Abstract
Considering the large q expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the t/q effects. The Krylov complexity naturally describes the “size” of the distribution while the higher cumulants encode richer information. We further consider the double-scaled limit of SYKq at infinite temperature, where q ~ \( \sqrt{N} \). In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be “hyperfast”, which is previously conjectured to be associated with scrambling in de Sitter space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
D.A. Roberts and B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].
L.F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E 81 (2010) 036206 [arXiv:0910.2985].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
X.-L. Qi and A. Streicher, Quantum epidemiology: operator growth, thermal effects, and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [INSPIRE].
Y.D. Lensky, X.-L. Qi and P. Zhang, Size of bulk fermions in the SYK model, JHEP 10 (2020) 053 [arXiv:2002.01961] [INSPIRE].
T. Schuster et al., Many-body quantum teleportation via operator spreading in the traversable wormhole protocol, Phys. Rev. X 12 (2022) 031013 [arXiv:2102.00010] [INSPIRE].
E.B. Rozenbaum, S. Ganeshan and V. Galitski, Lyapunov exponent and out-of-time-ordered correlator’s growth rate in a chaotic system, Phys. Rev. Lett. 118 (2017) 086801 [arXiv:1609.01707] [INSPIRE].
K. Hashimoto, K. Murata and R. Yoshii, Out-of-time-order correlators in quantum mechanics, JHEP 10 (2017) 138 [arXiv:1703.09435] [INSPIRE].
A. Nahum, S. Vijay and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].
T. Xu, T. Scaffidi and X. Cao, Does scrambling equal chaos?, Phys. Rev. Lett. 124 (2020) 140602 [arXiv:1912.11063] [INSPIRE].
T. Zhou and B. Swingle, Operator growth from global out-of-time-order correlators, Nature Commun. 14 (2023) 3411 [arXiv:2112.01562] [INSPIRE].
Y. Gu, A. Kitaev and P. Zhang, A two-way approach to out-of-time-order correlators, JHEP 03 (2022) 133 [arXiv:2111.12007] [INSPIRE].
D.E. Parker et al., A universal operator growth hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
J.L.F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On the evolution of operator complexity beyond scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space, JHEP 06 (2021) 062 [arXiv:2009.01862] [INSPIRE].
B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, Krylov complexity in saddle-dominated scrambling, JHEP 05 (2022) 174 [arXiv:2203.03534] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, https://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 27 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, https://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 7 May 2015.
A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2 (2020) 043234 [arXiv:1911.09672] [INSPIRE].
A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102 (2020) 085137 [arXiv:1912.12227] [INSPIRE].
S.-K. Jian, B. Swingle and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, JHEP 03 (2021) 014 [arXiv:2008.12274] [INSPIRE].
X. Cao, A statistical mechanism for operator growth, J. Phys. A 54 (2021) 144001 [arXiv:2012.06544] [INSPIRE].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [INSPIRE].
A. Kar, L. Lamprou, M. Rozali and J. Sully, Random matrix theory for complexity growth and black hole interiors, JHEP 01 (2022) 016 [arXiv:2106.02046] [INSPIRE].
P. Caputa, J.M. Magan and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4 (2022) 013041 [arXiv:2109.03824] [INSPIRE].
J. Kim, J. Murugan, J. Olle and D. Rosa, Operator delocalization in quantum networks, Phys. Rev. A 105 (2022) L010201 [arXiv:2109.05301] [INSPIRE].
J.M. Magán and J. Simón, On operator growth and emergent Poincaré symmetries, JHEP 05 (2020) 071 [arXiv:2002.03865] [INSPIRE].
P. Caputa and S. Datta, Operator growth in 2d CFT, JHEP 12 (2021) 188 [Erratum ibid. 09 (2022) 113] [arXiv:2110.10519] [INSPIRE].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, JHEP 12 (2022) 081 [arXiv:2207.05347] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov complexity from integrability to chaos, JHEP 07 (2022) 151 [arXiv:2207.07701] [INSPIRE].
W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B 984 (2022) 115948 [arXiv:2205.12815] [INSPIRE].
C. Liu, H. Tang and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5 (2023) 033085 [arXiv:2207.13603] [INSPIRE].
D. Patramanis, Probing the entanglement of operator growth, PTEP 2022 (2022) 063A01 [arXiv:2111.03424] [INSPIRE].
P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106 (2022) 195125 [arXiv:2205.05688] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov localization and suppression of complexity, JHEP 03 (2022) 211 [arXiv:2112.12128] [INSPIRE].
Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105 (2022) 062210 [arXiv:2202.07220] [INSPIRE].
Z.-Y. Fan, The growth of operator entropy in operator growth, JHEP 08 (2022) 232 [arXiv:2206.00855] [INSPIRE].
F.B. Trigueros and C.-J. Lin, Krylov complexity of many-body localization: operator localization in Krylov basis, SciPost Phys. 13 (2022) 037 [arXiv:2112.04722] [INSPIRE].
N. Hörnedal, N. Carabba, A.S. Matsoukas-Roubeas and A. del Campo, Ultimate speed limits to the growth of operator complexity, Commun. Phys. 5 (2022) 207 [arXiv:2202.05006] [INSPIRE].
V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) 046007 [arXiv:2202.06957] [INSPIRE].
B. Bhattacharjee, S. Sur and P. Nandy, Probing quantum scars and weak ergodicity breaking through quantum complexity, Phys. Rev. B 106 (2022) 205150 [arXiv:2208.05503] [INSPIRE].
R. Heveling, J. Wang and J. Gemmer, Numerically probing the universal operator growth hypothesis, Phys. Rev. E 106 (2022) 014152 [arXiv:2203.00533] [INSPIRE].
P. Caputa et al., Spread complexity and topological transitions in the Kitaev chain, JHEP 01 (2023) 120 [arXiv:2208.06311] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107 (2023) 126001 [arXiv:2208.08452] [INSPIRE].
M. Afrasiar et al., Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model, arXiv:2208.10520 [INSPIRE].
S. Guo, Operator growth in SU(2) Yang-Mills theory, arXiv:2208.13362 [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
J.M. Magan, Random free fermions: an analytical example of eigenstate thermalization, Phys. Rev. Lett. 116 (2016) 030401 [arXiv:1508.05339] [INSPIRE].
D. Anninos, T. Anous and F. Denef, Disordered quivers and cold horizons, JHEP 12 (2016) 071 [arXiv:1603.00453] [INSPIRE].
D.J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N. Lashkari et al., Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55 (1997) 4067 [cond-mat/9608116].
A.M. García-García and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents, Towards a full solution of the large N double-scaled SYK model, JHEP 03 (2019) 079 [arXiv:1811.02584] [INSPIRE].
M. Berkooz, V. Narovlansky and H. Raj, Complex Sachdev-Ye-Kitaev model in the double scaling limit, JHEP 02 (2021) 113 [arXiv:2006.13983] [INSPIRE].
H. Lin and L. Susskind, Infinite temperature’s not so hot, arXiv:2206.01083 [INSPIRE].
L. Susskind, De Sitter space, double-scaled SYK, and the separation of scales in the semiclassical limit, arXiv:2209.09999 [INSPIRE].
A.A. Rahman, dS JT gravity and double-scaled SYK, arXiv:2209.09997 [INSPIRE].
L. Susskind, Entanglement and chaos in de Sitter space holography: an SYK example, JHAP 1 (2021) 1 [arXiv:2109.14104] [INSPIRE].
L. Susskind, Scrambling in double-scaled SYK and de Sitter space, arXiv:2205.00315 [INSPIRE].
S. Chapman, D.A. Galante and E.D. Kramer, Holographic complexity and de Sitter space, JHEP 02 (2022) 198 [arXiv:2110.05522] [INSPIRE].
E. Jørstad, R.C. Myers and S.-M. Ruan, Holographic complexity in dSd+1, JHEP 05 (2022) 119 [arXiv:2202.10684] [INSPIRE].
V.S. Viswanath and G. Müller, The recursion method: application to many body dynamics, Springer, Berlin, Heidelberg, Germany (1994) [https://doi.org/10.1007/978-3-540-48651-0].
