- 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

Spread complexity and the saturation of wormhole size

Vijay Balasubramanian1,2,3,*, Javier M. Magan4,†, Poulami Nandi1,‡, and Qingyue Wu1,§

  • *Contact author: vijay@physics.upenn.edu
  • †Contact author: javier.magan.physics@gmail.com
  • ‡Contact author: pnandi@sas.upenn.edu
  • §Contact author: aqwalnut@sas.upenn.edu
PDF

Phys. Rev. D 113, 046004 – Published 9 February, 2026

DOI: https://doi.org/10.1103/vpyr-b3fb

Abstract

Recent proposals equate the size of Einstein-Rosen (ER) bridges in Jackiw-Teilboim (JT) gravity to spread complexity of a dual, double-scaled Sachdev-Ye-Kitaev theory (DSSYK). We show that the auxiliary “chord basis” of these proposals is an extrapolation from a subexponential part of the finite-dimensional physical Krylov basis of a spreading thermofield double state. The physical tridiagonal Hamiltonian coincides with the DSSYK approximation on the initial Krylov basis, but deviates markedly over an exponentially large part of the state space. We nonperturbatively extend the identification of ER bridge size and spread complexity to the complete Hilbert space, and show that it saturates at late times. We use methods for tridiagonalizing random Hamiltonians to study all universality classes to which large-N SYK theories and JT gravities can belong. The saturation dynamics depends on the universality class, and displays “white hole” physics at late times where the ER bridge shrinks from maximum size to a plateau. We describe extensions of our results to higher dimensions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (112)

