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Spectral and Krylov complexity in billiard systems

Hugo A. Camargo1,*, Viktor Jahnke1,†, Hyun-Sik Jeong2,3,‡, Keun-Young Kim1,4,§, and Mitsuhiro Nishida5,∥

  • 1Department of Physics and Photon Science, Gwangju Institute of Science and Technology, 123 Cheomdan-gwagiro, Gwangju 61005, Korea
  • 2Instituto de Física Teórica UAM/CSIC, Calle Nicolás Cabrera 13-15, 28049 Madrid, Spain
  • 3Departamento de Física Teórica, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
  • 4Research Center for Photon Science Technology, Gwangju Institute of Science and Technology, 123 Cheomdan-gwagiro, Gwangju 61005, Korea
  • 5Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea

  • *hugo.camargo@gist.ac.kr
  • †viktorjahnke@gist.ac.kr
  • ‡hyunsik.jeong@uam.es
  • §fortoe@gist.ac.kr
  • ∥nishida124@postech.ac.kr
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Phys. Rev. D 109, 046017 – Published 27 February, 2024

DOI: https://doi.org/10.1103/PhysRevD.109.046017

Abstract

In this work, we investigate spectral complexity and Krylov complexity in quantum billiard systems at finite temperature. We study both circle and stadium billiards as paradigmatic examples of integrable and nonintegrable quantum-mechanical systems, respectively. We show that the saturation value and timescale of spectral complexity may be used to probe the nonintegrability of the system since we find that when computed for the circle billiard, it saturates at a later timescale compared to the stadium billiards. This observation is verified for different temperatures. Furthermore, we study the Krylov complexity of the position operator and its associated Lanczos coefficients at finite temperature using the Wightman inner product. We find that the growth rate of the Lanczos coefficients saturates the conjectured universal bound at low temperatures. Additionally, we also find that even a subset of the Lanczos coefficients can potentially serve as an indicator of integrability, as they demonstrate erratic behavior specifically in the circle billiard case, in contrast to the stadium billiard. Finally, we also study Krylov entropy and verify its early-time logarithmic relation with Krylov complexity in both types of billiard systems.

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References (78)

  1. A. Einstein, Zum quantensatz von sommerfeld und epstein, Dtsch. Phys. Ges. Verh. 19, 82 (1917).
  2. M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. (N.Y.) 12, 343 (1971).
  3. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
  4. M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. A 356, 375 (1977).
  5. M. V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. A 400, 229 (1985).
  6. E. B. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69, 1477 (1992).
  7. J. Bolte, G. Steil, and F. Steiner, Arithmetical chaos and violation of universality in energy level statistics, Phys. Rev. Lett. 69, 2188 (1992).
  8. A. Prakash, J. H. Pixley, and M. Kulkarni, Universal spectral form factor for many-body localization, Phys. Rev. Res. 3, L012019 (2021).
  9. M. Winer, S.-K. Jian, and B. Swingle, An exponential ramp in the quadratic Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 125, 250602 (2020).
  10. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, J. Exp. Theor. Phys. 28, 6 (1969).
  11. S. Xu and B. Swingle, Scrambling dynamics and out-of-time ordered correlators in quantum many-body systems: A tutorial, arXiv:2202.07060.
  12. K. Richter, J. D. Urbina, and S. Tomsovic, Semiclassical roots of universality in many-body quantum chaos, J. Phys. A 55, 453001 (2022).
  13. 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).
  14. T. Xu, T. Scaffidi, and X. Cao, Does scrambling equal chaos?, Phys. Rev. Lett. 124, 140602 (2020).
  15. 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.
  16. N. Dowling, P. Kos, and K. Modi, Scrambling is necessary but not sufficient for chaos, Phys. Rev. Lett. 131, 180403 (2023).
  17. Y. Huang, F. G. S. L. Brandão, and Y.-L. Zhang, Finite-size scaling of out-of-time-ordered correlators at late times, Phys. Rev. Lett. 123, 010601 (2019).
  18. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  19. A. Avdoshkin, A. Dymarsky, and M. Smolkin, Krylov complexity in quantum field theory, and beyond, arXiv:2212.14429.
  20. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  21. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
  22. H. A. Camargo, V. Jahnke, K.-Y. Kim, and M. Nishida, Krylov complexity in free and interacting scalar field theories with bounded power spectrum, J. High Energy Phys. 05 (2023) 226.
  23. A. Kundu, V. Malvimat, and R. Sinha, State dependence of Krylov complexity in 2d CFTs, J. High Energy Phys. 09 (2023) 011.
  24. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  25. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  26. L. V. Iliesiu, M. Mezei, and G. Sárosi, The volume of the black hole interior at late times, J. High Energy Phys. 07 (2022) 073.
  27. J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, J. High Energy Phys. 10 (2019) 264.
  28. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  29. A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102, 085137 (2020).
  30. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator complexity: A journey to the edge of Krylov space, J. High Energy Phys. 06 (2021) 062.
  31. X. Cao, A statistical mechanism for operator growth, J. Phys. A 54, 144001 (2021).
  32. J. Kim, J. Murugan, J. Olle, and D. Rosa, Operator delocalization in quantum networks, Phys. Rev. A 105, L010201 (2022).
  33. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov localization and suppression of complexity, J. High Energy Phys. 03 (2022) 211.
  34. F. B. Trigueros and C.-J. Lin, Krylov complexity of many-body localization: Operator localization in Krylov basis, SciPost Phys. 13, 037 (2022).
  35. Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105, 062210 (2022).
  36. R. Heveling, J. Wang, and J. Gemmer, Numerically probing the universal operator growth hypothesis, Phys. Rev. E 106, 014152 (2022).
  37. B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Krylov complexity in saddle-dominated scrambling, J. High Energy Phys. 05 (2022) 174.
  38. P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106, 195125 (2022).
  39. W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B984, 115948 (2022).
  40. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, J. High Energy Phys. 07 (2022) 151.
  41. S. He, P. H. C. Lau, Z.-Y. Xian, and L. Zhao, Quantum chaos, scrambling and operator growth in TT¯ deformed SYK models, J. High Energy Phys. 12 (2022) 070.
  42. N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Ultimate physical limits to the growth of operator complexity, Commun. Phys. 5, 207 (2022).
  43. M. Alishahiha, On quantum complexity, Phys. Lett. B 842, 137979 (2023).
  44. M. Alishahiha and S. Banerjee, A universal approach to Krylov state and operator complexities, SciPost Phys. 15, 080 (2023).
  45. A. Bhattacharyya, D. Ghosh, and P. Nandi, Operator growth and Krylov complexity in Bose-Hubbard model, J. High Energy Phys. 12 (2023) 112.
  46. J. M. Magán and J. Simón, On operator growth and emergent Poincaré symmetries, J. High Energy Phys. 05 (2020) 071.
  47. S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, J. High Energy Phys. 03 (2021) 014.
  48. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  49. P. Caputa and S. Datta, Operator growth in 2d CFT, J. High Energy Phys. 12 (2021) 188.
  50. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  51. D. Patramanis, Probing the entanglement of operator growth, Prog. Theor. Exp. Phys. 2022, 063A01 (2022).
  52. D. Patramanis and W. Sybesma, Krylov complexity in a natural basis for the Schrödinger algebra, arXiv:2306.03133.
  53. A. Chattopadhyay, A. Mitra, and H. J. R. van Zyl, Spread complexity as classical dilaton solutions, Phys. Rev. D 108, 025013 (2023).
  54. N. Iizuka and M. Nishida, Krylov complexity in the IP matrix model, J. High Energy Phys. 11 (2023) 065.
  55. K. Pal, K. Pal, A. Gill, and T. Sarkar, Time evolution of spread complexity and statistics of work done in quantum quenches, Phys. Rev. B 108, 104311 (2023).
  56. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, J. High Energy Phys. 12 (2022) 081.
  57. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023).
  58. B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, An operator growth hypothesis for open quantum systems, J. High Energy Phys. 03 (2023) 054.
  59. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, On Krylov complexity in open systems: An approach via bi-Lanczos algorithm, J. High Energy Phys. 12 (2023) 066.
  60. L. Susskind, Entanglement is not enough, Fortschr. Phys. 64, 49 (2016).
  61. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016).
  62. V. Balasubramanian, M. Decross, A. Kar, and O. Parrikar, Quantum complexity of time evolution with chaotic Hamiltonians, J. High Energy Phys. 01 (2020) 134.
  63. V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li, and O. Parrikar, Complexity growth in integrable and chaotic models, J. High Energy Phys. 07 (2021) 011.
  64. J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Y. Halpern, Linear growth of quantum circuit complexity, Nat. Phys. 18, 528 (2022).
  65. Y. G. Sinai, Dynamical systems with elastic reflections, Russ. Math. Surv. 25, 137 (1970).
  66. L. A. Bunimovich, On ergodic properties of certain billiards, Funct. Anal. Appl. 8, 254 (1975).
  67. L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65, 295 (1979).
  68. L. A. Bunimovich, Y. G. Sinai, and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russ. Math. Surv. 46, 47 (1991).
  69. G. Benettin, Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy, Phys. Rev. A 17, 773 (1978).
  70. K. Hashimoto, K. Murata, N. Tanahashi, and R. Watanabe, Krylov complexity and chaos in quantum mechanics, J. High Energy Phys. 11 (2023) 040.
  71. K. Hashimoto, K. Murata, and R. Yoshii, Out-of-time-order correlators in quantum mechanics, J. High Energy Phys. 10 (2017) 138.
  72. G. Benettin and J. M. Strelcyn, Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy, Phys. Rev. A 17, 773 (1978).
  73. S. W. McDonald and A. N. Kaufman, Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories, Phys. Rev. Lett. 42, 1189 (1979).
  74. G. Cassati, F. Valz-Gris, and I. Guarnieri, On the connection between quantization of nonintegrable systems and statisctical theory of spectra, Lett. Nuovo Cimento 28, 279 (1980).
  75. M. C. Bañuls, J. I. Cirac, and M. B. Hastings, Strong and weak thermalization of infinite nonintegrable quantum systems, Phys. Rev. Lett. 106 (2011).
  76. T. Nosaka, D. Rosa, and J. Yoon, The Thouless time for mass-deformed SYK, J. High Energy Phys. 09 (2018) 041.
  77. B.-n. Du and M.-x. Huang, Krylov complexity in Calabi-Yau quantum mechanics, Int. J. Mod. Phys. A 38, 2350126 (2023).
  78. V. S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics (Springer, Berlin, Heidelberg, Germany, 1994).

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Phys. Rev. D 109, 046017– Published 27 February, 2024
  • Received 31 October 2023
  • Accepted 15 January 2024
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