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  • Open Access

Growth of block-diagonal operators and symmetry-resolved Krylov complexity

Pawel Caputa1,2,3, Giuseppe Di Giulio1,*, and Tran Quang Loc1

  • 1The Oscar Klein Centre and Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
  • 2Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
  • 3Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

  • *Contact author: giuseppe.di-giulio@fysik.su.se
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Phys. Rev. Research 7, 043055 – Published 15 October, 2025

DOI: https://doi.org/10.1103/9v9v-54zv

Abstract

This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure. For this purpose, we introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed. We find that, at early times, the complexity of the full operator is equal to the average of the symmetry-resolved contributions. At later times, however, the interplay among different charge sectors becomes more intricate. In general, the symmetry-resolved Krylov complexity depends on the charge sector, although in some cases this dependence disappears, leading to a form of Krylov complexity equipartition. Our analysis lays the groundwork for a broader application of symmetry structures in the study of Krylov space complexities with implications for thermalization and universality in many-body quantum systems.

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References (95)

  1. N. Laflorencie and S. Rachel, Spin-resolved entanglement spectroscopy of critical spin chains and Luttinger liquids, J. Stat. Mech. (2014) P11013.
  2. M. Goldstein and E. Sela, Symmetry-resolved entanglement in many-body systems, Phys. Rev. Lett. 120, 200602 (2018).
  3. J. C. Xavier, F. C. Alcaraz, and G. Sierra, Equipartition of the entanglement entropy, Phys. Rev. B 98, 041106 (2018).
  4. A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body localized system, Science 364, 256 (2019).
  5. S. Murciano, G. Di Giulio, and P. Calabrese, Entanglement and symmetry resolution in two dimensional free quantum field theories, J. High Energy Phys. 08 (2020) 073.
  6. R. Bonsignori and P. Calabrese, Boundary effects on symmetry resolved entanglement, J. Phys. A: Math. Theor. 54, 015005 (2021).
  7. L. Capizzi, P. Ruggiero, and P. Calabrese, Symmetry resolved entanglement entropy of excited states in a CFT, J. Stat. Mech. (2020) 073101.
  8. B. Estienne, Y. Ikhlef, and A. Morin-Duchesne, Finite-size corrections in critical symmetry-resolved entanglement, SciPost Phys. 10, 054 (2021).
  9. S. Zhao, C. Northe, and R. Meyer, Symmetry-resolved entanglement in AdS3/CFT2 coupled to U(1) Chern-Simons theory, J. High Energy Phys. 07 (2021) 030.
  10. D. X. Horvath, L. Capizzi, and P. Calabrese, U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap, J. High Energy Phys. 05 (2021) 197.
  11. L. Capizzi, D. X. Horváth, P. Calabrese, and O. A. Castro-Alvaredo, Entanglement of the 3-state Potts model via form factor bootstrap: Total and symmetry resolved entropies, J. High Energy Phys. 05 (2022) 113.
  12. P. Calabrese, J. Dubail, and S. Murciano, Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models, J. High Energy Phys. 10 (2021) 067.
  13. M. Ghasemi, Universal thermal corrections to symmetry-resolved entanglement entropy and full counting statistics, J. High Energy Phys. 05 (2023) 209.
  14. A. Foligno, S. Murciano, and P. Calabrese, Entanglement resolution of free Dirac fermions on a torus, J. High Energy Phys. 03 (2023) 096.
  15. L. Capizzi, S. Murciano, and P. Calabrese, Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects, J. Stat. Mech. (2023) 073102.
  16. M. Fossati, F. Ares, and P. Calabrese, Symmetry-resolved entanglement in critical non-Hermitian systems, Phys. Rev. B 107, 205153 (2023).
  17. Y. Kusuki, S. Murciano, H. Ooguri, and S. Pal, Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory, J. High Energy Phys. 11 (2023) 216.
  18. S. Murciano, G. Di Giulio, and P. Calabrese, Symmetry resolved entanglement in gapped integrable systems: A corner transfer matrix approach, SciPost Phys. 8, 046 (2020).
  19. R. Bonsignori, P. Ruggiero, and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, J. Phys. A: Math. Theor. 52, 475302 (2019).
  20. P. Calabrese, M. Collura, G. Di Giulio, and S. Murciano, Full counting statistics in the gapped XXZ spin chain, Europhys. Lett. 129, 60007 (2020).
  21. F. Ares, S. Murciano, and P. Calabrese, Symmetry-resolved entanglement in a long-range free-fermion chain, J. Stat. Mech. (2022) 063104.
  22. J. M. Magan, Proof of the universal density of charged states in QFT, J. High Energy Phys. 12 (2021) 100.
