178 (1989)

    work page 1989

  18. [18]

    How Algebraic Bethe Ansatz works for integrable model

    L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model” , hep-th/9605187, in: “Les Houches School of Physics: Astrophysical Sources of Gravit ational Radiation” , pp. pp. 149–219

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    Solving the Yang-Baxter, tetrahedron and higher simplex e quations using Clifford algebras

    P. Padmanabhan and V. Korepin, “Solving the Yang-Baxter, tetrahedron and higher simplex e quations using Clifford algebras” , Nucl. Phys. B 1007, 116664 (2024) , arxiv:2404.11501. 16

    work page arXiv 2024

  20. [20]

    Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions

    M. Gerstenhaber and A. Giaquinto, “Boundary solutions of the quantum Yang-Baxter equation an d solutions in three dimensions” , q-alg/9710033, https://arxiv.org/abs/q-alg/9710033

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    GENERALIZED YANG-BAXTER EQUATION

    R. M. KASHAEV and Y. G. STROGANOV, “GENERALIZED YANG-BAXTER EQUATION” , Modern Physics Letters A 08, 2299–2309 (1993) , http://dx.doi.org/10.1142/S0217732393003603

    work page doi:10.1142/s0217732393003603 1993

  22. [22]

    On Solutions to the Twisted Yang-Baxter equation

    V. Tarasov, “On Solutions to the Twisted Yang-Baxter equation ” , hep-th/9403011, https://arxiv.org/abs/hep-th/9403011

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Geometry and Classificatin of Solutions of the Classical Dy namical Yang-Baxter Equation

    P. Etingof and A. Varchenko, “Geometry and Classificatin of Solutions of the Classical Dy namical Yang-Baxter Equation ”, Communications in Mathematical Physics 192, 77–120 (1998) , http://dx.doi.org/10.1007/s002200050292. 17

    work page doi:10.1007/s002200050292 1998