- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Eigenstate thermalization hypothesis for off-diagonal matrix elements in integrable spin chains

Federico Rottoli* and Vincenzo Alba

  • *Contact author: federico.rottoli@df.unipi.it
PDF

Phys. Rev. B 113, 054308 – Published 17 February, 2026

DOI: https://doi.org/10.1103/m317-w58w

Abstract

We investigate off-diagonal matrix elements of local operators in integrable spin chains, focusing on the isotropic spin-1/2 Heisenberg chain (XXX chain). We employ state-of-the-art algebraic Bethe ansatz results, which allow us to efficiently compute matrix elements of operators with support up to two sites between generic energy eigenstates. We consider both matrix elements between eigenstates that are in the same thermodynamic macrostate, as well as eigenstates that belong to different macrostates. In the former case, focusing on thermal states we numerically show that matrix elements are compatible with the exponential decay as exp(−L|MijO|). The probability distribution functions of MijO depend on the observable and on the macrostate, and are well described by Gumbel distributions. On the other hand, matrix elements between eigenstates in different macrostates decay faster as exp(−|M′ijO|L2), with M′ijO, again, compatible with a Gumbel distribution.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  2. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  3. M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A: Math. Gen. 32, 1163 (1999).
  4. 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).
  5. S. Pappalardi, L. Foini, and J. Kurchan, Eigenstate thermalization hypothesis and free probability, Phys. Rev. Lett. 129, 170603 (2022).
  6. D. Bernard and L. Hruza, Exact entanglement in the driven quantum symmetric simple exclusion process, SciPost Phys. 15, 175 (2023).
  7. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  8. R. Steinigeweg, J. Herbrych, and P. Prelovšek, Eigenstate thermalization within isolated spin-chain systems, Phys. Rev. E 87, 012118 (2013).
  9. W. Beugeling, R. Moessner, and M. Haque, Off-diagonal matrix elements of local operators in many-body quantum systems, Phys. Rev. E 91, 012144 (2015).
  10. A. Chandran, M. D. Schulz, and F. J. Burnell, The eigenstate thermalization hypothesis in constrained Hilbert spaces: A case study in non-Abelian anyon chains, Phys. Rev. B 94, 235122 (2016).
  11. R. Mondaini and M. Rigol, Eigenstate thermalization in the two-dimensional transverse field Ising model. II. Off-diagonal matrix elements of observables, Phys. Rev. E 96, 012157 (2017).
  12. C. Nation and D. Porras, Off-diagonal observable elements from random matrix theory: Distributions, fluctuations, and eigenstate thermalization, New J. Phys. 20, 103003 (2018).
  13. T. Yoshizawa, E. Iyoda, and T. Sagawa, Numerical large deviation analysis of the eigenstate thermalization hypothesis, Phys. Rev. Lett. 120, 200604 (2018).
  14. M. Rigol and L. F. Santos, Quantum chaos and thermalization in gapped systems, Phys. Rev. A 82, 011604(R) (2010).
  15. G. Biroli, C. Kollath, and A. M. Läuchli, Effect of rare fluctuations on the thermalization of isolated quantum systems, Phys. Rev. Lett. 105, 250401 (2010).
  16. E. Khatami, G. Pupillo, M. Srednicki, and M. Rigol, Fluctuation-dissipation theorem in an isolated system of quantum dipolar bosons after a quench, Phys. Rev. Lett. 111, 050403 (2013).
  17. T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Entanglement and matrix elements of observables in interacting integrable systems, Phys. Rev. E 100, 062134 (2019).
  18. M. Brenes, J. Goold, and M. Rigol, Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain, Phys. Rev. B 102, 075127 (2020).
  19. T. LeBlond and M. Rigol, Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems, Phys. Rev. E 102, 062113 (2020).
  20. M. Mierzejewski and L. Vidmar, Quantitative impact of integrals of motion on the eigenstate thermalization hypothesis, Phys. Rev. Lett. 124, 040603 (2020).
  21. Y. Zhang, L. Vidmar, and M. Rigol, Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems, Phys. Rev. E 106, 014132 (2022).
  22. R. Patil and M. Rigol, Eigenstate thermalization in spin-1/2 systems with SU(2) symmetry, Phys. Rev. B 111, 205126 (2025).
  23. T. N. Ikeda, Y. Watanabe, and M. Ueda, Finite-size scaling analysis of the eigenstate thermalization hypothesis in a one-dimensional interacting Bose gas, Phys. Rev. E 87, 012125 (2013).
  24. V. Alba, M. Fagotti, and P. Calabrese, Entanglement entropy of excited states, J. Stat. Mech. (2009) P10020.
  25. V. Alba, Eigenstate thermalization hypothesis and integrability in quantum spin chains, Phys. Rev. B 91, 155123 (2015).
  26. F. H. L. Essler and A. J. J. M. de Klerk, Statistics of matrix elements of local operators in integrable models, Phys. Rev. X 14, 031048 (2024).
  27. J.-S. Caux, P. Calabrese, and N. A. Slavnov, One-particle dynamical correlations in the one-dimensional Bose gas, J. Stat. Mech. (2007) P01008.
  28. L. Piroli and P. Calabrese, Exact formulas for the form factors of local operators in the Lieb–Liniger model, J. Phys. A: Math. Theor. 48, 454002 (2015).
  29. M. Horvath, A. Bastianello, S. Dhar, R. Koch, Y. Guo, J.-S. Caux, M. Landini, and H.-C. Nägerl, Observing Bethe strings in an attractive Bose gas far from equilibrium, arXiv:2505.10550.
  30. N. Kitanine, J. M. Maillet, and V. Terras, Form factors of the XXZ Heisenberg spin-12 finite chain, Nucl. Phys. B 554, 647 (1999).
  31. N. Kitanine, J. M. Maillet, and V. Terras, Correlation functions of the XXZ Heisenberg spin-12 chain in a magnetic field, Nucl. Phys. B 567, 554 (2000).
  32. J.-S. Caux and J. M. Maillet, Computation of dynamical correlation functions of Heisenberg chains in a magnetic field, Phys. Rev. Lett. 95, 077201 (2005).
  33. J.-S. Caux, R. Hagemans, and J. M. Maillet, Computation of dynamical correlation functions of Heisenberg chains: The gapless anisotropic regime, J. Stat. Mech. (2005) P09003.
  34. J.-S. Caux and P. Calabrese, Dynamical density-density correlations in the one-dimensional Bose gas, Phys. Rev. A 74, 031605(R) (2006).
  35. J.-S. Caux, Correlation functions of integrable models: A description of the ABACUS algorithm, J. Math. Phys. 50, 095214 (2009).
  36. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, UK, 1999).
  37. H. Bethe, On the theory of metals, I. Eigenvalues and eigenfunctions of a linear chain of atoms, in Selected Works of Hans A Bethe, World Scientific Series in 20th Century Physics (World Scientific, Singapore, 1997), Vol. 18, pp. 155–183.
  38. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, UK, 1993).
  39. F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, UK, 2005).
  40. R. Hagemans and J.-S. Caux, Deformed strings in the Heisenberg model, J. Phys. A: Math. Theor. 40, 14605 (2007).
  41. P. Calabrese, F. H. L. Essler, and G. Mussardo, Introduction to “Quantum integrability in out of equilibrium systems”, J. Stat. Mech. (2016) 064001.
  42. V. Alba, Simulating the generalized Gibbs ensemble (GGE): A Hilbert space Monte Carlo approach, arXiv:1507.06994.
  43. V. Alba and P. Calabrese, The quench action approach in finite integrable spin chains, J. Stat. Mech. (2016) 043105.
  44. L. Foini, A. Dymarsky, and S. Pappalardi, Out-of-equilibrium eigenstate thermalization hypothesis, SciPost Phys. 18, 136 (2025).
  45. J. D. Nardis, L. Piroli, and J.-S. Caux, Relaxation dynamics of local observables in integrable systems, J. Phys. A: Math. Theor. 48, 43FT01 (2015).
  46. I. A. Maceira and A. M. Läuchli, Thermalization dynamics in closed quantum many body systems: A precision large scale exact diagonalization study, arXiv:2409.18863.
  47. J. Wang, R. Mishra, T.-H. Yang, L. V. Delacrétaz, and S. Pappalardi, Eigenstate thermalization hypothesis correlations via non-linear hydrodynamics, arXiv:2505.06869.
  48. F. Fritzsch, T. Prosen, and S. Pappalardi, Microcanonical free cumulants in lattice systems, Phys. Rev. B 111, 054303 (2025).
  49. O. Bouverot-Dupuis, S. Pappalardi, J. Kurchan, A. Polkovnikov, and L. Foini, Random matrix universality in dynamical correlation functions at late times, SciPost Phys. 19, 050 (2025).
  50. L. Capizzi, J. Wang, X. Xu, L. Mazza, and D. Poletti, Hydrodynamics and the eigenstate thermalization hypothesis, Phys. Rev. X 15, 011059 (2025).
  51. N. A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the alcebraic Bethe ansatz, Theor. Math. Phys. 79, 502 (1989).

Outline

Information

Phys. Rev. B 113, 054308– Published 17 February, 2026
  • Received 6 June 2025
  • Revised 15 October 2025
  • Accepted 23 January 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