Abstract
We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order of N, the standard result for the q-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter, k. We present an algorithm to sum the diagrams at any given order of 1/(kq)n. We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finite N becomes more significant for larger values of N.
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S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].
K. Jensen, Chaos in AdS2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nature Phys. 14 (2018) 988 [INSPIRE].
A.M. García-García, B. Loureiro, A. Romero-Bermúdez and M. Tezuka, Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model, Phys. Rev. Lett. 120 (2018) 241603 [arXiv:1707.02197] [INSPIRE].
Y. Gu and A. Kitaev, On the relation between the magnitude and exponent of OTOCs, JHEP 02 (2019) 075 [arXiv:1812.00120] [INSPIRE].
W. Fischler, T. Guglielmo and P. Nguyen, Quantum chaos in a weakly-coupled field theory with nonlocality, JHEP 09 (2022) 097 [arXiv:2111.10895] [INSPIRE].
S. Xu and B. Swingle, Scrambling Dynamics and Out-of-Time Ordered Correlators in Quantum Many-Body Systems: a Tutorial, arXiv:2202.07060 [INSPIRE].
I. García-Mata, R.A. Jalabert and D.A. Wisniacki, Out-of-time-order correlators and quantum chaos, Scholarpedia 18 (2023) 55237 [arXiv:2209.07965] [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, JETP 28 (1969) 1200.
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
S. Xu, L. Susskind, Y. Su and B. Swingle, A Sparse Model of Quantum Holography, arXiv:2008.02303 [INSPIRE].
A.M. García-García, Y. Jia, D. Rosa and J.J.M. Verbaarschot, Sparse Sachdev-Ye-Kitaev model, quantum chaos and gravity duals, Phys. Rev. D 103 (2021) 106002 [arXiv:2007.13837] [INSPIRE].
E. Cáceres, A. Misobuchi and A. Raz, Spectral form factor in sparse SYK models, JHEP 08 (2022) 236 [arXiv:2204.07194] [INSPIRE].
E. Cáceres, A. Misobuchi and R. Pimentel, Sparse SYK and traversable wormholes, JHEP 11 (2021) 015 [arXiv:2108.08808] [INSPIRE].
D. Jafferis et al., Traversable wormhole dynamics on a quantum processor, Nature 612 (2022) 51 [INSPIRE].
B. Kobrin et al., Many-Body Chaos in the Sachdev-Ye-Kitaev Model, Phys. Rev. Lett. 126 (2021) 030602 [arXiv:2002.05725] [INSPIRE].
M. Tezuka et al., Binary-coupling sparse Sachdev-Ye-Kitaev model: An improved model of quantum chaos and holography, Phys. Rev. B 107 (2023) L081103 [arXiv:2208.12098] [INSPIRE].
I. Dumitriu and Y. Zhu, Spectra of Random Regular Hypergraphs, Electron. J. Comb. 28 (2021) P3.36.
N.Y. Yao et al., Interferometric Approach to Probing Fast Scrambling, arXiv:1607.01801 [INSPIRE].
É. Lantagne-Hurtubise, S. Plugge, O. Can and M. Franz, Diagnosing quantum chaos in many-body systems using entanglement as a resource, Phys. Rev. Res. 2 (2020) 013254 [arXiv:1907.01628] [INSPIRE].
H. Shen, P. Zhang, R. Fan and H. Zhai, Out-of-Time-Order Correlation at a Quantum Phase Transition, Phys. Rev. B 96 (2017) 054503 [arXiv:1608.02438] [INSPIRE].
A. Keleş, E. Zhao and W.V. Liu, Scrambling dynamics and many-body chaos in a random dipolar spin model, Phys. Rev. A 99 (2019) 053620 [arXiv:1810.03815] [INSPIRE].
G.D. Kahanamoku-Meyer and J. Wei, GregDMeyer/dynamite: v0.3.1, (2023), https://doi.org/10.5281/ZENODO.7706785.
T.J. Park and J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction, J. Chem. Phys. 85 (1986) 5870.
S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zanghi, Canonical Typicality, Phys. Rev. Lett. 96 (2006) 050403 [cond-mat/0511091] [INSPIRE].
D.J. Luitz and Y.B. Lev, The ergodic side of the many-body localization transition, Annalen Phys. 529 (2017) 1600350 [arXiv:1610.08993] [INSPIRE].
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].
A.S. Shankar, M. Fremling, S. Plugge and L. Fritz, Lyapunov exponents in a Sachdev-Ye-Kitaev-type model with population imbalance in the conformal limit and beyond, arXiv:2302.08876 [INSPIRE].
R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading (1989).
D. Stanzione et al., Frontera: The Evolution of Leadership Computing at the National Science Foundation, in the proceedings of the PEARC ’20: Practice and Experience in Advanced Research Computing, New York, NY, U.S.A., Association for Computing Machinery (2020), p. 106–111 [https://doi.org/10.1145/3311790.3396656].
Acknowledgments
The authors would like to thank Javier Mas, Alfonso Ramallo, Brian Swingle and Shenglong Xu for useful discussions. The work of EC, BK, and AM was supported by the National Science Foundation under Grant Number PHY-2112725. BK is also supported by NSF PHY-2210562 and TG by PHY-1914679. The work of TG was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. EC thanks the Instituto de Física Teórica (IFT) at UAM, Madrid, for hospitality during the last stages of this work. The authors also acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. URL: http://www.tacc.utexas.edu.
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Cáceres, E., Guglielmo, T., Kent, B. et al. Out-of-time-order correlators and Lyapunov exponents in sparse SYK. J. High Energ. Phys. 2023, 88 (2023). https://doi.org/10.1007/JHEP11(2023)088
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DOI: https://doi.org/10.1007/JHEP11(2023)088