  1. J. M. Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
  2. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016); 64, 44(A) (2016).
  3. D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90, 126007 (2014).
  4. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116, 191301 (2016).
  5. A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Does complexity equal anything?, Phys. Rev. Lett. 128, 081602 (2022).
  6. A. R. Brown, L. Susskind, and Y. Zhao, Quantum complexity and negative curvature, Phys. Rev. D 95, 045010 (2017).
  7. R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, J. High Energy Phys. 10 (2017) 107.
  8. S. Chapman, M. P. Heller, H. Marrochio, and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120, 121602 (2018).
  9. J. M. Magán, Black holes, complexity and quantum chaos, J. High Energy Phys. 09 (2018) 043.
  10. P. Caputa and J. M. Magan, Quantum computation as gravity, Phys. Rev. Lett. 122, 231302 (2019).
  11. V. Balasubramanian, M. DeCross, A. Kar, and O. Parrikar, Binding complexity and multiparty entanglement, J. High Energy Phys. 02 (2019) 069.
  12. M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Inf. Comput. 6, 213 (2006).
  13. M. R. Dowling and M. A. Nielsen, The geometry of quantum computation, Quantum Inf. Comput. 8, 0861 (2008).
  14. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311, 1133 (2006).
  15. V. Balasubramanian, M. Decross, A. Kar, and O. Parrikar, Quantum complexity of time evolution with chaotic Hamiltonians, J. High Energy Phys. 01 (2020) 134.
  16. P. Bueno, J. M. Magan, and C. S. Shahbazi, Complexity measures in QFT and constrained geometric actions, J. High Energy Phys. 09 (2021) 200.
  17. V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li, and O. Parrikar, Complexity growth in integrable and chaotic models, J. High Energy Phys. 07 (2021) 011.
  18. A. R. Brown, Polynomial equivalence of complexity geometries, Quantum 8, 1391 (2024).
  19. V. Balasubramanian, A. Lawrence, J. M. Magan, and M. Sasieta, Microscopic origin of the entropy of black holes in general relativity, Phys. Rev. X 14, 011024 (2024).
  20. V. Balasubramanian, A. Lawrence, J. M. Magan, and M. Sasieta, Microscopic origin of the entropy of astrophysical black holes, Phys. Rev. Lett. 132, 141501 (2024).
  21. P. Saad, Late Time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311.
  22. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  23. P.-S. Hsin, L. V. Iliesiu, and Z. Yang, A violation of global symmetries from replica wormholes and the fate of black hole remnants, Classical Quantum Gravity 38, 194004 (2021).
  24. J. Chandra and T. Hartman, Coarse graining pure states in AdS/CFT, J. High Energy Phys. 10 (2023) 030.
  25. J. Boruch, L. V. Iliesiu, and C. Yan, Constructing all BPS black hole microstates from the gravitational path integral, J. High Energy Phys. 09 (2024) 058.
  26. S. Antonini, M. Sasieta, and B. Swingle, Cosmology from random entanglement, J. High Energy Phys. 11 (2023) 188.
  27. A. Climent, R. Emparan, J. M. Magan, M. Sasieta, and A. Vilar López, Universal construction of black hole microstates, Phys. Rev. D 109, 086024 (2024).
  28. L. V. Iliesiu, A. Levine, H. W. Lin, H. Maxfield, and M. Mezei, On the non-perturbative bulk Hilbert space of JT gravity, J. High Energy Phys. 10 (2024) 220.
  29. J. Boruch, L. V. Iliesiu, G. Lin, and C. Yan, How the Hilbert space of two-sided black holes factorises, J. High Energy Phys. 06 (2025) 092.
  30. V. Balasubramanian, B. Craps, J. Hernandez, M. Khramtsov, and M. Knysh, Factorization of the Hilbert space of eternal black holes in general relativity, J. High Energy Phys. 01 (2025) 046.
  31. V. Balasubramanian, P. Caputa, J. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  32. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  33. 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).
  34. L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17, 441 (2014).
  35. J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, J. High Energy Phys. 05 (2017) 118.
  36. M. Berkooz, P. Narayan, and J. Simón, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, J. High Energy Phys. 08 (2018) 192.
  37. M. Berkooz, M. Isachenkov, V. Narovlansky, and G. Torrents, Towards a full solution of the large N double-scaled SYK model, J. High Energy Phys. 03 (2019) 079.
  38. P. Caputa, B. Chen, R. W. McDonald, J. Simón, and B. Strittmatter, Spread complexity rate as proper momentum, arXiv:2410.23334.
  39. Z.-Y. Fan, Momentum-Krylov complexity correspondence, arXiv:2411.04492.
  40. H. W. Lin, The bulk Hilbert space of double scaled SYK, J. High Energy Phys. 11 (2022) 060.
  41. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  42. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  43. D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24, 1475 (2020).
  44. V. Balasubramanian, J. M. Magan, and Q. Wu, Quantum chaos, integrability, and late times in the Krylov basis, Phys. Rev. E 111, 014218 (2025).
  45. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  46. D. Stanford and Z. Yang, Firewalls from wormholes, arXiv:2208.01625.
  47. S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg ferromagnet, Phys. Rev. Lett. 70, 3339 (1993).
  48. A. Kitaev, A simple model of quantum holography, Talks at KITP, April 7, 2015 and May 27, 2015.
  49. S. Sachdev, Bekenstein-Hawking entropy and strange metals, Phys. Rev. X 5, 041025 (2015).
  50. J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, J. High Energy Phys. 04 (2016) 001.
  51. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).
  52. K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117, 111601 (2016).
  53. J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-sitter space, arXiv:1606.01857.
  54. A. M. García-García and J. J. M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 126010 (2016).
  55. A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, J. High Energy Phys. 05 (2018) 183.
  56. G. Sárosi, AdS2 holography and the SYK model, Proc. Sci., Modave2017 (2018) 001 [arXiv:1711.08482].
  57. D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, J. High Energy Phys. 10 (2017) 008.
  58. T. G. Mertens, G. J. Turiaci, and H. L. Verlinde, Solving the Schwarzian via the conformal bootstrap, J. High Energy Phys. 08 (2017) 136.
  59. J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comput. 29, 215 (1975).
  60. P. Flajolet and M. Noy, Analytic combinatorics of chord diagrams, in Formal Power Series and Algebraic Combinatorics, edited by D. Krob, A. A. Mikhalev, and A. V. Mikhalev (Springer, Berlin, Heidelberg, 2000), pp. 191–201.
  61. M. E. Ismail, D. Stanton, and G. Viennot, The combinatorics of q-hermite polynomials and the Askey—Wilson integral, Eur. J. Combinatorics 8, 379 (1987).
  62. D. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, J. High Energy Phys. 02 (2020) 177.
  63. Y.-Z. You, A. W. W. Ludwig, and C. Xu, Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry protected topological states, Phys. Rev. B 95, 115150 (2017).
  64. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. (N.Y.) 3, 140 (1962).
  65. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  66. M. L. Mehta, Random Matrices, 3rd ed. (Academic Press, Boston, MA, 1991).
  67. T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller, Random matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  68. F. Haake, Quantum Signatures of Chaos (Springer-Verlag Berlin Heidelberg, 2001), Vol. 54.
  69. G. Akemann, J. Baik, and P. Di Francesco, The Oxford Handbook of Random Matrix Theory, Oxford Handbooks in Mathematics (Oxford University Press, Oxford, 2011).
  70. T. Li, J. Liu, Y. Xin, and Y. Zhou, Supersymmetric SYK model and random matrix theory, J. High Energy Phys. 06 (2017) 111.
  71. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  72. R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Complexity growth and the Krylov-Wigner function, J. High Energy Phys. 05 (2024) 264.
  73. V. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics, Lecture Notes in Physics Monographs (Springer, Berlin, Heidelberg, 2008).
  74. N. S. Witte and D. Bessis, The Lanczos algorithm for extensive many-body systems in the thermodynamic limit, J. Math. Phys. (N.Y.) 40, 4975 (1999).
  75. L. C. L. Hollenberg, Plaquette expansion in lattice Hamiltonian models, Phys. Rev. D 47, 1640 (1993).
  76. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  77. W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B984, 115948 (2022).
  78. P. Caputa, J. M. Magan, D. Patramanis, and E. Tonni, Krylov complexity of modular Hamiltonian evolution, Phys. Rev. D 109, 086004 (2024).
  79. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. 45, 255 (1950).
  80. V. Balasubramanian, R. N. Das, J. Erdmenger, and Z.-Y. Xian, Chaos and integrability in triangular billiards, J. Stat. Mech. (2025) 033202.
  81. K. Papadodimas and S. Raju, Local operators in the eternal black hole, Phys. Rev. Lett. 115, 211601 (2015).
  82. A. del Campo, J. Molina-Vilaplana, and J. Sonner, Scrambling the spectral form factor: Unitarity constraints and exact results, Phys. Rev. D 95, 126008 (2017).
  83. H. Verlinde, Deconstructing the wormhole: Factorization, entanglement, decoherence, arXiv:2105.02142.
  84. P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115.
  85. C. V. Johnson, Nonperturbative Jackiw-Teitelboim gravity, Phys. Rev. D 101, 106023 (2020).
  86. H. Maxfield and G. J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, J. High Energy Phys. 01 (2021) 118.
  87. T. G. Mertens and G. J. Turiaci, Liouville quantum gravity—holography, JT and matrices, J. High Energy Phys. 01 (2021) 073.
  88. E. Witten, Matrix models and deformations of JT gravity, Proc. R. Soc. A 476, 20200582 (2020).
  89. W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57, 97 (1970).
  90. P. Saad, S. H. Shenker, and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840.
  91. Y. Chen, V. Ivo, and J. Maldacena, Comments on the double cone wormhole, J. High Energy Phys. 04 (2024) 124.
  92. S. D. Mathur, The information paradox: A pedagogical introduction, Classical Quantum Gravity 26, 224001 (2009).
  93. A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Black holes: Complementarity or firewalls?, J. High Energy Phys. 02 (2013) 062.
  94. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, and J. Sully, An apologia for firewalls, J. High Energy Phys. 09 (2013) 018.
  95. D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev. Lett. 111, 171301 (2013).
  96. R. Bousso, Complementarity is not enough, Phys. Rev. D 87, 124023 (2013).
  97. E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, J. High Energy Phys. 10 (2013) 107.
  98. K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, J. High Energy Phys. 10 (2013) 212.
  99. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61, 781 (2013).
  100. J. de Boer, R. van Breukelen, S. F. Lokhande, K. Papadodimas, and E. Verlinde, On the interior geometry of a typical black hole microstate, J. High Energy Phys. 05 (2019) 010.
  101. B. Concepcion, Y. Nomura, K. Ritchie, and S. Weiss, Complementarity for a dynamical black hole, Phys. Rev. D 110, 086002 (2024).
  102. S. Banerjee, P. Basteiro, R. N. Das, and M. Dorband, Geometric quantum discord signals non-factorization, J. High Energy Phys. 08 (2023) 104.
  103. J. M. Magan and Q. Wu, Two types of quantum chaos: Testing the limits of the Bohigas-Giannoni-Schmit conjecture, arXiv:2411.08186.
  104. S. Banerjee, J. Erdmenger, and J. Karl, Non-locality induces isometry and factorisation in holography, Phys. Rev. D 112, L021902 (2025).
  105. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55, 5112 (1997).
  106. V. Balasubramanian, R. Gopakumar, and F. Larsen, Gauge theory, geometry and the large N limit, Nucl. Phys. B526, 415 (1998).
  107. J. Polchinski, M theory and the light cone, Prog. Theor. Phys. Suppl. 134, 158 (1999).
  108. V. Balasubramanian, A. Naqvi, and J. Simon, A multiboundary AdS orbifold and DLCQ holography: A universal holographic description of extremal black hole horizons, J. High Energy Phys. 08 (2004) 023.
  109. M. Miyaji, Non-perturbative discrete spectrum of interior length and timeshift in two-sided black hole, J. High Energy Phys. 04 (2025) 190.
  110. 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.
  111. P. Nandy, Tridiagonal Hamiltonians modeling the density of states of the double-scaled SYK model, J. High Energy Phys. 01 (2025) 072.
  112. https://inspirehep.net/literature/2855422.

Outline

Information

Phys. Rev. D 113, 046004– Published 9 February, 2026
  • Received 14 January 2025
  • Accepted 15 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation