- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Holographic Krylov complexity for charged, composite, and extended probes

Horatiu Nastase1, Carlos Nunez2, and Dibakar Roychowdhury3

PDF

Phys. Rev. D 114, 086001 – Published 1 October, 2026

DOI: https://doi.org/10.1103/3lxz-17p9

Abstract

We study a proposed holographic representation of spread/Krylov complexity for states or operators dual to charged, composite, and extended probes. Our analysis builds on the relation between the rate of spread complexity and the proper momentum of an infalling bulk excitation. A conserved Noether charge requires particular care: unitary spread complexity satisfies C˙K(0)=0, and therefore the velocity conjugate to a fixed charge cannot be counted as an independent spreading direction. We consequently eliminate cyclic coordinates by a partial Legendre transform and compute the proper momentum from the fixed-charge Routhian. We first apply this prescription to an R-charged particle in AdS5×S5. The charge enters through the reduced effective mass and energy, while the resulting complexity has the required quadratic onset. We next study probes that are pointlike from the boundary perspective but possess nontrivial bulk structure. These are baryon-vertex configurations and giant gravitons. For the giant graviton the fixed-charge prescription retains the effects of the Born-Infeld and Wess-Zumino couplings without reintroducing the cyclic angular velocity into the proper momentum. We also analyze a fundamental string falling in anti–de Sitter (AdS) while stretched along a boundary spatial direction as a collective model of a nonlocal excitation. Within the rigid-probe Poincaré-AdS examples considered here, the proper-momentum rate is linear at late times, whereas charges, compositeness, and spatial extension affect its normalization and its subleading or intermediate-time structure. We interpret this as a robust pattern in the tested class (not as a general universality theorem). The fixed-charge Routhian provides a candidate bulk counterpart of sector selection. An explicit symmetry-resolved boundary Lanczos construction remains an open problem.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  2. J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, J. High Energy Phys. 10 (2019) 264.
  3. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  4. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
  5. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  6. 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).
  7. S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science, Phys. Rep. 1159, 1 (2026).
  8. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity, arXiv:2507.06286.
  9. 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).
  10. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator complexity: A journey to the edge of Krylov space, J. High Energy Phys. 06 (2021) 062.
  11. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, J. High Energy Phys. 07 (2022) 151.
  12. M. Baggioli, K.-B. Huh, H.-S. Jeong, K.-Y. Kim, and J. F. Pedraza, Krylov complexity as an order parameter for quantum chaotic-integrable transitions, Phys. Rev. Res. 7, 023028 (2025).
  13. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016).
  14. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116, 191301 (2016).
  15. V. Balasubramanian, P. Caputa, and J. Simón, Variations on a theme of Krylov, J. High Energy Phys. 04 (2026) 172.
  16. P. Caputa, B. Chen, R. W. McDonald, J. Simón, and B. Strittmatter, Spread complexity rate as proper momentum, Phys. Rev. D 113, L041901 (2026).
  17. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  18. K.-B. Huh, H.-S. Jeong, and J. F. Pedraza, Spread complexity in saddle-dominated scrambling, J. High Energy Phys. 05 (2024) 137.
  19. Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105, 062210 (2022).
  20. W. Mück, Krylov complexity has it all, arXiv:2605.28681.
  21. D. Chatzis, M. Hammond, C. Nunez, A. V. Ramallo, and R. T. Santamaria, Holographic spread complexity from branes and strings, arXiv:2607.00074.
  22. Z.-Y. Fan, Momentum-Krylov complexity correspondence, arXiv:2411.04492.
  23. P.-Z. He, Revisit the relationship between spread complexity rate and radial momentum, arXiv:2411.19172.
  24. H.-S. Jeong, Krylov subspace dynamics as near-horizon AdS2 holography, arXiv:2602.11627.
  25. D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, Strings in flat space and pp waves from N=4 superYang-Mills, J. High Energy Phys. 04 (2002) 013.
  26. E. Witten, Baryons and branes in anti-de Sitter space, J. High Energy Phys. 07 (1998) 006.
  27. J. McGreevy, L. Susskind, and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, J. High Energy Phys. 06 (2000) 008.
  28. M. T. Grisaru, R. C. Myers, and O. Tafjord, SUSY and goliath, J. High Energy Phys. 08 (2000) 040.
  29. A. Fatemiabhari, H. Nastase, C. Nunez, and D. Roychowdhury, Holographic Krylov complexity for conformal quiver gauge theories, Nucl. Phys. B1025, 117402 (2026).
  30. A. Fatemiabhari and C. Nunez, Krylov Complexity, confinement and universality, J. High Energy Phys. 07 (2026) 035.
  31. A. Fatemiabhari, C. Nunez, and R. T. Santamaria, Complexity and operator growth in holographic 6d SCFTs, Nucl. Phys. B1028, 117496 (2026).
  32. D. Roychowdhury, Holographic Krylov complexity for Yang-Baxter deformed supergravity backgrounds, Eur. Phys. J. C 86, 1005 (2026).
  33. P. Caputa, G. Di Giulio, and T. Q. Loc, Growth of block-diagonal operators and symmetry-resolved Krylov complexity, Phys. Rev. Res. 7, 043055 (2025).
  34. P. Caputa, G. Di Giulio, and T. Q. Loc, Symmetry-resolved spread complexity, J. High Energy Phys. 02 (2026) 189.
  35. B. Craps, O. Evnin, and G. Pascuzzi, Multiseed Krylov complexity, Phys. Rev. Lett. 134, 050402 (2025).
  36. S. PG, J. B. Kannan, R. Modak, and S. Aravinda, Dependence of Krylov complexity saturation on the initial operator and state, Phys. Rev. E 112, L032203 (2025).
  37. C. G. Callan, Jr., A. Guijosa, and K. G. Savvidy, Baryons and string creation from the five-brane world volume action, Nucl. Phys. B547, 127 (1999).
  38. J. Gomis, A. V. Ramallo, J. Simon, and P. K. Townsend, Supersymmetric baryonic branes, J. High Energy Phys. 11 (1999) 019.
  39. B. Janssen, Y. Lozano, and D. Rodriguez-Gomez, The Baryon vertex with magnetic flux, J. High Energy Phys. 11 (2006) 082.
  40. A. Fatemiabhari, H. Nastase, and D. Roychowdhury, Holographic Krylov complexity in N=4 SYM, Phys. Rev. D 113, 106033 (2026).
  41. Z. Li and J. Tian, The holography of spread complexity: A story of observers, J. High Energy Phys. 07 (2026) 187.
  42. D. Zoakos, Holographic Krylov complexity in the Coulomb branch of N=4 SYM, J. High Energy Phys. 06 (2026) 066.
  43. Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, AdS3 solutions in Massive IIA with small N=(4,0) supersymmetry, J. High Energy Phys. 01 (2020) 129.
  44. Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, 1/4 BPS solutions and the AdS3/CFT2 correspondence, Phys. Rev. D 101, 026014 (2020).
  45. Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, Two dimensional N=(0,4) quivers dual to AdS3 solutions in massive IIA, J. High Energy Phys. 01 (2020) 140.
  46. Y. Lozano, C. Nunez, A. Ramirez, and S. Speziali, M-strings and AdS3 solutions to M-theory with small N=(0,4) supersymmetry, J. High Energy Phys. 08 (2020) 118.
  47. Y. Lozano, C. Nunez, A. Ramirez, and S. Speziali, New AdS2 backgrounds and N=4 conformal quantum mechanics, J. High Energy Phys. 03 (2021) 277.
  48. J. M. Camino, A. Paredes, and A. V. Ramallo, Stable wrapped branes, J. High Energy Phys. 05 (2001) 011.
  49. M. M. Caldarelli and P. J. Silva, Multi-giant graviton systems, SUSY breaking and CFT, J. High Energy Phys. 02 (2004) 052.
  50. R. N. Das, S. Demulder, J. Erdmenger, and C. Northe, Spread complexity for the planar limit of holography, J. High Energy Phys. 06 (2025) 166.
  51. A. Anabalon and S. F. Ross, Supersymmetric solitons and a degeneracy of solutions in AdS/CFT, J. High Energy Phys. 07 (2021) 015.
  52. C. Nunez, M. Oyarzo, and R. Stuardo, Confinement in (1+1) dimensions: A holographic perspective from I-branes, J. High Energy Phys. 09 (2023) 201.
  53. C. Nunez, M. Oyarzo, and R. Stuardo, Confinement and D5-branes, J. High Energy Phys. 03 (2024) 080.
  54. D. Chatzis, A. Fatemiabhari, C. Nunez, and P. Weck, SCFT deformations via uplifted solitons, Nucl. Phys. B1006, 116659 (2024).
  55. D. Chatzis, A. Fatemiabhari, C. Nunez, and P. Weck, Conformal to confining SQFTs from holography, J. High Energy Phys. 08 (2024) 041.
  56. D. Chatzis, M. Hammond, G. Itsios, C. Nunez, and D. Zoakos, Supersymmetric AdS solitons, Coulomb branch flows and twisted compactifications, J. High Energy Phys. 04 (2026) 184.
  57. A. Anabalón and H. Nastase, Universal IR holography, scalar fluctuations, and glueball spectra, Phys. Rev. D 109, 066011 (2024).
  58. A. Anabalón, H. Nastase, and M. Oyarzo, Supersymmetric AdS solitons and the interconnection of different vacua of N=4 super Yang-Mills, J. High Energy Phys. 05 (2024) 217.
  59. A. Anabalón, H. Nastase, C. Nunez, M. Oyarzo, and R. Stuardo, Moduli space of N=4 Super Yang-Mills from AdS/CFT, J. High Energy Phys. 05 (2026) 251.
  60. A. Fatemiabhari, H. Nastase, C. Nunez, and D. Roychowdhury, Holographic Krylov complexity in confining gauge theories, J. High Energy Phys. 09 (2026) 155.

Outline

Information

Phys. Rev. D 114, 086001– Published 1 October, 2026
  • Received 13 July 2026
  • Accepted 16 September 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation