- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Real-time quantum dynamics on the fuzzy sphere: Chaos and entanglement

S. Kürkcüoğlu* and B. Özcan†

  • *Contact author: kseckin@metu.edu.tr
  • †Contact author: berk@metu.edu.tr
PDF

Phys. Rev. D 114, 065014 – Published 11 September, 2026

DOI: https://doi.org/10.1103/194z-9gqd

Abstract

We study the real-time quantum dynamics of a matrix model consisting two bosonic fields on the fuzzy sphere SF2 using the Gaussian state approximation. Starting from the Hamiltonian formulation and using Wick’s theorem, we derive a closed set of coupled nonlinear differential equations governing the time evolution of the one- and two-point correlation functions. The thermal equation of state is found by maximizing the von Neumann entropy over Gaussian states and solving the algebraic self-consistency equation(s) leading to a complete determination of the symplectic spectrum of the covariance matrix. We identify near-thermal initial conditions and use them to solve the equations of motion and employ our findings to probe chaos by calculating the largest Lyapunov exponent at various temperatures. Our results demonstrate that the latter tends to zero at a finite temperature indicating that the quantum dynamics respect the Maldacena-Shenker-Stanford bound across all temperatures, while approaching toward the classically chaotic regime at high temperatures. Finally, we examine the entanglement dynamics of the model in real-time by considering a sequence of bipartitions of the Hilbert space and computing the entanglement entropy and clearly exhibit the fast scrambling features that emerge in due detail.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. Y. Sekino and L. Susskind, Fast scramblers, J. High Energy Phys. 10 (2008) 065.
  2. C. Asplund, D. Berenstein, and D. Trancanelli, Evidence for fast thermalization in the plane-wave matrix model, Phys. Rev. Lett. 107, 171602 (2011).
  3. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  4. G. Gur-Ari, M. Hanada, and S. H. Shenker, Chaos in classical D0-Brane mechanics, J. High Energy Phys. 02 (2016) 091.
  5. D. Berenstein and D. Kawai, Smallest matrix black hole model in the classical limit, Phys. Rev. D 95, 106004 (2017).
  6. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  7. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).
  8. S. Aoki, M. Hanada, and N. Iizuka, Quantum black hole formation in the BFSS matrix model, J. High Energy Phys. 07 (2015) 029.
  9. Y. Asano, D. Kawai, and K. Yoshida, Chaos in the BMN matrix model, J. High Energy Phys. 06 (2015) 191.
  10. E. Berkowitz, E. Rinaldi, M. Hanada, G. Ishiki, S. Shimasaki, and P. Vranas, Precision lattice test of the gauge/gravity duality at large-N, Phys. Rev. D 94, 094501 (2016).
  11. E. Berkowitz, M. Hanada, and J. Maltz, Chaos in matrix models and black hole evaporation, Phys. Rev. D 94, 126009 (2016).
  12. C. T. Asplund and D. Berenstein, Entanglement entropy converges to classical entropy around periodic orbits, Ann. Phys. (Amsterdam) 366, 113 (2016).
  13. P. Buividovich, M. Hanada, and A. Schäfer, Real-time dynamics of matrix quantum mechanics beyond the classical approximation, EPJ Web Conf. 175, 08006 (2018).
  14. P. V. Buividovich, M. Hanada, and A. Schäfer, Quantum chaos, thermalization, and entanglement generation in real-time simulations of the banks Fischler Shenker Susskind matrix model, Phys. Rev. D 99, 046011 (2019).
  15. Ü. H. Coşkun, S. Kürkçüoğlu, G. C. Toga, and G. Ünal, Chaos from equivariant fields on fuzzy S4, J. High Energy Phys. 12 (2018) 015.
  16. K. Başkan, S. Kürkçüoğlu, O. Oktay, and C. Taşc𝚤, Chaos from massive deformations of Yang-Mills matrix models, J. High Energy Phys. 10 (2020) 003.
  17. K. Başkan, S. Kürkçüoğlu, and C. Taşc𝚤, Chaotic dynamics of the mass deformed ABJM model, Phys. Rev. D 107, 066006 (2023).
  18. S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105, 151602 (2010).
  19. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55, 5112 (1997).
  20. N. Iizuka, D. Kabat, S. Roy, and D. Sarkar, Black hole formation at the correspondence point, Phys. Rev. D 87, 126010 (2013).
  21. K. Başkan and S. Kürkçüoğlu, Chaos in the SU(2) Yang-Mills Chern-Simons matrix model, Phys. Rev. D 104, 066006 (2021).
  22. P. Amore, L. A. Pando Zayas, J. F. Pedraza, N. Quiroz, and C. A. Terrero-Escalante, Fuzzy spheres in stringy matrix models: Quantifying chaos in a mixed phase space, J. High Energy Phys. 06 (2025) 031.
  23. N. Kawahara, J. Nishimura, and S. Takeuchi, Phase structure of matrix quantum mechanics at finite temperature, J. High Energy Phys. 10 (2007) 097.
  24. R. Delgadillo-Blando, D. O’Connor, and B. Ydri, Geometry in transition: A model of emergent geometry, Phys. Rev. Lett. 100, 201601 (2008).
  25. R. Delgadillo-Blando, D. O’Connor, and B. Ydri, Matrix models, gauge theory and emergent geometry, J. High Energy Phys. 05 (2009) 049.
  26. V. G. Filev and D. O’Connor, The BFSS model on the lattice, J. High Energy Phys. 05 (2016) 167.
  27. D. O’Connor and V. G. Filev, Membrane Matrix models and non-perturbative checks of gauge/gravity duality, Proc. Sci. CORFU2015 (2016) 111 [arXiv:1605.01611].
