Scrambling at the genesis of chaos

Michel, Thomas; Steinhuber, Mathias; Urbina, Juan Diego; Schlagheck, Peter

doi:10.1103/p978-5r2s

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Scrambling at the genesis of chaos

Michel, Thomas; Steinhuber, Mathias; Urbina, Juan Diego et al.

2026 • In Physical Review. E, 114 (3)

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10.1103/p978-5r2s

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Disciplines :

Physics

Author, co-author :

Michel, Thomas ; Université de Liège - ULiège > Complex and Entangled Systems from Atoms to Materials (CESAM)

Steinhuber, Mathias; Universität Regensburg

Urbina, Juan Diego ; Universität Regensburg

Schlagheck, Peter ; Université de Liège - ULiège > Complex and Entangled Systems from Atoms to Materials (CESAM)

Language :

English

Title :

Scrambling at the genesis of chaos

Publication date :

15 September 2026

Journal title :

Physical Review. E

ISSN :

2470-0045

eISSN :

2470-0053

Publisher :

American Physical Society (APS)

Volume :

114

Issue :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.aps.org/article/10.1103/p978-5r2s

Funders :

Studienstiftung des Deutschen Volkes

Available on ORBi :

since 29 September 2026

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Bibliography

E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, Cambridge, 2002).
M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, New York, 1989).
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer, New York, 1992).
M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
F. Haake, Quantum Signatures of Chaos, 3rd ed. (Springer, Heidelberg, 2010).
O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984) 0031-9007 10.1103/PhysRevLett.52.1.
J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991) 1050-2947 10.1103/PhysRevA.43.2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994) 1063-651X 10.1103/PhysRevE.50.888.
A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A 30, 1610 (1984) 0556-2791 10.1103/PhysRevA.30.1610.
R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Phys. Rev. Lett. 86, 2490 (2001) 0031-9007 10.1103/PhysRevLett.86.2490.
B. Yan, L. Cincio, and W. H. Zurek, Information scrambling and Loschmidt echo, Phys. Rev. Lett. 124, 160603 (2020) 0031-9007 10.1103/PhysRevLett.124.160603.
S. PG, V. Madhok, and A. Lakshminarayan, Out-of-time-ordered correlators and the Loschmidt echo in the quantum kicked top: How low can we go? J. Phys. D: Appl. Phys. 54, 274004 (2021) 0022-3727 10.1088/1361-6463/abf8f3.
A. Bhattacharyya, W. Chemissany, S. S. Haque, and B. Yan, Towards the web of quantum chaos diagnostics, Eur. Phys. J. C 82, 87 (2022) 1434-6044 10.1140/epjc/s10052-022-10035-3.
A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28, 1200 (1969).
B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nat. Phys. 14, 988 (2018) 1745-2473 10.1038/s41567-018-0295-5.
R. J. Garcia, K. Bu, and A. Jaffe, Resource theory of quantum scrambling, Proc. Natl. Acad. Sci. USA 120, e2217031120 (2023) 0027-8424 10.1073/pnas.2217031120.
J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106 1029-8479 10.1007/JHEP08(2016)106.
J. Rammensee, J. D. Urbina, and K. Richter, Many-body quantum interference and the saturation of out-of-time-order correlators, Phys. Rev. Lett. 121, 124101 (2018) 0031-9007 10.1103/PhysRevLett.121.124101.
S. Xu and B. Swingle, Scrambling dynamics and out-of-time-ordered correlators in quantum many-body systems, PRX Quantum 5, 010201 (2024) 2691-3399 10.1103/PRXQuantum.5.010201.
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, 086801 (2017) 0031-9007 10.1103/PhysRevLett.118.086801.
R. A. Jalabert, I. García-Mata, and D. A. Wisniacki, Semiclassical theory of out-of-time-order correlators for low-dimensional classically chaotic systems, Phys. Rev. E 98, 062218 (2018) 2470-0045 10.1103/PhysRevE.98.062218.
T. R. Michel, J. D. Urbina, and P. Schlagheck, Quasiclassical theory of out-of-time-ordered correlators, J. Phys. A: Math. Theor. 58, 275303 (2025) 1751-8113 10.1088/1751-8121/ade57d.
S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva, and R. Fazio, Scrambling and entanglement spreading in long-range spin chains, Phys. Rev. B 98, 134303 (2018) 2469-9950 10.1103/PhysRevB.98.134303.