T. Chihara, An introduction to orthogonal polynomials, reprint edition, Dover Publications (2011).
D. Parker, Local operators and quantum chaos, Ph.D. thesis, University of California, Berkeley, CA, U.S.A. (2020).
Y. Jia and J.J.M. Verbaarschot, Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 [arXiv:1806.03271] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
C. Choi, M. Mezei and G. Sárosi, Exact four point function for large q SYK from Regge theory, JHEP 05 (2021) 166 [arXiv:1912.00004] [INSPIRE].
J. Jiang and Z. Yang, Thermodynamics and many body chaos for generalized large q SYK models, JHEP 08 (2019) 019 [arXiv:1905.00811] [INSPIRE].
M. Berkooz, N. Brukner, S.F. Ross and M. Watanabe, Going beyond ER=EPR in the SYK model, JHEP 08 (2022) 051 [arXiv:2202.11381] [INSPIRE].
H.W. Lin, The bulk Hilbert space of double scaled SYK, JHEP 11 (2022) 060 [arXiv:2208.07032] [INSPIRE].
M. Khramtsov and E. Lanina, Spectral form factor in the double-scaled SYK model, JHEP 03 (2021) 031 [arXiv:2011.01906] [INSPIRE].
H. Gharibyan, M. Hanada, S.H. Shenker and M. Tezuka, Onset of random matrix behavior in scrambling systems, JHEP 07 (2018) 124 [Erratum ibid. 02 (2019) 197] [arXiv:1803.08050] [INSPIRE].
J.B. French and S.S.M. Wong, Some random-matrix level and spacing distributions for fixed-particle-rank interactions, Phys. Lett. B 35 (1971) 5 [INSPIRE].
L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17 (2014) 441 [arXiv:1407.1552] [INSPIRE].
A.M. García-García and J.J.M. Verbaarschot, Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N, Phys. Rev. D 96 (2017) 066012 [arXiv:1701.06593] [INSPIRE].
A.M. García-García, Y. Jia and J.J.M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2, JHEP 04 (2018) 146 [arXiv:1801.02696] [INSPIRE].
T. Carleman, Les fonctions quasi analytiques, Gauthier-Villars, Paris, France (1926).
G. Tarnopolsky, Large q expansion in the Sachdev-Ye-Kitaev model, Phys. Rev. D 99 (2019) 026010 [arXiv:1801.06871] [INSPIRE].
K. Schmüdgen, Ten lectures on the moment problem, arXiv:2008.12698.
D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Matrix models for eigenstate thermalization, arXiv:2209.02130 [INSPIRE].
X. Dong, E. Silverstein and G. Torroba, De Sitter holography and entanglement entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
M. Berkooz, P. Narayan and J. Simon, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 [arXiv:1806.04380] [INSPIRE].
S. He, P.H.C. Lau, Z.-Y. Xian and L. Zhao, Quantum chaos, scrambling and operator growth in \( T\overline{T} \) deformed SYK models, JHEP 12 (2022) 070 [arXiv:2209.14936] [INSPIRE].
W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [Addendum ibid. 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].
M. Berkooz, N. Brukner, V. Narovlansky and A. Raz, The double scaled limit of super-symmetric SYK models, JHEP 12 (2020) 110 [arXiv:2003.04405] [INSPIRE].
Acknowledgments
We would like to thank Pawel Caputa, Anatoly Dymarsky, Chethan Krishnan, Tatsuma Nishioka, Prasanth Raman, Tadashi Takayanagi, and Masataka Watanabe for fruitful discussions and comments on the draft. We especially thank Leonard Susskind for the correspondence regarding [64]. We also thank the anonymous referee to point out several subtle issues which improve the quality of the paper. Part of the work was presented (by P.N.) in the Extreme Universe circular meeting. B.B. is supported by the Ministry of Human Resource Development (MHRD), Government of India, through the Prime Ministers’ Research Fellowship. The work of P.N. is supported by the JSPS Grant-in-Aid for Transformative Research Areas (A) “Extreme Universe” No. 21H05190.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2210.02474
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bhattacharjee, B., Nandy, P. & Pathak, T. Krylov complexity in large q and double-scaled SYK model. J. High Energ. Phys. 2023, 99 (2023). https://doi.org/10.1007/JHEP08(2023)099
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2023)099