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Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions

Akihiro Hokkyo1,*, Mizuki Sanatani2,†, and Yuuya Chiba3,‡

  • 1Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8654, Japan
  • 2Department of Basic Science, Department of Multidisciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
  • 3Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, Pioneering Research Institute (PRI), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

  • *Contact author: hokkyo@cat.phys.s.u-tokyo.ac.jp
  • †Contact author: yamaguchi-q@g.ecc.u-tokyo.ac.jp
  • ‡Contact author: yuya.chiba@riken.jp
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Phys. Rev. Research 7, 043297 – Published 18 December, 2025

DOI: https://doi.org/10.1103/vjtq-sz21

Abstract

We provide a complete classification of the integrability and nonintegrability of the spin-1 bilinear-biquadratic model with a uniaxial anisotropic field, which includes the Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki model. It is rigorously shown that, within this class, the only integrable systems are those that have been solved by the Bethe ansatz method, and that all other systems are nonintegrable, in the sense that they do not have nontrivial local conserved quantities. Here, “nontrivial” excludes quantities like the Hamiltonian or the total magnetization, and “local” refers to sums of operators that act only on sites within a finite distance. This result establishes the nonintegrability of the Affleck-Kennedy-Lieb-Tasaki model and, consequently, demonstrates that the quantum many-body scars observed in this model emerge independently of any conservation laws of local quantities. Furthermore, we extend the proof of nonintegrability to more general spin-1 models that encompass anisotropic extensions of the bilinear-biquadratic Hamiltonian and completely classified the integrability of generic Hamiltonians that possess translational symmetry, U(1) symmetry, time-reversal symmetry, and spin-flip symmetry. Our result has accomplished a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.

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References (81)

  1. H. Bethe, Zur theorie der metalle: I. Eigenwerte und eigenfunktionen der linearen atomkette, Z. Phys. 71, 205 (1931).
  2. E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantum inverse problem method. I, Theor. Math. Phys. 40, 688 (1979).
  3. L. A. Takhtadzhan and L. D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surv. 34, 11 (1979).
  4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, San Diego, 1982).
  5. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, 1993).
  6. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, 1999).
  7. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  8. T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: A theoretical overview, J. Phys. B: At. Mol. Opt. Phys. 51, 112001 (2018).
  9. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  10. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  11. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  12. M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007).
  13. L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. (2016) 064007.
  14. M. S. Green, Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids, J. Chem. Phys. 22, 398 (1954).
  15. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  16. P. Mazur, Non-ergodicity of phase functions in certain systems, Physica 43, 533 (1969).
  17. M. Suzuki, Ergodicity, constants of motion, and bounds for susceptibilities, Physica 51, 277 (1971).
  18. J. Sirker, Transport in one-dimensional integrable quantum systems, SciPost Phys. Lect. Notes 17, 17 (2020).
  19. Y. Chiba, K. Asano, and A. Shimizu, Anomalous behavior of magnetic susceptibility obtained by quench experiments in isolated quantum systems, Phys. Rev. Lett. 124, 110609 (2020).
  20. Y. Chiba and A. Shimizu, Key observable for linear thermalization, Phys. Rev. Res. 5, 033037 (2023).
  21. K. Saito, Strong evidence of normal heat conduction in a one-dimensional quantum system, Europhys. Lett. 61, 34 (2003).
  22. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  23. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  24. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018).
  25. S. Moudgalya, S. Rachel, B. A. Bernevig, and N. Regnault, Exact excited states of nonintegrable models, Phys. Rev. B 98, 235155 (2018).
  26. D. K. Mark, C.-J. Lin, and O. I. Motrunich, Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models, Phys. Rev. B 101, 195131 (2020).
  27. S. Moudgalya, E. O’Brien, B. A. Bernevig, P. Fendley, and N. Regnault, Large classes of quantum scarred Hamiltonians from matrix product states, Phys. Rev. B 102, 085120 (2020).
  28. N. O’Dea, F. Burnell, A. Chandran, and V. Khemani, From tunnels to towers: Quantum scars from lie algebras and q-deformed lie algebras, Phys. Rev. Res. 2, 043305 (2020).
  29. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  30. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Commun. Math. Phys. 115, 477 (1988).
  31. S. Moudgalya, N. Regnault, and B. A. Bernevig, Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis, Phys. Rev. B 98, 235156 (2018).
  32. S. R. White and D. A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S = 1 Heisenberg chain, Phys. Rev. B 48, 3844 (1993).
  33. G. V. Uimin, One-Dimensional problem for S = 1 with modified antiferromagnetic Hamiltonian, JETP Lett. 12, 225 (1970).
  34. C. K. Lai, Lattice gas with nearest-neighbor interaction in one dimension with arbitrary statistics, J. Math. Phys. 15, 1675 (1974).
  35. B. Sutherland, Model for a multicomponent quantum system, Phys. Rev. B 12, 3795 (1975).
  36. A. Schmitt, K.-H. Mütter, and M. Karbach, The spin-1 Lai- sutherland model with external and internal fields: I. The phase diagram, J. Phys. A: Math. Gen. 29, 3951 (1996).
  37. L. A. Takhtajan, The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87, 479 (1982).
  38. H. M. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S, Phys. Lett. A 90, 479 (1982).
  39. J. B. Parkinson, On the integrability of the S = 1 Quantum spin chain with pure biquadratic exchange, J. Phys. C: Solid State Phys. 20, L1029 (1987).
  40. A. Klümper, New results for q-state vertex models and the pure biquadratic spin-1 Hamiltonian, Europhys. Lett. 9, 815 (1989).
  41. M. N. Barber and M. T. Batchelor, Spectrum of the biquadratic spin-1 Antiferromagnetic chain, Phys. Rev. B 40, 4621 (1989).
  42. H.-J. Mikeska and A. K. Kolezhuk, One-Dimensional magnetism, in Quantum Magnetism, edited by U. Schollwöck, J. Richter, D. J. J. Farnell, and R. F. Bishop (Springer, Berlin, 2004), pp. 1–83.
  43. K. Rodríguez, A. Argüelles, A. K. Kolezhuk, L. Santos, and T. Vekua, Field-Induced phase transitions of repulsive spin-1 Bosons in optical lattices, Phys. Rev. Lett. 106, 105302 (2011).
  44. G. De Chiara, M. Lewenstein, and A. Sanpera, Bilinear-biquadratic spin-1 chain undergoing quadratic Zeeman effect, Phys. Rev. B 84, 054451 (2011).
  45. T. Quella, Symmetry-protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain, Phys. Rev. B 103, 054404 (2021).
  46. M. V. Rakov and M. Weyrauch, Bilinear-biquadratic spin-1 model in the Haldane and dimerized phases, Phys. Rev. B 105, 024424 (2022).
  47. A. Dhar, D. Jaschke, and L. D. Carr, Dynamics for the Haldane phase in the bilinear-biquadratic model, Phys. Rev. B 105, 094309 (2022).