  23. K. Weisenberger, S. Zhao, C. Northe, and R. Meyer, Symmetry-resolved entanglement for excited states and two entangling intervals in AdS3/CFT2, J. High Energy Phys. 12 (2021) 104.
  24. S. Zhao, C. Northe, K. Weisenberger, and R. Meyer, Charged moments in W3 higher spin holography, J. High Energy Phys. 05 (2022) 166.
  25. G. Di Giulio, R. Meyer, C. Northe, H. Scheppach, and S. Zhao, On the boundary conformal field theory approach to symmetry-resolved entanglement, SciPost Phys. Core 6, 049 (2023).
  26. C. Northe, Entanglement resolution with respect to conformal symmetry, Phys. Rev. Lett. 131, 151601 (2023).
  27. V. Benedetti, H. Casini, Y. Kawahigashi, R. Longo, and J. M. Magan, Modular invariance as completeness, Phys. Rev. D 110, 125004 (2024).
  28. H. Casini, M. Huerta, J. M. Magán, and D. Pontello, Entanglement entropy and superselection sectors. Part I. Global symmetries, J. High Energy Phys. 02 (2020) 014.
  29. H. Casini, M. Huerta, J. M. Magan, and D. Pontello, Entropic order parameters for the phases of QFT, J. High Energy Phys. 04 (2021) 277.
  30. H. Casini and J. M. Magan, On completeness and generalized symmetries in quantum field theory, Mod. Phys. Lett. A 36, 2130025 (2021).
  31. F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nat. Commun. 14, 2036 (2023).
  32. F. Ares, P. Calabrese, and S. Murciano, The quantum Mpemba effects, Nat. Rev. Phys. 7, 451 (2025).
  33. O. A. Castro-Alvaredo and L. Santamaría-Sanz, Symmetry-resolved measures in quantum field theory: A short review, Mod. Phys. Lett. B 39, 2430002 (2025).
  34. E. Cornfeld, M. Goldstein, and E. Sela, Imbalance entanglement: Symmetry decomposition of negativity, Phys. Rev. A 98, 032302 (2018).
  35. S. Murciano, R. Bonsignori, and P. Calabrese, Symmetry decomposition of negativity of massless free fermions, SciPost Phys. 10, 111 (2021).
  36. L. Capizzi and P. Calabrese, Symmetry resolved relative entropies and distances in conformal field theory, J. High Energy Phys. 10 (2021) 195.
  37. H.-H. Chen, Charged Rényi negativity of massless free bosons, J. High Energy Phys. 02 (2022) 117.
  38. H.-H. Chen, Symmetry decomposition of relative entropies in conformal field theory, J. High Energy Phys. 07 (2021) 084.
  39. F. Ares, P. Calabrese, G. Di Giulio, and S. Murciano, Multi-charged moments of two intervals in conformal field theory, J. High Energy Phys. 09 (2022) 051.
  40. H.-H. Chen, Dynamics of charge imbalance resolved negativity after a global quench in free scalar field theory, J. High Energy Phys. 08 (2022) 146; 10 (2022) 157(E).
  41. A. Rath, V. Vitale, S. Murciano, M. Votto, J. Dubail, R. Kueng, C. Branciard, P. Calabrese, and B. Vermersch, Entanglement barrier and its symmetry resolution: Theory and experimental observation, PRX Quantum 4, 010318 (2023).
  42. G. Parez, Symmetry-resolved Rényi fidelities and quantum phase transitions, Phys. Rev. B 106, 235101 (2022).
  43. G. Parez, R. Bonsignori, and P. Calabrese, Dynamics of charge-imbalance-resolved entanglement negativity after a quench in a free-fermion model, J. Stat. Mech. (2022) 053103.
  44. S. Murciano, J. Dubail, and P. Calabrese, More on symmetry resolved operator entanglement, J. Phys. A: Math. Theor. 57, 145002 (2024).
  45. G. Di Giulio and J. Erdmenger, Symmetry-resolved modular correlation functions in free fermionic theories, J. High Energy Phys. 07 (2023) 058.
  46. F. Yan, S. Murciano, P. Calabrese, and R. M. Konik, On symmetry-resolved generalized entropies, SciPost Phys. 18, 211 (2025).
  47. L. Nie, Operator growth and symmetry-resolved coefficient entropy in quantum maps, arXiv:2111.08729.
  48. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  49. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  50. D. A. Roberts, D. Stanford, and A. Streicher, Operator growth in the SYK model, J. High Energy Phys. 06 (2018) 122.
  51. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, J. High Energy Phys. 07 (2022) 151.
  52. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  53. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  54. T. Q. Loc, Lanczos spectrum for random operator growth, arXiv:2402.07980.
  55. P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106, 195125 (2022).
  56. P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan, and H. J. R. Van Zyl, Spread complexity and topological transitions in the Kitaev chain, J. High Energy Phys. 01 (2023) 120.
  57. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
  58. A. Avdoshkin, A. Dymarsky, and M. Smolkin, Krylov complexity in quantum field theory, and beyond, J. High Energy Phys. 06 (2024) 066.
