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Long-time Freeness in the Kicked Top

Quantum Journal

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AbstractRecent work highlighted the importance of higher-order correlations in quantum dynamics for a deeper understanding of quantum chaos and thermalization. The full Eigenstate Thermalization Hypothesis, the framework encompassing correlations, can be formalized using the language of Free Probability theory. In this context, chaotic dynamics at long times are proposed to lead to free independence or "freeness" of observables. In this work, we investigate these issues in a paradigmatic semiclassical model – the kicked top – which exhibits a transition from integrability to chaos. Despite its simplicity, we identify several non-trivial features. By numerically studying 2n-point out-of-time-order correlators, we show that in the fully chaotic regime, long-time freeness is reached exponentially fast. These considerations lead us to introduce a large deviation theory for freeness that enables us to define and analyze the associated time scale. The numerical results confirm the existence of a hierarchy of different time scales, indicating a multifractal approach to freeness in this model. Our findings provide novel insights into the long-time behavior of chaotic dynamics and may have broader implications for the study of many-body quantum dynamics.Featured image: A multifractal route to Freeness.Popular summaryChaos is often associated with unpredictability and randomness. In quantum systems, however, understanding how randomness emerges in deterministic dynamics remains a subtle problem. To characterize this regime, we employ concepts from Free Probability theory, a mathematical framework that extends ordinary probability theory to non-commuting quantities. In this setting, chaotic dynamics is expected to drive observables towards freeness, a generalized form of statistical independence that can be probed through higher-order correlation functions. In this work, we investigate higher-order correlations in the kicked top, a prototypical model of quantum chaos. Our numerical results show that, in the chaotic regime, the kicked top approaches freeness exponentially fast at long times. Moreover, we find a hierarchy of timescales for which increasingly higher-order correlations require progressively longer times to show freeness. This points to a multifractal route to randomness. These findings provide new insight into the long-time behavior of quantum chaotic systems and suggest that freeness may offer a powerful framework for understanding thermalization in a broad range of many-body quantum systems.► BibTeX data@article{Vallini2026longtimefreenessin, doi = {10.22331/q-2026-06-08-2129}, url = {https://doi.org/10.22331/q-2026-06-08-2129}, title = {Long-time {F}reeness in the {K}icked {T}op}, author = {Vallini, Elisa and Pappalardi, Silvia}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2129}, month = jun, year = {2026} }► References [1] Philippe Delsarte, Jean-Marie Goethals, and Johan Jacob Seidel. ``Spherical codes and designs''. In Geometry and Combinatorics. Pages 68–93. Elsevier (1991). https://doi.org/10.1007/BF03187604 [2] Christoph Dankert. ``Efficient simulation of random quantum states and operators'' (2005). arXiv:quant-ph/0512217. arXiv:quant-ph/0512217 [3] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80, 012304 (2009). https://doi.org/10.1103/PhysRevA.80.012304 [4] Jordan S Cotler, Daniel K Mark, Hsin-Yuan Huang, Felipe Hernandez, Joonhee Choi, Adam L Shaw, Manuel Endres, and Soonwon Choi. ``Emergent quantum state designs from individual many-body wave functions''. PRX Quantum 4, 010311 (2023). https://doi.org/10.1103/PRXQuantum.4.010311 [5] Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Ran Finkelstein, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Anant Kale, Hannes Pichler, Fernando G. S. L. Brandão, Soonwon Choi, and Manuel Endres. ``Preparing random states and benchmarking with many-body quantum chaos''. Nature 613, 468–473 (2023). https://doi.org/10.1038/s41586-022-05442-1 [6] Wen Wei Ho and Soonwon Choi. ``Exact emergent quantum state designs from quantum chaotic dynamics''. Phys. Rev. Lett. 128, 060601 (2022). https://doi.org/10.1103/PhysRevLett.128.060601 [7] Pieter W Claeys and Austen Lamacraft. ``Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics''. Quantum 6, 738 (2022). https://doi.org/10.22331/q-2022-06-15-738 [8] Matteo Ippoliti and Wen Wei Ho. ``Solvable model of deep thermalization with distinct design times''. Quantum 6, 886 (2022). https://doi.org/10.22331/q-2022-12-29-886 [9] Maxime Lucas, Lorenzo Piroli, Jacopo De Nardis, and Andrea De Luca. ``Generalized deep thermalization for free fermions''. Phys. Rev. A 107, 032215 (2023). https://doi.org/10.1103/PhysRevA.107.032215 [10] Tanmay Bhore, Jean-Yves Desaules, and Zlatko Papić. ``Deep thermalization in constrained quantum systems''. Phys. Rev. B 108, 104317 (2023). https://doi.org/10.1103/PhysRevB.108.104317 [11] Max McGinley and Michele Fava. ``Shadow tomography from emergent state designs in analog quantum simulators''. Phys. Rev. Lett. 131, 160601 (2023). https://doi.org/10.1103/PhysRevLett.131.160601 [12] Saúl Pilatowsky-Cameo, Ceren B. Dag, Wen Wei Ho, and Soonwon Choi. ``Complete hilbert-space ergodicity in quantum dynamics of generalized fibonacci drives''. Phys. Rev. Lett. 131, 250401 (2023). https://doi.org/10.1103/PhysRevLett.131.250401 [13] Daniel K Mark, Federica Surace, Andreas Elben, Adam L Shaw, Joonhee Choi, Gil Refael, Manuel Endres, and Soonwon Choi. ``A maximum entropy principle in deep thermalization and in hilbert-space ergodicity'' (2024). arXiv:2403.11970. https://doi.org/10.1103/PhysRevX.14.041051 arXiv:2403.11970 [14] Anatoly Dymarsky, Nima Lashkari, and Hong Liu. ``Subsystem eigenstate thermalization hypothesis''. Phys. Rev. E 97, 012140 (2018). https://doi.org/10.1103/PhysRevE.97.012140 [15] Yichen Huang. ``Universal entanglement of mid-spectrum eigenstates of chaotic local hamiltonians''. Nuclear Physics B 966, 115373 (2021). https://doi.org/10.1016/j.nuclphysb.2021.115373 [16] Tsung-Cheng Lu and Tarun Grover. ``Renyi entropy of chaotic eigenstates''. Phys. Rev. E 99, 032111 (2019). https://doi.org/10.1103/PhysRevE.99.032111 [17] Chaitanya Murthy and Mark Srednicki. ``Structure of chaotic eigenstates and their entanglement entropy''. Physical Review E 100, 022131 (2019). https://doi.org/10.1103/PhysRevE.100.022131 [18] Zhengyan Darius Shi, Shreya Vardhan, and Hong Liu. ``Local dynamics and the structure of chaotic eigenstates''. Physical Review B 108, 224305 (2023). https://doi.org/10.1103/PhysRevB.108.224305 [19] Dominik Hahn, David J. Luitz, and J. T. Chalker. ``Eigenstate correlations, the eigenstate thermalization hypothesis, and quantum information dynamics in chaotic many-body quantum systems''. Phys. Rev. X 14, 031029 (2024). https://doi.org/10.1103/PhysRevX.14.031029 [20] Siddharth Jindal and Pavan Hosur. ``Generalized free cumulants for quantum chaotic systems'' (2024). arXiv:2401.13829. https://doi.org/10.1007/JHEP09(2024)066 arXiv:2401.13829 [21] Benjamin Doyon and Jason Myers. ``Fluctuations in ballistic transport from euler hydrodynamics''.

In Annales Henri Poincaré. Volume 21, pages 255–302. Springer (2020). https://doi.org/10.1007/s00023-019-00860-w [22] Jason Myers, Joe Bhaseen, Rosemary J Harris, and Benjamin Doyon. ``Transport fluctuations in integrable models out of equilibrium''. SciPost Physics 8, 007 (2020). https://doi.org/10.21468/SciPostPhys.8.1.007 [23] Michele Fava, Sounak Biswas, Sarang Gopalakrishnan, Romain Vasseur, and SA Parameswaran. ``Hydrodynamic nonlinear response of interacting integrable systems''. Proceedings of the National Academy of Sciences 118, e2106945118 (2021). https://doi.org/10.1073/pnas.2106945118 [24] Luca V Delacrétaz and Ruchira Mishra. ``Nonlinear response in diffusive systems''. SciPost Physics 16, 047 (2024). https://doi.org/10.21468/SciPostPhys.16.2.047 [25] Taishi Kawamoto. ``A strategy for proving the strong eigenstate thermalization hypothesis: Chaotic systems and holography'' (2024). arXiv:2411.09746. https://doi.org/10.1007/JHEP01(2025)095 arXiv:2411.09746 [26] Juan Maldacena, Stephen H Shenker, and Douglas Stanford. ``A bound on chaos''. Journal of High Energy Physics 2016, 106 (2016). https://doi.org/10.1007/JHEP08(2016)106 [27] Pavan Hosur, Xiao-Liang Qi, Daniel A Roberts, and Beni Yoshida. ``Chaos in quantum channels''. Journal of High Energy Physics 2016, 004 (2016). https://doi.org/10.1007/JHEP02(2016)004 [28] Shenglong Xu and Brian Swingle. ``Scrambling dynamics and out-of-time ordered correlators in quantum many-body systems: a tutorial'' (2022). arXiv:2202.07060. https://doi.org/10.1103/PRXQuantum.5.010201 arXiv:2202.07060 [29] Ignacio García-Mata, Rodolfo A. Jalabert, and Diego A. Wisniacki. ``Out-of-time-order correlators and quantum chaos'' (2022). arXiv:2209.07965. https://doi.org/10.4249/scholarpedia.55237 arXiv:2209.07965 [30] A. I. Larkin and Yu. N. Ovchinnikov. ``Quasiclassical method in the theory of superconductivity''. Soviet Physics JETP 28, 1200–1205 (1969). url: https://jetp.ras.ru/cgi-bin/dn/e_028_06_1200.pdf. https://jetp.ras.ru/cgi-bin/dn/e_028_06_1200.pdf [31] A. Kitaev. ``Talk given at the fundamental physics prize symposium''. YouTube. (20145). url: https://www.youtube.com/watch?v=OQ9qN8j7EZI. https://www.youtube.com/watch?v=OQ9qN8j7EZI [32] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. Journal of High Energy Physics 2017, 121 (2017). https://doi.org/10.1007/JHEP04(2017)121 [33] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and Beni Yoshida. ``Chaos, complexity, and random matrices''. Journal of High Energy Physics 2017, 1–60 (2017). https://doi.org/10.1007/JHEP11(2017)048 [34] Naoto Tsuji, Tomohiro Shitara, and Masahito Ueda. ``Bound on the exponential growth rate of out-of-time-ordered correlators''. Physical Review E 98, 012216 (2018). https://doi.org/10.1103/PhysRevE.98.012216 [35] Jordan Cotler and Nicholas Hunter-Jones. ``Spectral decoupling in many-body quantum chaos''. Journal of High Energy Physics 2020, 1–62 (2020). https://doi.org/10.1007/JHEP12(2020)205 [36] Lorenzo Leone, Salvatore FE Oliviero, You Zhou, and Alioscia Hamma. ``Quantum chaos is quantum''. Quantum 5, 453 (2021). https://doi.org/10.22331/q-2021-05-04-453 [37] Silvia Pappalardi and Jorge Kurchan. ``Quantum bounds on the generalized lyapunov exponents''. Entropy 25, 246 (2023). https://doi.org/10.3390/e25020246 [38] Dmitrii A Trunin. ``Quantum chaos without false positives''. Physical Review D 108, L101703 (2023). https://doi.org/10.1103/PhysRevD.108.L101703 [39] Dmitrii A Trunin. ``Refined quantum lyapunov exponents from replica out-of-time-order correlators''. Physical Review D 108, 105023 (2023). https://doi.org/10.1103/PhysRevD.108.105023 [40] Mark Srednicki. ``Chaos and quantum thermalization''. Physical Review E 50, 888–901 (1994). https://doi.org/10.1103/PhysRevE.50.888 [41] Mark Srednicki. ``The approach to thermal equilibrium in quantized chaotic systems''. Journal of Physics A: Mathematical and General 32, 1163–1175 (1999). https://doi.org/10.1088/0305-4470/32/7/007 [42] Luca D'Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics''. Advances in Physics 65, 239–362 (2016). https://doi.org/10.1080/00018732.2016.1198134 [43] Laura Foini and Jorge Kurchan. ``Eigenstate thermalization hypothesis and out of time order correlators''. Physical Review E 99, 042139 (2019). https://doi.org/10.1103/PhysRevE.99.042139 [44] Tomaz Prosen. ``Ergodic properties of a generic nonintegrable quantum many-body system in the thermodynamic limit''. Phys. Rev. E 60, 3949–3968 (1999). https://doi.org/10.1103/PhysRevE.60.3949 [45] Julian Sonner and Manuel Vielma. ``Eigenstate thermalization in the sachdev-ye- model''. Journal of High Energy Physics 2017, 1–28 (2017). https://doi.org/10.1007/JHEP11(2017)149 [46] Laura Foini and Jorge Kurchan. ``Eigenstate thermalization and rotational invariance in ergodic quantum systems''. Phys. Rev. Lett. 123, 260601 (2019). https://doi.org/10.1103/PhysRevLett.123.260601 [47] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Eigenstate correlations, thermalization, and the butterfly effect''. Phys. Rev. Lett. 122, 220601 (2019). https://doi.org/10.1103/PhysRevLett.122.220601 [48] Chaitanya Murthy and Mark Srednicki. ``Bounds on chaos from the eigenstate thermalization hypothesis''.

Physical Review Letters 123, 230606 (2019). https://doi.org/10.1103/PhysRevLett.123.230606 [49] Jonas Richter, Anatoly Dymarsky, Robin Steinigeweg, and Jochen Gemmer. ``Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies''. Physical Review E 102, 042127 (2020). https://doi.org/10.1103/PhysRevE.102.042127 [50] Jiaozi Wang, Mats H Lamann, Jonas Richter, Robin Steinigeweg, Anatoly Dymarsky, and Jochen Gemmer. ``Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time'' (2021). arXiv:2110.04085. https://doi.org/10.1103/PhysRevLett.128.180601 arXiv:2110.04085 [51] Marlon Brenes, Silvia Pappalardi, Mark T Mitchison, John Goold, and Alessandro Silva. ``Out-of-time-order correlations and the fine structure of eigenstate thermalization''. Physical Review E 104, 034120 (2021). https://doi.org/10.1103/PhysRevE.104.034120 [52] Anatoly Dymarsky. ``Bound on eigenstate thermalization from transport''. Phys. Rev. Lett. 128, 190601 (2022). https://doi.org/10.1103/PhysRevLett.128.190601 [53] Zohar Nussinov and Saurish Chakrabarty. ``Exact universal chaos, speed limit, acceleration, planckian transport coefficient,“collapse” to equilibrium, and other bounds in thermal quantum systems''. Annals of Physics 443, 168970 (2022). https://doi.org/10.1016/j.aop.2022.168970 [54] Daniel Louis Jafferis, David K Kolchmeyer, Baur Mukhametzhanov, and Julian Sonner. ``Matrix models for eigenstate thermalization'' (2022). arXiv:2209.02130. arXiv:2209.02130 [55] Daniel Louis Jafferis, David K Kolchmeyer, Baur Mukhametzhanov, and Julian Sonner. ``Jt gravity with matter, generalized eth, and random matrices'' (2022). arXiv:2209.02131. arXiv:2209.02131 [56] Jiaozi Wang, Jonas Richter, Mats H Lamann, Robin Steinigeweg, Jochen Gemmer, and Anatoly Dymarsky. ``Emergence of unitary symmetry of microcanonically truncated operators in chaotic quantum systems''. Physical Review E 110, L032203 (2024). https://doi.org/10.1103/PhysRevE.110.L032203 [57] Michele Fava, Jorge Kurchan, and Silvia Pappalardi. ``Designs via free probability''. Physical Review X 15, 011031 (2025). https://doi.org/10.1103/PhysRevX.15.011031 [58] Silvia Pappalardi, Felix Fritzsch, and Tomaž Prosen. ``Full eigenstate thermalization via free cumulants in quantum lattice systems'' (2023). arXiv:2303.00713. https://doi.org/10.1103/PhysRevLett.134.140404 arXiv:2303.00713 [59] Silvia Pappalardi, Laura Foini, and Jorge Kurchan. ``Microcanonical windows on quantum operators''. Quantum 8, 1227 (2024). https://doi.org/10.22331/q-2024-01-11-1227 [60] Felix Fritzsch, Tomaž Prosen, and Silvia Pappalardi. ``Microcanonical free cumulants in lattice systems''. Physical Review B 111, 054303 (2025). https://doi.org/10.1103/PhysRevB.111.054303 [61] Silvia Pappalardi, Laura Foini, and Jorge Kurchan. ``Eigenstate thermalization hypothesis and free probability''.

Physical Review Letters 129, 170603 (2022). https://doi.org/10.1103/PhysRevLett.129.170603 [62] Dan Voiculescu. ``Limit laws for random matrices and free products''. Inventiones Mathematicae 104, 201–220 (1991). https://doi.org/10.1007/BF01245072 [63] Dan V Voiculescu, Ken J Dykema, and Alexandru Nica. ``Free random variables''.

American Mathematical Society. (1992). https://doi.org/10.1090/crmm/001 [64] Roland Speicher. ``Free probability theory and non-crossing partitions''. Séminaire Lotharingien de Combinatoire 39 (1997). url: https://www.emis.de/journals/SLC/wpapers/s39speicher.pdf. https://www.emis.de/journals/SLC/wpapers/s39speicher.pdf [65] Edouard Brézin, Claude Itzykson, Giorgio Parisi, and Jean-Bernard Zuber. ``Planar diagrams''. Communications in Mathematical Physics 59, 35–51 (1978). https://doi.org/10.1007/BF01614153 [66] Predrag Cvitanović. ``Planar perturbation expansion''. Physics Letters B 99, 49–52 (1981). https://doi.org/10.1016/0370-2693(81)90801-7 [67] Predrag Cvitanović, P G Lauwers, and P N Scharbach. ``The planar sector of field theories''. Nuclear Physics B 203, 385–412 (1982). https://doi.org/10.1016/0550-3213(82)90320-0 [68] Benoı̂t Collins and Ion Nechita. ``Random matrix techniques in quantum information theory''. Journal of Mathematical Physics 57, 015215 (2016). https://doi.org/10.1063/1.4936880 [69] Benoı̂t Collins and Ion Nechita. ``Random quantum channels i: Graphical calculus and the bell state phenomenon''. Communications in Mathematical Physics 297, 345–370 (2010). https://doi.org/10.1007/s00220-010-1012-0 [70] Jonah Kudler-Flam, Vladimir Narovlansky, and Shinsei Ryu. ``Distinguishing random and black hole microstates''. PRX Quantum 2, 040340 (2021). https://doi.org/10.1103/PRXQuantum.2.040340 [71] Jonah Kudler-Flam, Vladimir Narovlansky, and Shinsei Ryu. ``Negativity spectra in random tensor networks and holography''. Journal of High Energy Physics 2022, 1–74 (2022). https://doi.org/10.1007/JHEP02(2022)076 [72] Newton Cheng, Cécilia Lancien, Geoff Penington, Michael Walter, and Freek Witteveen. ``Random tensor networks with non-trivial links''.

In Annales Henri Poincaré. Volume 25, pages 2107–2212. Springer (2024). https://doi.org/10.1007/s00023-023-01358-2 [73] Ramis Movassagh and Alan Edelman. ``Isotropic entanglement'' (2010). arXiv:1012.5039. arXiv:1012.5039 [74] Ramis Movassagh and Alan Edelman. ``Density of states of quantum spin systems from isotropic entanglement''.

Physical Review Letters 107, 097205 (2011). https://doi.org/10.1103/PhysRevLett.107.097205 [75] Jiahao Chen, Eric Hontz, Jeremy Moix, Matthew Welborn, Troy Van Voorhis, Alberto Suárez, Ramis Movassagh, and Alan Edelman. ``Error analysis of free probability approximations to the density of states of disordered systems''. Phys. Rev. Lett. 109, 036403 (2012). https://doi.org/10.1103/PhysRevLett.109.036403 [76] Ludwig Hruza and Denis Bernard. ``Coherent fluctuations in noisy mesoscopic systems, the open quantum SSEP, and free probability''. Physical Review X 13 (2023). https://doi.org/10.1103/physrevx.13.011045 [77] Michel Bauer, Denis Bernard, Philippe Biane, and Ludwig Hruza. ``Bernoulli variables, classical exclusion processes and free probability''.