  48. L. F. Santos, F. Pérez-Bernal, and E. J. Torres-Herrera, Speck of chaos, Phys. Rev. Res. 2, 043034 (2020).
  49. E. Ilievski and T. Prosen, Exact steady state manifold of a boundary driven spin-1 Lai-Sutherland chain, Nucl. Phys. B 882, 485 (2014).
  50. T. Prosen, Matrix product solutions of boundary driven quantum chains, J. Phys. A: Math. Theor. 48, 373001 (2015).
  51. L. Piroli and E. Vernier, Quasi-local conserved charges and spin transport in spin-1 integrable chains, J. Stat. Mech. (2016) 053106.
  52. M. Dupont and J. E. Moore, Universal spin dynamics in infinite-temperature one-dimensional quantum magnets, Phys. Rev. B 101, 121106(R) (2020).
  53. L. S. Lima, Spin transport in the bilinear–biquadratic Heisenberg model on square lattice, Solid State Commun. 368, 115177 (2023).
  54. N. Shiraishi, Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field, Europhys. Lett. 128, 17002 (2019).
  55. Y. Chiba, Proof of absence of local conserved quantities in the mixed-field Ising chain, Phys. Rev. B 109, 035123 (2024).
  56. H. K. Park and S. Lee, Graph-theoretical proof of nonintegrability in quantum many-body systems: Application to the PXP model, Phys. Rev. B 111, L081101 (2025).
  57. N. Shiraishi, Absence of local conserved quantity in the Heisenberg model with next-nearest-neighbor interaction, J. Stat. Phys. 191, 114 (2024).
  58. M. Sanatani, Y. Chiba, and N. Shiraishi, Complete classification of integrability and non-integrability for spin-1/2 Chain with symmetric nearest-neighbor interaction, arXiv:2411.02162.
  59. J. S. Caux and J. Mossel, Remarks on the notion of quantum integrability, J. Stat. Mech. (2011) P02023.
  60. M. Sanatani, Y. Chiba, and N. Shiraishi, Proof of the absence of local conserved quantities in general spin-1/2 chains with symmetric nearest-neighbor interaction, arXiv:2411.02163.
  61. We call k-local conserved quantities “local” ones even when k=N/2 because the corresponding k-local quantities in much larger systems also become conserved quantities, while it is not the case when k>N/2. For instance, we consider a spin chain with N sites and nearest-neighbor interactions, and define a 3-local quantity Q[N]=∑i=1NSixSi+1ySi+2z. If it is the 3-local conserved quantity for N=6, then it is also a conserved quantity of a larger system N>6. On the other hand, although H2 is an (N1/2+2)-local conserved quantity in the nearest-neighbor-interacting systems on an N=N1 chain, the corresponding (N1/2+2)-local quantity (which is different from the square of the Hamiltonian for the large system) is not conserved when N>N1 in general. See also Sec. VI A in Ref. [55].
  62. T. Kennedy, Solutions of the Yang-Baxter equation for isotropic quantum spin chains, J. Phys. A: Math. Gen. 25, 2809 (1992).
  63. Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  64. M. L. Mehta, Random Matrices (Academic Press, New York, 2004), 3rd ed.
  65. M. Gell-Mann, Symmetries of baryons and mesons, Phys. Rev. 125, 1067 (1962).
  66. J. J. Sylvester and F. Franklin, A constructive theory of partitions, arranged in three acts, an interact and an exodion, Am. J. Math. 5, 251 (1882).
  67. H. V. Weyl, Quantenmechanik und gruppentheorie, Z. Phys. 46, 1 (1927).
  68. J. Madore, The fuzzy sphere, Class. Quantum Grav. 9, 69 (1992).
  69. These relations can be seen from the fact that subscript m represents the eigenvalue with respect to the adjoint action by E0: [E0,•].
  70. Here, we used the fact that k≤N/2. Notice that for k>N/2, the list of {Ajk} contributing to Bik becomes more complicated. For a detail discussion, see Sec. VI A in Ref. [55].
  71. A. B. Zamolodchikov and V. A. Fateev, Model factorized s-matrix and an integrable spin-1 Heisenberg chain, Sov. J. Nucl. Phys. 32, 298 (1980).
  72. A. Klümper, A. Schadschneider, and J. Zittartz, Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets, Europhys. Lett. 24, 293 (1993).
  73. E. Bartel, A. Schadschneider, and J. Zittartz, Excitations of anisotropic spin-1 chains with matrix product ground state, Eur. Phys. J. B 31, 209 (2003).
  74. L. Mezincescu, R. I. Nepomechie, and V. Rittenberg, Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147, 70 (1990).
  75. Precisely speaking, the first solution appears in Refs. [76, 77] only when h=2ε1−2ε39. However, since the corresponding R matrix commutes with F0⊗I+I⊗F0, adding a term proportional to ∑i=1NF0,i does not affect integrability.
  76. M. Idzumi, T. Tokihiro, and M. Arai, Solvable nineteen-vertex models and quantum spin chains of spin one, J. Phys. I 4, 1151 (1994).
  77. P. N. Bibikov and A. G. Nuramatov, R-matrices for integrable axially symmetric S=1 spin chains, arXiv:1408.4385.
  78. M. Lüscher, Dynamical charges in the quantized renormalized massive Thirring model, Nucl. Phys. B 117, 475 (1976).
  79. H. K. Park and S. Lee, Proof of nonintegrability of the spin-1 bilinear-biquadratic chain model, Phys. Rev. B 111, 134444 (2025).
  80. In fact, one of the authors (A.H.) has recently shown that this condition is equivalent to each other for all 3≤k≤N/2 [81].
  81. A. Hokkyo, Rigorous test for quantum integrability and nonintegrability, arXiv:2501.18400.

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Phys. Rev. Research 7, 043297– Published 18 December, 2025
  • Received 21 January 2025
  • Accepted 3 November 2025
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