  28. Y. Asano, V. G. Filev, S. Kováčik, and D. O’Connor, The non-perturbative phase diagram of the BMN matrix model, J. High Energy Phys. 07 (2018) 152.
  29. B. Ydri, Review of M(atrix)-Theory, Type IIB matrix model and matrix string theory, arXiv:1708.00734.
  30. B. Ydri, Lectures on matrix field theory, Lect. Notes Phys. 929, 1 (2017).
  31. D. O’Connor and C. Saemann, Fuzzy scalar field theory as a multitrace matrix model, J. High Energy Phys. 08 (2007) 066.
  32. V. P. Nair, A. P. Polychronakos, and J. Tekel, Fuzzy spaces and new random matrix ensembles, Phys. Rev. D 85, 045021 (2012).
  33. E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar diagrams, Commun. Math. Phys. 59, 35 (1978).
  34. J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739, 3 (2004).
  35. J. Broeckhove, L. Lathouwers, and P. van Leuven, The hamilton formalism of gaussian wave-packet dynamics, J. Mol. Struct. 199, 245 (1989).
  36. E. J. Heller, Time‐dependent approach to semiclassical dynamics, J. Chem. Phys. 62, 1544 (1975).
  37. N. Mukunda, R. Simon, and G. Sudarshan, Gaussian pure states in quantum mechanics and the symplectic group, Phys. Rev. A 37, 3028 (1988).
  38. R. Simon, N. Mukunda, and B. Dutta, Quantum-noise matrix for multimode systems: U(N) invariance, squeezing, and normal forms, Phys. Rev. A 49, 1567 (1994).
  39. R. Bertlmann and N. Friis, Modern Quantum Theory (Oxford University Press, New York, 2023).
  40. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  41. J. Madore, The Fuzzy sphere, Classical Quantum Gravity 9, 69 (1992).
  42. A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya, Lectures on Fuzzy and Fuzzy SUSY Physics (World-Scientific, Singapore, 2007).
  43. J. Berges, S. Floerchinger, and R. Venugopalan, Dynamics of entanglement in expanding quantum fields, J. High Energy Phys. 04 (2018) 145.
  44. L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34, 373 (1986).
  45. R. D. Sorkin, Expressing entropy globally in terms of (4D) field-correlations, J. Phys. Conf. Ser. 484, 012004 (2014).
  46. M. Saravani, R. D. Sorkin, and Y. K. Yazdi, Spacetime entanglement entropy in 1+1 dimensions, Classical Quantum Gravity 31, 214006 (2014).
  47. J. Y. L. Jones and Y. K. Yazdi, Spectral spacetime entropy for quasifree theories, arXiv:2602.16782.
  48. J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math. 58, 141 (1936).
  49. D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, Strings in flat space and pp waves from N=4 superYang-Mills, J. High Energy Phys. 04 (2002) 013.
  50. K. Dasgupta, M. M. Sheikh-Jabbari, and M. Van Raamsdonk, Matrix perturbation theory for M theory on a pp wave, J. High Energy Phys. 05 (2002) 056.
  51. K. Dasgupta, M. M. Sheikh-Jabbari, and M. Van Raamsdonk, Protected multiplets of M theory on a plane wave, J. High Energy Phys. 09 (2002) 021.
  52. S. Kürkcüoğlu and B. Özcan (to be published).
  53. S. Vaidya, Perturbative dynamics on the fuzzy S**2 and RP**2, Phys. Lett. B 512, 403 (2001).
  54. C. S. Chu, J. Madore, and H. Steinacker, Scaling limits of the fuzzy sphere at one loop, J. High Energy Phys. 08 (2001) 038.
  55. B. P. Dolan, D. O’Connor, and P. Presnajder, Matrix phi**4 models on the fuzzy sphere and their continuum limits, J. High Energy Phys. 03 (2002) 013.
  56. S. Vaidya and B. Ydri, On the origin of the UV-IR mixing in noncommutative matrix geometry, Nucl. Phys. B671, 401 (2003).
  57. S. Minwalla, M. Van Raamsdonk, and N. Seiberg, Noncommutative perturbative dynamics, J. High Energy Phys. 02 (2000) 020.
  58. J. D. Bekenstein, A universal upper bound on the entropy to energy ratio for bounded systems, Phys. Rev. D 23, 287 (1981).
  59. D. N. Page, The Bekenstein bound, in Jacob Bekenstein (World Scientific, Singapore, 2019).
  60. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71, 666 (1993).
  61. H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42, 504007 (2009).
  62. M. Huerta, Numerical determination of the entanglement entropy for free fields in the cylinder, Phys. Lett. B 710, 691 (2012).
  63. T. Nishioka, Entanglement entropy: Holography and renormalization group, Rev. Mod. Phys. 90, 035007 (2018).
  64. E. Witten, Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit?, in Dialogues Between Physics and Mathematics (Springer, Cham, 2022).
  65. E. Witten, A mini-introduction to information theory, Riv. Nuovo Cimento 43, 187 (2020).
  66. T. Kunihiro, B. Muller, A. Ohnishi, and A. Schafer, Towards a theory of entropy production in the little and big bang, Prog. Theor. Phys. 121, 555 (2009).
  67. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  68. V. Balasubramanian, P. Caputa, J. M. Magán, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  69. K. B. Huh, H. S. Jeong, L. A. Pando Zayas, and J. F. Pedraza, Krylov complexity in mixed phase space, Phys. Rev. D 111, L121902 (2025).
  70. 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).
  71. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific Publishing Company, Singapore, 1988).

Outline

Information

Phys. Rev. D 114, 065014– Published 11 September, 2026
  • Received 29 May 2026
  • Accepted 6 August 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