Annales Henri Poincaré 24, 1–48 (2023). https://doi.org/10.1007/s00023-023-01320-2 [78] Denis Bernard and Ludwig Hruza. ``Exact entanglement in the driven quantum symmetric simple exclusion process''. SciPost Physics 15, 175 (2023). https://doi.org/10.21468/SciPostPhys.15.4.175 [79] Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, and Genis Torrents. ``Towards a full solution of the large n double-scaled syk model''. Journal of High Energy Physics 2019, 079 (2019). https://doi.org/10.1007/JHEP03(2019)079 [80] Geoff Penington, Stephen H Shenker, Douglas Stanford, and Zhenbin Yang. ``Replica wormholes and the black hole interior''. Journal of High Energy Physics 2022, 205 (2022). https://doi.org/10.1007/JHEP03(2022)205 [81] Jinzhao Wang. ``Beyond islands: a free probabilistic approach''. Journal of High Energy Physics 2023, 40 (2023). https://doi.org/10.1007/JHEP10(2023)040 [82] Shuang Wu. ``Non-commutative probability insights into the double-scaling limit syk model with constant perturbations: moments, cumulants and q-independence''. Journal of Physics A: Mathematical and Theoretical 57, 325203 (2024). https://doi.org/10.1088/1751-8121/ad65a6 [83] Venkatesa Chandrasekaran, Geoff Penington, and Edward Witten. ``Large n algebras and generalized entropy''. Journal of High Energy Physics 2023, 009 (2023). https://doi.org/10.1007/JHEP04(2023)009 [84] Giorgio Cipolloni, László Erdős, and Dominik Schröder. ``Thermalisation for wigner matrices''. Journal of Functional Analysis 282, 109394 (2022). https://doi.org/10.1016/j.jfa.2022.109394 [85] Hyaline Junhe Chen and Jonah Kudler-Flam. ``Free independence and the noncrossing partition lattice in dual-unitary quantum circuits'' (2024). arXiv:2409.17226. https://doi.org/10.1103/PhysRevB.111.014311 arXiv:2409.17226 [86] F. Haake, M. Kuś, and R. Scharf. ``Classical and quantum chaos for a kicked top''. Z. Phys. B-Condensed Matter 65, 381–395 (1987). https://doi.org/10.1007/BF01303727 [87] M. Kuś, R. Scharf, and F. Haake. ``Symmetry versus degree of level repulsion for kicked quantum systems''. Zeitschrift für Physik B Condensed Matter 66, 129–134 (1987). https://doi.org/10.1007/BF01312770 [88] Roberto Benzi, Giovanni Paladin, Giorgio Parisi, and Angelo Vulpiani. ``On the multifractal nature of fully developed turbulence and chaotic systems''. Journal of Physics A: Mathematical and General 17, 3521 (1984). https://doi.org/10.1142/9789812799050_0017 [89] Giovanni Paladin and Angelo Vulpiani. ``Anomalous scaling laws in multifractal objects''. Physics Reports 156, 147–225 (1987). https://doi.org/10.1016/0370-1573(87)90110-4 [90] Giovanni Paladin and Angelo Vulpiani. ``Intermittency in chaotic systems and rényi entropies''. Journal of Physics A: Mathematical and General 19, L997 (1986). https://doi.org/10.1088/0305-4470/19/16/009 [91] Andrea Crisanti, Giovanni Paladin, and Angelo Vulpiani. ``Generalized lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices''. Journal of Statistical Physics 53, 583–601 (1988). https://doi.org/10.1007/BF01014215 [92] F. Evers and A. D. Mirlin. ``Fluctuations of the inverse participation ratio at the anderson transition''. Phys. Rev. Lett. 84, 3690–3693 (2000). https://doi.org/10.1103/PhysRevLett.84.3690 [93] Nicolas Macé, Fabien Alet, and Nicolas Laflorencie. ``Multifractal scalings across the many-body localization transition''. Phys. Rev. Lett. 123, 180601 (2019). https://doi.org/10.1103/PhysRevLett.123.180601 [94] Piotr Sierant and Xhek Turkeshi. ``Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions''. Phys. Rev. Lett. 128, 130605 (2022). https://doi.org/10.1103/PhysRevLett.128.130605 [95] Elisa Vallini and Silvia Pappalardi. ``Codes and data for: Long-time Freeness in the Kicked Top''. Zenodo (2026). https://doi.org/10.5281/zenodo.19698543 [96] S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen. ``Quantum signatures of chaos in a kicked top''. Nature 461, 768–771 (2009). https://doi.org/10.1038/nature08396 [97] Qian Wang and Marko Robnik. ``Multifractality in quasienergy space of coherent states as a signature of quantum chaos''. Entropy 23, 1347 (2021). https://doi.org/10.3390/e23101347 [98] Silvia Pappalardi, Angelo Russomanno, Bojan Žunkovič, Fernando Iemini, Alessandro Silva, and Rosario Fazio. ``Scrambling and entanglement spreading in long-range spin chains''. Phys. Rev. B 98, 134303 (2018). https://doi.org/10.1103/PhysRevB.98.134303 [99] Akshay Seshadri, Vaibhav Madhok, and Arul Lakshminarayan. ``Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos''. Phys. Rev. E 98, 052205 (2018). https://doi.org/10.1103/PhysRevE.98.052205 [100] Lukas M. Sieberer, Tobias Olsacher, Andreas Elben, Markus Heyl, Philipp Hauke, Fritz Haake, and Peter Zoller. ``Digital quantum simulation, trotter errors, and quantum chaos of the kicked top''. npj Quantum Information 5 (2019). https://doi.org/10.1038/s41534-019-0192-5 [101] Saúl Pilatowsky-Cameo, Jorge Chávez-Carlos, Miguel A. Bastarrachea-Magnani, Pavel Stránský, Sergio Lerma-Hernández, Lea F. Santos, and Jorge G. Hirsch. ``Positive quantum lyapunov exponents in experimental