- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Complete classification of integrability and nonintegrability for spin-12 chain with symmetric nearest-neighbor interaction

Mizuki Sanatani1,*, Yuuya Chiba2,†, and Naoto Shiraishi1,‡

  • 1Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
  • 2Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

  • *Contact author: yamaguchi-q@g.ecc.u-tokyo.ac.jp
  • †Contact author: yuya.chiba@riken.jp
  • ‡Contact author: shiraishi@phys.c.u-tokyo.ac.jp
PDF

Phys. Rev. Research 8, 033240 – Published 28 August, 2026

DOI: https://doi.org/10.1103/k8xr-brs7

Abstract

General spin-12 chains with symmetric nearest-neighbor interaction are studied. We rigorously prove that all spin models in this class, except for known integrable systems, are nonintegrable in the sense that they possess no nontrivial local conserved quantities. This result confirms that there are no missing integrable systems; i.e., integrable systems in this class are exactly those that are already known. In addition, this result excludes the possibility of intermediate systems that have a finite number of nontrivial local conserved quantities. Our findings support the expectation that integrable systems are exceptional in quantum many-body systems and most systems are nonintegrable.

View figure in article

Physics Subject Headings (PhySH)

See Also

Proof of the absence of local conserved quantities in general spin-12 chains with symmetric nearest-neighbor interaction

Mizuki Sanatani, Yuuya Chiba, and Naoto Shiraishi
Phys. Rev. B 114, 094437 (2026)

Article Text

References (45)

  1. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Elsevier, Amsterdam, 2016).
  2. M. Jimbo and T. Miwa, Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Mathematics, Vol. 85 (American Mathematical Society, Providence, RI, 1994).
  3. L. D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Symétries Quantiques, edited by A. Connes et al., Les Houches, 1995 (North-Holland, Amsterdam, 1998), pp. 149–219.
  4. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, 1999).
  5. H. A. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Zeit. für Physik 71, 205 (1931).
  6. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16, 407 (1961).
  7. C.-N. Yang and C.-P. Yang, One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system, Phys. Rev. 150, 321 (1966).
  8. E. H. Lieb and F.-Y. Wu, Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett. 20, 1445 (1968).
  9. R. Baxter, One-dimensional anisotropic Heisenberg chain, Phys. Rev. Lett. 26, 834 (1971).
  10. Z. Maassarani, The XXC models, Phys. Lett. A 244, 160 (1998).
  11. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  12. Y. Yanagihara and K. Minami, Exact solution of a cluster model with next-nearest-neighbor interaction, Prog. Theor. Exp. Phys. 2020, 113A01 (2020).
  13. B. Sutherland, Model for a multicomponent quantum system, Phys. Rev. B 12, 3795 (1975).
  14. L. A. Takhtajan, The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87, 479 (1982).
  15. H. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S, Phys. Lett. A 90, 479 (1982).
  16. M. N. Barber and M. T. Batchelor, Spectrum of the biquadratic spin-1 antiferromagnetic chain, Phys. Rev. B 40, 4621 (1989).
  17. J.-S. Caux and J. Mossel, Remarks on the notion of quantum integrability, J. Stat. Mech.: Theory Exp. (2011) P02023.
  18. C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016).
  19. M. Grabowski and P. Mathieu, Structure of the conservation laws in quantum integrable spin chains with short range interactions, Ann. Phys. 243, 299 (1995).
  20. M. Fagotti and Fabian H. L. Essler, Reduced density matrix after a quantum quench, Phys. Rev. B 87, 245107 (2013).
  21. 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).
  22. B. Pozsgay, The generalized Gibbs ensemble for Heisenberg spin chains, J. Stat. Mech.: Theory Exp. (2013) P07003.
  23. L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech.: Theory Exp. (2016) 064007.
  24. T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature (London) 440, 900 (2006).
  25. T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015).
  26. F. Reiter, F. Lange, S. Jain, M. Grau, J. P. Home, and Z. Lenarčič, Engineering generalized Gibbs ensembles with trapped ions, Phys. Rev. Res. 3, 033142 (2021).
  27. P. Mazur, Non-ergodicity of phase functions in certain systems, Physica 43, 533 (1969).
  28. M. Suzuki, Ergodicity, constants of motion, and bounds for susceptibilities, Physica 51, 277 (1971).
  29. X. Zotos, F. Naef, and P. Prelovsek, Transport and conservation laws, Phys. Rev. B 55, 11029 (1997).
  30. K. Saito, Strong evidence of normal heat conduction in a one-dimensional quantum system, Europhys. Lett. 61, 34 (2003).
  31. 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).
  32. N. Shiraishi, Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field, Europhys. Lett. 128, 17002 (2019).
  33. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3, 140 (1962).
  34. E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions I, in The Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, edited by A. S. Wightman (Springer-Verlag, Berlin/Heidelberg, 1993), pp. 524–540.
  35. M. Rigol, Breakdown of thermalization in finite one-dimensional systems, Phys. Rev. Lett. 103, 100403 (2009).
  36. L. F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E 81, 036206 (2010).
  37. 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).
  38. Y. Chiba, Proof of absence of local conserved quantities in the mixed-field Ising chain, Phys. Rev. B 109, 035123 (2024).
  39. 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).
  40. N. Shiraishi, Absence of local conserved quantity in the Heisenberg model with next-nearest-neighbor interaction, J. Stat. Phys. 191, 114 (2024).
  41. A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys. 37, 1871 (1996).
  42. M. Sanatani, Y. Chiba, and N. Shiraishi, companion paper, Proof of the absence of local conserved quantities in general spin-12 chains with symmetric nearest-neighbor interaction, Phys. Rev. B 114, 094437 (2026).
  43. M. P. Grabowski and P. Mathieu, Integrability test for spin chains, J. Phys. A: Math. Gen. 28, 4777 (1995).
  44. Y. Nozawa and K. Fukai, Explicit construction of local conserved quantities in the XYZ spin-1/2 chain, Phys. Rev. Lett. 125, 090602 (2020).
  45. K. Fukai, All local conserved quantities of the one-dimensional Hubbard model, Phys. Rev. Lett. 131, 256704 (2023).

Outline

Information

Phys. Rev. Research 8, 033240– Published 28 August, 2026
  • Received 12 February 2026
  • Accepted 9 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation