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  • Open Access

Origin of Exponential Operator Growth in Hilbert Space

Vijay Ganesh Sadhasivam1,2,*, Jan M. Rost1,†, and Stuart C. Althorpe2,‡

  • *Contact author: vgs23@cam.ac.uk
  • †Contact author: rost@pks.mpg.de
  • ‡Contact author: sca10@cam.ac.uk
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Phys. Rev. Lett. 137, 060404 – Published 6 August, 2026

DOI: https://doi.org/10.1103/81ws-xw2t

Abstract

The question of thermalization in quantum many-body systems has long been studied through the properties of matrix elements of operators corresponding to local observables. More recently, the focus has shifted to the dynamics of operators, which lead to seminal works proposing universal bounds on the rate of operator growth. In this Letter, we unify these two approaches: we show that exponential operator growth in Hilbert space, as measured by Krylov complexity, is governed by an exponential off-diagonal decay of the operator matrix elements in the system eigenbasis. When this decay is algebraic or slower, the growth rate saturates the universal bound, thereby establishing a microscopic origin of operator growth which is independent of chaos, dimensionality or the presence of many-body interactions.

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References (64)

  1. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
  2. J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015).
  3. S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Nat. Phys. 8, 325 (2012).
  4. A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  5. M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008).
  6. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
  7. J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018).
  8. M. Srednicki, Phys. Rev. E 50, 888 (1994).
  9. W. Beugeling, R. Moessner, and M. Haque, Phys. Rev. E 91, 012144 (2015).
  10. T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Phys. Rev. E 100, 062134 (2019).
  11. C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Phys. Rev. X 8, 021013 (2018).
  12. A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X 8, 021014 (2018).
  13. S. Xu and B. Swingle, Phys. Rev. X 9, 031048 (2019).
  14. S. H. Shenker and D. Stanford, J. High Energy Phys. 03 (2014) 067.
  15. J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106.
  16. B. Swingle, Nat. Phys. 14, 988 (2018).
  17. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, Phys. Rev. X 9, 041017 (2019).
  18. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Phys. Rep. 1125, 1 (2025).
  19. H. A. Camargo, V. Jahnke, H.-S. Jeong, K.-Y. Kim, and M. Nishida, Phys. Rev. D 109, 046017 (2024).
  20. K.-B. Huh, H.-S. Jeong, and J. F. Pedraza, J. High Energy Phys. 05 (2024) 137.
  21. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, J. High Energy Phys. 07 (2022) 151.
  22. K. Hashimoto, K. Murata, N. Tanahashi, and R. Watanabe, J. High Energy Phys. 11 (2023) 040.
  23. A. Dymarsky and M. Smolkin, Phys. Rev. D 104, L081702 (2021).
  24. A. Avdoshkin, A. Dymarsky, and M. Smolkin, J. High Energy Phys. 06 (2024) 066.
  25. A. Avdoshkin and A. Dymarsky, Phys. Rev. Res. 2, 043234 (2020).
  26. P. H. Bento, A. del Campo, and L. C. Céleri, Phys. Rev. B 109, 224304 (2024).
  27. P. Suchsland, R. Moessner, and P. W. Claeys, Phys. Rev. B 111, 014309 (2025).
  28. C. Liu, H. Tang, and H. Zhai, Phys. Rev. Res. 5, 033085 (2023).
  29. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, J. High Energy Phys. 12 (2023) 066.
  30. T. Xu, T. Scaffidi, and X. Cao, Phys. Rev. Lett. 124, 140602 (2020).
  31. B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, J. High Energy Phys. 03 (2023) 054.
  32. R. A. Kidd, A. Safavi-Naini, and J. F. Corney, Phys. Rev. A 102, 023330 (2020).
  33. V. G. Sadhasivam, L. Meuser, D. R. Reichman, and S. C. Althorpe, Proc. Natl. Acad. Sci. U.S.A. 120, e2312378120 (2023).
  34. T. R. Michel, J. D. Urbina, and P. Schlagheck, J. Phys. A 58, 275303 (2025).
  35. K. Hashimoto, K.-B. Huh, K.-Y. Kim, and R. Watanabe, J. High Energy Phys. 11 (2020) 068.
  36. N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Commun. Phys. 5, 207 (2022).