  59. H. A. Camargo, V. Jahnke, K.-Y. Kim, and M. Nishida, Krylov complexity in free and interacting scalar field theories with bounded power spectrum, J. High Energy Phys. 05 (2023) 226.
  60. P. Caputa and S. Datta, Operator growth in 2d CFT, J. High Energy Phys. 12 (2021) 188; 09 (2022) 113(E).
  61. P. Caputa, J. M. Magan, D. Patramanis, and E. Tonni, Krylov complexity of modular Hamiltonian evolution, Phys. Rev. D 109, 086004 (2024).
  62. V. Malvimat, S. Porey, and B. Roy, Krylov complexity in 2d CFTs with SL(2, R) deformed Hamiltonians, J. High Energy Phys. 02 (2025) 035.
  63. P. Caputa and G. Di Giulio, Local quenches from a Krylov perspective, J. High Energy Phys. 07 (2025) 164.
  64. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023).
  65. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, J. High Energy Phys. 12 (2022) 081.
  66. E. Medina-Guerra, I. V. Gornyi, and Y. Gefen, Phase transitions in a non-Hermitian Su-Schrieffer-Heeger model via Krylov spread complexity, Phys. Rev. B 112, 035427 (2025).
  67. E. Medina-Guerra, I. V. Gornyi, and Y. Gefen, Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field, Phys. Rev. B 111, 174207 (2025).
  68. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rep. 1125–1128, 1 (2025).
  69. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  70. V. Balasubramanian, J. M. Magan, P. Nandi, and Q. Wu, Spread complexity and the saturation of wormhole size, arXiv:2412.02038.
  71. M. Miyaji, S.-M. Ruan, S. Shibuya, and K. Yano, Non-perturbative overlaps in JT gravity: From spectral form factor to generating functions of complexity, J. High Energy Phys. 06 (2025) 251.
  72. P. Caputa, B. Chen, R. W. McDonald, J. Simón, and B. Strittmatter, Spread complexity rate as proper momentum, arXiv:2410.23334.
  73. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  74. J. Berges, S. Borsanyi, and C. Wetterich, Prethermalization, Phys. Rev. Lett. 93, 142002 (2004).
  75. M. Kollar, F. Alexander Wolf, and M. Eckstein, Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems, Phys. Rev. B 84, 054304 (2011).
  76. T. Langen, T. Gasenzer, and J. Schmiedmayer, Prethermalization and universal dynamics in near-integrable quantum systems, J. Stat. Mech. (2016) 064009.
  77. J. Choi et al., Preparing random states and benchmarking with many-body quantum chaos, Nature (London) 613, 468 (2023).
  78. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernandez, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
  79. R.-A. Chang, H. Shrotriya, W. W. Ho, and M. Ippoliti, Deep thermalization under charge-conserving quantum dynamics, PRX Quantum 6, 020343 (2025).
  80. 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).
  81. S. Choi, C. J. Turner, H. Pichler, W. W. Ho, A. A. Michailidis, Z. Papić, M. Serbyn, M. D. Lukin, and D. A. Abanin, Emergent SU(2) dynamics and perfect quantum many-body scars, Phys. Rev. Lett. 122, 220603 (2019).
  82. V. S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics (Springer, Berlin, Heidelberg, Germany, 1994).
  83. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. B 45, 255 (1950).
  84. B. Craps, O. Evnin, and G. Pascuzzi, Multiseed Krylov complexity, Phys. Rev. Lett. 134, 050402 (2025).
  85. P. Caputa, H.-S. Jeong, S. Liu, J. F. Pedraza, and L.-C. Qu, Krylov complexity of density matrix operators, J. High Energy Phys. 05 (2024) 337.
  86. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, Boca Raton, Florida, 2017).
  87. P. Caputa, G. Di Giulio, and T. Q. Loc, Symmetry-resolved spread complexity, arXiv:2509.12992.
  88. A. Kar, L. Lamprou, M. Rozali, and J. Sully, Random matrix theory for complexity growth and black hole interiors, J. High Energy Phys. 01 (2022) 016.
  89. A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102, 085137 (2020).
  90. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  91. S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller, and N. Y. Halpern, Quantum complexity in gravity, quantum field theory, and quantum information science, arXiv:2503.10753.
  92. P. Zanardi, Entanglement of quantum evolutions, Phys. Rev. A 63, 040304 (2001).
  93. T. Prosen and I. Pižorn, Operator space entanglement entropy in a transverse Ising chain, Phys. Rev. A 76, 032316 (2007).
  94. J. Dubail, Entanglement scaling of operators: A conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d, J. Phys. A: Math. Theor. 50, 234001 (2017).
  95. Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105, 062210 (2022).

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Phys. Rev. Research 7, 043055– Published 15 October, 2025
  • Received 15 July 2025
  • Accepted 22 September 2025
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