systems with a regular classical limit''. Phys. Rev. E 101, 010202 (2020). https://doi.org/10.1103/PhysRevE.101.010202 [102] Alessio Lerose and Silvia Pappalardi. ``Bridging entanglement dynamics and chaos in semiclassical systems''. Phys. Rev. A 102, 032404 (2020). https://doi.org/10.1103/PhysRevA.102.032404 [103] J. M. Deutsch. ``Quantum statistical mechanics in a closed system''. Physical Review A 43, 2046–2049 (1991). https://doi.org/10.1103/PhysRevA.43.2046 [104] Roland Speicher. ``Free probability theory''. Jahresbericht der Deutschen Mathematiker-Vereinigung 119 (2016). https://doi.org/10.1365/s13291-016-0150-5 [105] Xiang-Gen Xia. ``A simple introduction to free probability theory and its application to random matrices'' (2019). arXiv:1902.10763. arXiv:1902.10763 [106] James A Mingo and Roland Speicher. ``Free probability and random matrices''. Springer. (2017). https://doi.org/10.1007/978-1-4939-6942-5 [107] Neil Dowling, Jacopo De Nardis, Markus Heinrich, Xhek Turkeshi, and Silvia Pappalardi. ``Free independence and unitary design from random matrix product unitaries'' (2025). arXiv:2508.00051. arXiv:2508.00051 [108] Noah Linden, Sandu Popescu, Anthony J Short, and Andreas Winter. ``Quantum mechanical evolution towards thermal equilibrium''. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 79, 061103 (2009). https://doi.org/10.1103/PhysRevE.79.061103 [109] Roland Speicher. ``Lecture notes on "free probability theory"'' (2025). arXiv:1908.08125. arXiv:1908.08125 [110] Yichen Huang, Fernando G. S. L. Brandão, and Yong-Liang Zhang. ``Finite-size scaling of out-of-time-ordered correlators at late times''. Phys. Rev. Lett. 123, 010601 (2019). https://doi.org/10.1103/PhysRevLett.123.010601 [111] Felix Fritzsch and Tomaz Prosen. ``Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions''. Phys. Rev. E 103, 062133 (2021). https://doi.org/10.1103/PhysRevE.103.062133 [112] Ignacio García-Mata, Marcos Saraceno, Rodolfo A. Jalabert, Augusto J. Roncaglia, and Diego A. Wisniacki. ``Chaos signatures in the short and long time behavior of the out-of-time ordered correlator''. Phys. Rev. Lett. 121, 210601 (2018). https://doi.org/10.1103/PhysRevLett.121.210601 [113] Tomás Notenson, Ignacio García-Mata, Augusto J Roncaglia, and Diego A Wisniacki. ``Classical approach to equilibrium of out-of-time ordered correlators in mixed systems''. Physical Review E 107, 064207 (2023). https://doi.org/10.1103/PhysRevE.107.064207 [114] Bryce Kobrin, Zhenbin Yang, Gregory D. Kahanamoku-Meyer, Christopher T. Olund, Joel E. Moore, Douglas Stanford, and Norman Y. Yao. ``Many-body chaos in the sachdev-ye-kitaev model''. Phys. Rev. Lett. 126, 030602 (2021). https://doi.org/10.1103/PhysRevLett.126.030602 [115] Joseph Polchinski. ``Chaos in the black hole s-matrix'' (2015). arXiv:1505.08108. arXiv:1505.08108 [116] Felix Fritzsch and Pieter W. Claeys. ``Free probability in a minimal quantum circuit model'' (2025). arXiv:2506.11197. arXiv:2506.11197 [117] Shreya Vardhan and Jinzhao Wang. ``Free mutual information and higher-point otocs'' (2025). arXiv:2509.13406. arXiv:2509.13406 [118] Felix Fritzsch, Gabriel O. Alves, Michael A. Rampp, and Pieter W. Claeys. ``Free cumulants and full eigenstate thermalization from boundary scrambling'' (2025). arXiv:2509.08060. arXiv:2509.08060 [119] Viktor Jahnke, Pratik Nandy, Kuntal Pal, Hugo A. Camargo, and Keun-Young Kim. ``Free probability approach to spectral and operator statistics in rosenzweig-porter random matrix ensembles''. Journal of High Energy Physics 2025 (2025). https://doi.org/10.1007/jhep12(2025)002 [120] Jordan S Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, Phil Saad, Stephen H Shenker, Douglas Stanford, Alexandre Streicher, and Masaki Tezuka. ``Black holes and random matrices''. Journal of High Energy Physics 2017, 1–54 (2017). https://doi.org/10.1007/JHEP05(2017)118 [121] Luca V Delacretaz. ``Heavy operators and hydrodynamic tails''. SciPost Physics 9, 034 (2020). https://doi.org/10.21468/SciPostPhys.9.3.034 [122] Alexander Altland, Dmitry Bagrets, Pranjal Nayak, Julian Sonner, and Manuel Vielma. ``From operator statistics to wormholes''. Phys. Rev. Res. 3, 033259 (2021). https://doi.org/10.1103/PhysRevResearch.3.033259 [123] Takato Yoshimura, Samuel J Garratt, and J T Chalker. ``Operator dynamics in floquet many-body systems'' (2023). arXiv:2312.14234. https://doi.org/10.1103/PhysRevB.111.094316 arXiv:2312.14234 [124] Oscar Bouverot-Dupuis, Silvia Pappalardi, Jorge Kurchan, Anatoli Polkovnikov, and Laura Foini. ``Random matrix universality in dynamical correlation functions at late times'' (2024). arXiv:2407.12103. https://doi.org/10.21468/SciPostPhys.19.2.050 arXiv:2407.12103Cited by[1] Elisa Vallini, Laura Foini, and Silvia Pappalardi, "Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability", arXiv:2511.23217, (2025). [2] Luca Lumia, Emanuele Tirrito, Mario Collura, Fabian H. L. Essler, and Rosario Fazio, "Complexity of Quantum Trajectories", arXiv:2602.00232, (2026). [3] Felix Fritzsch, Gabriel O. Alves, Michael A. Rampp, and Pieter W. Claeys, "Free Cumulants and Full Eigenstate Thermalization from Boundary Scrambling", arXiv:2509.08060, (2025). [4] Felix Fritzsch and Pieter W. Claeys, "Free Probability in a Minimal Quantum Circuit Model", arXiv:2506.11197, (2025). [5] Gabriel O. Alves, Felix Fritzsch, and Pieter W. Claeys, "Probes of Full Eigenstate Thermalization in Ergodicity-Breaking Quantum Circuits", Quantum 9, 1949 (2025). [6] Jiaozi Wang, Ruchira Mishra, Tian-Hua Yang, Luca V. Delacrétaz, and Silvia Pappalardi, "Eigenstate Thermalization Hypothesis Correlations via Nonlinear Hydrodynamics", Physical Review Letters 136 13, 130402 (2026). [7] Jerónimo Duarte, Ignacio García-Mata, and Diego A. Wisniacki, "Ruelle-Pollicott decay of out-of-time-order correlators in many-body systems", Physical Review E 113 2, 024209 (2026). [8] Hugo A. Camargo, Yichao Fu, Viktor Jahnke, Keun-Young Kim, and Kuntal Pal, "Quantum signatures of chaos from free probability", Journal of High Energy Physics 2025 10, 138 (2025). [9] Neil Dowling, Jacopo De Nardis, Markus Heinrich, Xhek Turkeshi, and Silvia Pappalardi, "Free Independence and Unitary Design from Random Matrix Product Unitaries", arXiv:2508.00051, (2025). [10] Taishi Kawamoto, "A strategy for proving the strong eigenstate thermalization hypothesis: chaotic systems and holography", Journal of High Energy Physics 2025 1, 95 (2025). [11] Piotr Sierant, Xhek Turkeshi, and Poetri Sonya Tarabunga, "Theory of the Matchgate Commutant", arXiv:2603.12392, (2026). [12] Viktor Jahnke, Pratik Nandy, Kuntal Pal, Hugo A. Camargo, and Keun-Young Kim, "Free probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles", Journal of High Energy Physics 2025 12, 2 (2025). [13] Shreya Vardhan and Jinzhao Wang, "Free mutual information and higher-point OTOCs", arXiv:2509.13406, (2025). [14] Neil Dowling and Silvia Pappalardi, "Page Curve for Local-Operator Entanglement from Free Probability", arXiv:2605.02995, (2026). [15] Pallab Basu, Suman Das, and Pratik Nandy, "Complexity of quadratic quantum chaos", Journal of High Energy Physics 2026 4, 81 (2026). [16] Nikita Kolganov and Dmitrii A. Trunin, "Streamlined Krylov construction and classification of ergodic Floquet systems", Physical Review E 111 5, L052202 (2025). [17] Tabea Herrmann, Felix Fritzsch, and Arnd Bäcker, "Timescales for Deep and Full Thermalization", arXiv:2604.27749, (2026). [18] Keun-Young Kim and Kuntal Pal, "Density of states of quantum systems from free probability theory: a brief overview", arXiv:2512.03850, (2025). [19] Merlin Füllgraf, Jochen Gemmer, and Jiaozi Wang, "Scaling of free cumulants in closed system-bath setups", arXiv:2511.11333, (2025). [20] Merlin Füllgraf, Jochen Gemmer, and Jiaozi Wang, "Scaling of free cumulants in closed system-bath setups", SciPost Physics Core 9 2, 025 (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-08 13:43:04). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-08 13:43:03: Could not fetch cited-by data for 10.22331/q-2026-06-08-2129 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractRecent work highlighted the importance of higher-order correlations in quantum dynamics for a deeper understanding of quantum chaos and thermalization. The full Eigenstate Thermalization Hypothesis, the framework encompassing correlations, can be formalized using the language of Free Probability theory. In this context, chaotic dynamics at long times are proposed to lead to free independence or "freeness" of observables. In this work, we investigate these issues in a paradigmatic semiclassical model – the kicked top – which exhibits a transition from integrability to chaos. Despite its simplicity, we identify several non-trivial features. By numerically studying 2n-point out-of-time-order correlators, we show that in the fully chaotic regime, long-time freeness is reached exponentially fast. These considerations lead us to introduce a large deviation theory for freeness that enables us to define and analyze the associated time scale. The numerical results confirm the existence of a hierarchy of different time scales, indicating a multifractal approach to freeness in this model. Our findings provide novel insights into the long-time behavior of chaotic dynamics and may have broader implications for the study of many-body quantum dynamics.Featured image: A multifractal route to Freeness.Popular summaryChaos is often associated with unpredictability and randomness. In quantum systems, however, understanding how randomness emerges in deterministic dynamics remains a subtle problem. To characterize this regime, we employ concepts from Free Probability theory, a mathematical framework that extends ordinary probability theory to non-commuting quantities. In this setting, chaotic dynamics is expected to drive observables towards freeness, a generalized form of statistical independence that can be probed through higher-order correlation functions. In this work, we investigate higher-order correlations in the kicked top, a prototypical model of quantum chaos. Our numerical results show that, in the chaotic regime, the kicked top approaches freeness exponentially fast at long times. Moreover, we find a hierarchy of timescales for which increasingly higher-order correlations require progressively longer times to show freeness. This points to a multifractal route to randomness. These findings provide new insight into the long-time behavior of quantum chaotic systems and suggest that freeness may offer a powerful framework for understanding thermalization in a broad range of many-body quantum systems.