  37. Technically, the equivalence between (7) and (8) is valid only when the {bn} sequence depends smoothly on n, as pointed out in [24, 25]. In some cases, the Lanczos sequence can split into two bands with their own distinct slopes, and the Krylov complexity does not grow exponentially (and may even be bounded).

  38. See Supplemental Material at http://link.aps.org/supplemental/10.1103/81ws-xw2t for derivations of the matrix-element decay for the harmonic oscillator [Eq. (19)] and for one-dimensional power-law potentials [Eqs. (24) and (25)], a discussion of Lanczos staggering from operator localization, and additional numerical details, which includes Refs. [6,39–51].
  39. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Elsevier, New York, 2013), Vol. 3.
  40. J. Cornwall and G. Tiktopoulos, Ann. Phys. (Berlin) 228, 365 (1993).
  41. M. Voloshin, Phys. Rev. D 43, 1726 (1991).
  42. P. W. Claeys, Introduction to the Eigenstate Thermalization Hypothesis, Lecture notes, Max Planck Institute for the Physics of Complex Systems (Max Planck Institute, Dresden, 2023).
  43. F. P. Simonotti, E. Vergini, and M. Saraceno, Phys. Rev. E 56, 3859 (1997).
  44. A. Bäcker, R. Schubert, and P. Stifter, J. Phys. A 30, 6783 (1997).
  45. E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).
  46. B. Li and B. Hu, J. Phys. A 31, 483 (1998).
  47. Q. Hu, W.-Y. Zhang, Y. Han, and W.-L. You, Phys. Rev. B 111, 165106 (2025).
  48. S. Nandy, B. Mukherjee, A. Bhattacharyya, and A. Banerjee, J. Phys. Condens. Matter 36, 155601 (2024).
  49. X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55, 11029 (1997).
  50. R. Steinigeweg, J. Herbrych, and P. Prelovšek, Phys. Rev. E 87, 012118 (2013).
  51. A. Klümper and K. Sakai, J. Phys. A 35, 2173 (2002).
  52. H. Tang, arXiv:2312.17416.
  53. E. Khatami, G. Pupillo, M. Srednicki, and M. Rigol, Phys. Rev. Lett. 111, 050403 (2013).
  54. S. Pappalardi, L. Foini, and J. Kurchan, SciPost Phys. 12, 130 (2022).
  55. C. Murthy and M. Srednicki, Phys. Rev. Lett. 123, 230606 (2019).
  56. V. G. Sadhasivam, A. C. Hunt, L. Meuser, Y. Litman, and S. C. Althorpe, Phys. Rev. E 110, L012204 (2024).
  57. K. Kudo and T. Deguchi, J. Phys. Soc. Jpn. 74, 1992 (2005).
  58. P. Weinberg and M. Bukov, SciPost Phys. 2, 003 (2017).
  59. D. Jansen, J. Stolpp, L. Vidmar, and F. Heidrich-Meisner, Phys. Rev. B 99, 155130 (2019).
  60. K. Mallayya and M. Rigol, Phys. Rev. Lett. 123, 240603 (2019).
  61. K. Richter, J. D. Urbina, and S. Tomsovic, J. Phys. A 55, 453001 (2022).
  62. Y. Liao, A. Vikram, and V. Galitski, Phys. Rev. Lett. 125, 250601 (2020).
  63. S. Will, T. Best, U. Schneider, L. Hackermüller, D.-S. Lühmann, and I. Bloch, Nature (London) 465, 197 (2010).
  64. V. Sadhasivam, J. M. Rost, and S. C. Althorpe, Research data supporting “Origin of exponential operator growth in Hilbert space,” Dataset, 10.17863/CAM.123074 (2026).

Outline

Information

Phys. Rev. Lett. 137, 060404– Published 6 August, 2026
  • Received 14 November 2025
  • Revised 13 April 2026
  • Accepted 7 July 2026
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