Q. Hummel, B. Geiger, J. D. Urbina, and K. Richter, Reversible quantum information spreading in many-body systems near criticality, Phys. Rev. Lett. 123, 160401 (2019) 0031-9007 10.1103/PhysRevLett.123.160401.
E. B. Rozenbaum, L. A. Bunimovich, and V. Galitski, Early-time exponential instabilities in nonchaotic quantum systems, Phys. Rev. Lett. 125, 014101 (2020) 0031-9007 10.1103/PhysRevLett.125.014101.
T. Xu, T. Scaffidi, and X. Cao, Does scrambling equal chaos? Phys. Rev. Lett. 124, 140602 (2020) 0031-9007 10.1103/PhysRevLett.124.140602.
K. Hashimoto, K.-B. Huh, K.-Y. Kim, and R. Watanabe, Exponential growth of out-of-time-order correlator without chaos: Inverted harmonic oscillator, J. High Energy Phys. 11 (2020) 068 1029-8479 10.1007/JHEP11(2020)068.
T. Morita, Extracting classical Lyapunov exponent from one-dimensional quantum mechanics, Phys. Rev. D 106, 106001 (2022) 2470-0010 10.1103/PhysRevD.106.106001.
F. Meier, M. Steinhuber, J. D. Urbina, D. Waltner, and T. Guhr, Signatures of the interplay between chaos and local criticality on the dynamics of scrambling in many-body systems, Phys. Rev. E 107, 054202 (2023) 2470-0045 10.1103/PhysRevE.107.054202.
D. A. Trunin, Quantum chaos without false positives, Phys. Rev. D 108, L101703 (2023) 2470-0010 10.1103/PhysRevD.108.L101703.
M. Steinhuber, P. Schlagheck, J. D. Urbina, and K. Richter, Dynamical transition from localized to uniform scrambling in locally hyperbolic systems, Phys. Rev. E 108, 024216 (2023) 2470-0045 10.1103/PhysRevE.108.024216.
B. K. Melnikov, Lines of force of a magnetic field, Dokl. Akad Nauk SSSR 144, 747 (1962).
V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156, 9 (1964).
B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52, 263 (1979) 0370-1573 10.1016/0370-1573(79)90023-1.
I. Goldhirsch, P.-L. Sulem, and S. A. Orszag, Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method, Physica D 27, 311 (1987) 0167-2789 10.1016/0167-2789(87)90034-0.
K. Richter, J. D. Urbina, and S. Tomsovic, Semiclassical roots of universality in many-body quantum chaos, J. Phys. A: Math. Theor. 55, 453001 (2022) 1751-8113 10.1088/1751-8121/ac9e4e.
A. Polkovnikov, Phase space representation of quantum dynamics, Ann. Phys. (NY) 325, 1790 (2010) 0003-4916 10.1016/j.aop.2010.02.006.
A. M. Ozorio de Almeida, Tunneling and the semiclassical spectrum for an isolated classical resonance, J. Phys. Chem. 88, 6139 (1984) 0022-3654 10.1021/j150669a017.
O. Brodier, P. Schlagheck, and D. Ullmo, Resonance-assisted tunneling, Ann. Phys. (NY) 300, 88 (2002) 0003-4916 10.1006/aphy.2002.6281.
N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russ. Math. Surv. 32, 1 (1977) 0036-0279 10.1070/RM1977v032n06ABEH003859.
J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z. 213, 187 (1993) 0025-5874 10.1007/BF03025718.
The threshold was set at 0.5, but its value is of little importance since any threshold will yield an exponential scaling.
A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16, 285 (1985) 0167-2789 10.1016/0167-2789(85)90011-9.
M. P. Strzys, E. M. Graefe, and H. J. Korsch, Kicked Bose–Hubbard systems and kicked tops—Destruction and stimulation of tunneling, New J. Phys. 10, 013024 (2008) 1367-2630 10.1088/1367-2630/10/1/013024.
C. Liang, Y. Zhang, and S. Chen, Statistical and dynamical aspects of quantum chaos in a kicked Bose-Hubbard dimer, Phys. Rev. A 109, 033316 (2024) 2469-9926 10.1103/PhysRevA.109.033316.
E. J. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett. 53, 1515 (1984) 0031-9007 10.1103/PhysRevLett.53.1515.
S. Sinha, S. Ray, and S. Sinha, Fingerprint of chaos and quantum scars in kicked Dicke model: An out-of-time-order correlator study, J. Phys.: Condens. Matter 33, 174005 (2021) 0953-8984 10.1088/1361-648X/abe26b.
Q. Hummel, K. Richter, and P. Schlagheck, Genuine many-body quantum scars along unstable modes in Bose-Hubbard systems, Phys. Rev. Lett. 130, 250402 (2023) 0031-9007 10.1103/PhysRevLett.130.250402.
We will keep the first order only for the position and the momentum, but as obtaining the latter is done by time derivating the former, this introduces some (Formula presented) prefactor. To be consistent in our approximation, we must keep the (Formula presented) and neglect them only after the derivation.