► BibTeX data@article{Vallini2026longtimefreenessin, doi = {10.22331/q-2026-06-08-2129}, url = {https://doi.org/10.22331/q-2026-06-08-2129}, title = {Long-time {F}reeness in the {K}icked {T}op}, author = {Vallini, Elisa and Pappalardi, Silvia}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2129}, month = jun, year = {2026} }► References [1] Philippe Delsarte, Jean-Marie Goethals, and Johan Jacob Seidel. ``Spherical codes and designs''. In Geometry and Combinatorics. Pages 68–93. Elsevier (1991). https://doi.org/10.1007/BF03187604 [2] Christoph Dankert. ``Efficient simulation of random quantum states and operators'' (2005). arXiv:quant-ph/0512217. arXiv:quant-ph/0512217 [3] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80, 012304 (2009). https://doi.org/10.1103/PhysRevA.80.012304 [4] Jordan S Cotler, Daniel K Mark, Hsin-Yuan Huang, Felipe Hernandez, Joonhee Choi, Adam L Shaw, Manuel Endres, and Soonwon Choi. ``Emergent quantum state designs from individual many-body wave functions''. PRX Quantum 4, 010311 (2023). https://doi.org/10.1103/PRXQuantum.4.010311 [5] Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Ran Finkelstein, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Anant Kale, Hannes Pichler, Fernando G. S. L. Brandão, Soonwon Choi, and Manuel Endres. ``Preparing random states and benchmarking with many-body quantum chaos''. Nature 613, 468–473 (2023). https://doi.org/10.1038/s41586-022-05442-1 [6] Wen Wei Ho and Soonwon Choi. ``Exact emergent quantum state designs from quantum chaotic dynamics''. Phys. Rev. Lett. 128, 060601 (2022). https://doi.org/10.1103/PhysRevLett.128.060601 [7] Pieter W Claeys and Austen Lamacraft. ``Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics''. Quantum 6, 738 (2022). https://doi.org/10.22331/q-2022-06-15-738 [8] Matteo Ippoliti and Wen Wei Ho. ``Solvable model of deep thermalization with distinct design times''. Quantum 6, 886 (2022). https://doi.org/10.22331/q-2022-12-29-886 [9] Maxime Lucas, Lorenzo Piroli, Jacopo De Nardis, and Andrea De Luca. ``Generalized deep thermalization for free fermions''. Phys. Rev. A 107, 032215 (2023). https://doi.org/10.1103/PhysRevA.107.032215 [10] Tanmay Bhore, Jean-Yves Desaules, and Zlatko Papić. ``Deep thermalization in constrained quantum systems''. Phys. Rev. B 108, 104317 (2023). https://doi.org/10.1103/PhysRevB.108.104317 [11] Max McGinley and Michele Fava. ``Shadow tomography from emergent state designs in analog quantum simulators''. Phys. Rev. Lett. 131, 160601 (2023). https://doi.org/10.1103/PhysRevLett.131.160601 [12] Saúl Pilatowsky-Cameo, Ceren B. Dag, Wen Wei Ho, and Soonwon Choi. ``Complete hilbert-space ergodicity in quantum dynamics of generalized fibonacci drives''. Phys. Rev. Lett. 131, 250401 (2023). https://doi.org/10.1103/PhysRevLett.131.250401 [13] Daniel K Mark, Federica Surace, Andreas Elben, Adam L Shaw, Joonhee Choi, Gil Refael, Manuel Endres, and Soonwon Choi. ``A maximum entropy principle in deep thermalization and in hilbert-space ergodicity'' (2024). arXiv:2403.11970. https://doi.org/10.1103/PhysRevX.14.041051 arXiv:2403.11970 [14] Anatoly Dymarsky, Nima Lashkari, and Hong Liu. ``Subsystem eigenstate thermalization hypothesis''. Phys. Rev. E 97, 012140 (2018). https://doi.org/10.1103/PhysRevE.97.012140 [15] Yichen Huang. ``Universal entanglement of mid-spectrum eigenstates of chaotic local hamiltonians''. Nuclear Physics B 966, 115373 (2021). https://doi.org/10.1016/j.nuclphysb.2021.115373 [16] Tsung-Cheng Lu and Tarun Grover. ``Renyi entropy of chaotic eigenstates''. Phys. Rev. E 99, 032111 (2019). https://doi.org/10.1103/PhysRevE.99.032111 [17] Chaitanya Murthy and Mark Srednicki. ``Structure of chaotic eigenstates and their entanglement entropy''. Physical Review E 100, 022131 (2019). https://doi.org/10.1103/PhysRevE.100.022131 [18] Zhengyan Darius Shi, Shreya Vardhan, and Hong Liu. ``Local dynamics and the structure of chaotic eigenstates''. Physical Review B 108, 224305 (2023). https://doi.org/10.1103/PhysRevB.108.224305 [19] Dominik Hahn, David J. Luitz, and J. T. Chalker. ``Eigenstate correlations, the eigenstate thermalization hypothesis, and quantum information dynamics in chaotic many-body quantum systems''. Phys. Rev. X 14, 031029 (2024). https://doi.org/10.1103/PhysRevX.14.031029 [20] Siddharth Jindal and Pavan Hosur. ``Generalized free cumulants for quantum chaotic systems'' (2024). arXiv:2401.13829. https://doi.org/10.1007/JHEP09(2024)066 arXiv:2401.13829 [21] Benjamin Doyon and Jason Myers. ``Fluctuations in ballistic transport from euler hydrodynamics''.

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