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  • Open Access

Sticky eigenstates in systems with sharply divided phase space

Hua Yan*

  • Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia

  • *Contact author: yanhua@ustc.edu.cn
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Phys. Rev. E 114, 034222 – Published 25 September, 2026

DOI: https://doi.org/10.1103/dwhh-42vk

Abstract

We investigate mixed eigenstates in systems with sharply divided phase space, using different piecewise-linear maps whose regular-chaotic boundaries are formed by marginally unstable periodic orbits (MUPOs) or by quasiperiodic orbits. With the overlap index and the entropy localization length, we classify mixed eigenstates and show that the contribution from dynamical tunneling scales as ∼ℏexp(−b/ℏ), with b>0 associated with the relative size of the regular region. The dominant fraction of states that remain sticky to the boundaries, referred to as sticky eigenstates, scales approximately as ℏ1/2 in the MUPO case and oscillates around this algebraic behavior in the quasiperiodic case. This behavior generalizes established predictions for hierarchical states in Kolmogorov-Arnold-Moser (KAM) systems, which scale as ℏ1−1/γ, with γ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative recurrence-time distributions t−γ. For the piecewise-linear maps studied here, γ=2. These results reveal a clear quantum signature of classical stickiness in non-KAM systems.

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References (69)

  1. I. Percival, Regular and irregular spectra, J. Phys. B 6, L229 (1973).
  2. M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A: Math. Gen. 10, 2083 (1977).
  3. M. Robnik, Topics in quantum chaos of generic systems, Nonlinear Phenom. Complex Syst. 1, 1 (1998).
  4. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer Science & Business Media, Cham, 2013), Vol. 1.
  5. F. Haake, Quantum signatures of chaos, in Quantum Coherence in Mesoscopic Systems, edited by B. Kramer (Springer, Berlin, 1991), pp. 583–595.
  6. H.-J. Stöckmann, Quantum Chaos An Introduction (Cambridge University Press, Cambridge, UK, 2007).
  7. M. V. Berry and M. Robnik, Semiclassical level spacings when regular and chaotic orbits coexist, J. Phys. A: Math. Gen. 17, 2413 (1984).
  8. T. Prosen and M. Robnik, Semiclassical energy level statistics in the transition region between integrability and chaos: Transition from Brody-like to Berry-Robnik behaviour, J. Phys. A: Math. Gen. 27, 8059 (1994).
  9. H. Yan, Spacing ratios in mixed-type systems, Phys. Rev. E 111, 054213 (2025).
  10. O. Bohigas, S. Tomsovic, and D. Ullmo, Manifestations of classical phase space structures in quantum mechanics, Phys. Rep. 223, 43 (1993).
  11. S. Tomsovic and D. Ullmo, Chaos-assisted tunneling, Phys. Rev. E 50, 145 (1994).
  12. S. Keshavamurthy and P. Schlagheck, Dynamical Tunneling: Theory and Experiment (CRC Press, Boca Raton, FL, 2011).
  13. F. Izrailev, Quantum localization and statistics of quasienergy spectrum in a classically chaotic system, Phys. Lett. A 134, 13 (1988).
  14. B. Chirikov, F. Izrailev, and D. Shepelyansky, Quantum chaos: Localization vs ergodicity, Physica D 33, 77 (1988).
  15. B. Batistić and M. Robnik, Quantum localization of chaotic eigenstates and the level spacing distribution, Phys. Rev. E 88, 052913 (2013).
  16. V. A. Podolskiy and E. E. Narimanov, Semiclassical description of chaos-assisted tunneling, Phys. Rev. Lett. 91, 263601 (2003).
  17. A. Bäcker, R. Ketzmerick, S. Löck, and L. Schilling, Regular-to-chaotic tunneling rates using a fictitious integrable system, Phys. Rev. Lett. 100, 104101 (2008).
  18. O. Brodier, P. Schlagheck, and D. Ullmo, Resonance-assisted tunneling in near-integrable systems, Phys. Rev. Lett. 87, 064101 (2001).
  19. E. G. Altmann, J. S. E. Portela, and T. Tél, Leaking chaotic systems, Rev. Mod. Phys. 85, 869 (2013).
  20. J. D. Meiss and E. Ott, Markov-tree model of intrinsic transport in Hamiltonian systems, Phys. Rev. Lett. 55, 2741 (1985).
  21. G. Cristadoro and R. Ketzmerick, Universality of algebraic decays in Hamiltonian systems, Phys. Rev. Lett. 100, 184101 (2008).
  22. R. Venegeroles, Universality of algebraic laws in Hamiltonian systems, Phys. Rev. Lett. 102, 064101 (2009).
  23. R. Ketzmerick, Fractal conductance fluctuations in generic chaotic cavities, Phys. Rev. B 54, 10841 (1996).
  24. Y. Alhassid, The statistical theory of quantum dots, Rev. Mod. Phys. 72, 895 (2000).
  25. B. Huckestein, R. Ketzmerick, and C. H. Lewenkopf, Quantum transport through ballistic cavities: Soft vs hard quantum chaos, Phys. Rev. Lett. 84, 5504 (2000).
  26. R. Ketzmerick, L. Hufnagel, F. Steinbach, and M. Weiss, New class of eigenstates in generic Hamiltonian systems, Phys. Rev. Lett. 85, 1214 (2000).
  27. L. A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11, 802 (2001).
  28. E. G. Altmann, A. E. Motter, and H. Kantz, Stickiness in mushroom billiards, Chaos 15, 033105 (2005).
  29. A. Bäcker, R. Ketzmerick, S. Löck, M. Robnik, G. Vidmar, R. Höhmann, U. Kuhl, and H.-J. Stöckmann, Dynamical tunneling in mushroom billiards, Phys. Rev. Lett. 100, 174103 (2008).
  30. A. H. Barnett and T. Betcke, Quantum mushroom billiards, Chaos 17, 043125 (2007).
  31. C. P. Dettmann and O. Georgiou, Open mushrooms: Stickiness revisited, J. Phys. A: Math. Theor. 44, 195102 (2011).
  32. M. Orel, Č. Lozej, M. Robnik, and H. Yan, Quantum chaos and semiclassical behavior in mushroom billiards I: Spectral statistics, Phys. Rev. E 113, 044217 (2026).
  33. E. G. Altmann, A. E. Motter, and H. Kantz, Stickiness in Hamiltonian systems: From sharply divided to hierarchical phase space, Phys. Rev. E 73, 026207 (2006).
  34. E. G. Altmann, Intermittent chaos in Hamiltonian dynamical systems, Ph.D. thesis, Verlag nicht ermittelbar, 2007.
  35. A. Akaishi and A. Shudo, Accumulation of unstable periodic orbits and the stickiness in the two-dimensional piecewise linear map, Phys. Rev. E 80, 066211 (2009).
  36. Č. Lozej, D. Lukman, and M. Robnik, Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structure, Phys. Rev. E 106, 054203 (2022).
  37. H. Yan and M. Robnik, Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model. II. Phenomenology of quantum eigenstates, Phys. Rev. E 109, 054211 (2024).
  38. H. Yan, Q. Wang, and M. Robnik, Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space, Phys. Rev. E 110, 064222 (2024).
  39. Q. Wang and M. Robnik, Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space, Phys. Rev. E 108, 054217 (2023).
  40. M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  41. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  42. K.-C. Lee, What makes chaos border sticky? Physica D 35, 186 (1989).
  43. J. Malovrh and T. Prosen, Spectral statistics of a system with sharply divided phase space, J. Phys. A: Math. Gen. 35, 2483 (2002).
  44. A. Ishii, A. Akaishi, A. Shudo, and H. Schomerus, Weyl law for open systems with sharply divided mixed phase space, Phys. Rev. E 85, 046203 (2012).
  45. R. Devaney, An Introduction to Chaotic Dynamical Systems (CRC Press, Boca Raton, FL, 2018).
  46. I. Dana, N. W. Murray, and I. C. Percival, Resonances and diffusion in periodic Hamiltonian maps, Phys. Rev. Lett. 62, 233 (1989).
  47. S. Vaienti, Ergodic properties of the discontinuous sawtooth map, J. Stat. Phys. 67, 251 (1992).
  48. M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behaviour, Commun. Math. Phys. 80, 453 (1981).
  49. C. H. Skokos, G. A. Gottwald, and J. Laskar, Chaos Detection and Predictability (Springer, Berlin, 2016), Vol. 1.
  50. A. M. Mancho, S. Wiggins, J. Curbelo, and C. Mendoza, Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 18, 3530 (2013).
  51. C. Lopesino, F. Balibrea, S. Wiggins, and A. M. Mancho, Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps, Commun. Nonlinear Sci. Numer. Simul. 27, 40 (2015).
  52. M. Hillebrand, S. Zimper, A. Ngapasare, M. Katsanikas, S. Wiggins, and C. Skokos, Quantifying chaos using Lagrangian descriptors, Chaos 32, 123122 (2022).
  53. P. Leboeuf, J. Kurchan, M. Feingold, and D. P. Arovas, Phase-space localization: Topological aspects of quantum chaos, Phys. Rev. Lett. 65, 3076 (1990).
  54. M. Saraceno and A. Voros, Towards a semiclassical theory of the quantum baker's map, Physica D 79, 206 (1994).
  55. J. Keating, F. Mezzadri, and J. Robbins, Quantum boundary conditions for torus maps, Nonlinearity 12, 579 (1999).
  56. J. H. Hannay and M. V. Berry, Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating, Physica D 1, 267 (1980).
  57. J. P. Keating, The cat maps: Quantum mechanics and classical motion, Nonlinearity 4, 309 (1991).
  58. A. Bäcker, in The Mathematical Aspects of Quantum Maps (Springer, Berlin, 2003), pp. 91–144.
  59. T. A. Brun and R. Schack, Realizing the quantum baker's map on a NMR quantum computer, Phys. Rev. A 59, 2649 (1999).
  60. Y. S. Weinstein, S. Lloyd, J. Emerson, and D. G. Cory, Experimental implementation of the quantum baker's map, Phys. Rev. Lett. 89, 157902 (2002).
  61. A. M. Patoary, A. Vikram, L. Shou, and V. Galitski, Chaotic roots of the modular multiplication dynamical system in Shor's algorithm, Phys. Rev. Res. 6, L032046 (2024).
  62. G. Benenti, G. Casati, J. Gong, and Z. Zou, The quantum kicked rotor: A paradigm of quantum chaos. Foundational aspects and new perspectives, arXiv:2604.12316.
  63. B. Batistić and M. Robnik, Dynamical localization of chaotic eigenstates in the mixed-type systems: Spectral statistics in a billiard system after separation of regular and chaotic eigenstates, J. Phys. A: Math. Theor. 46, 315102 (2013).
  64. M. Robnik, Recent developments in quantum chaos of mixed-type systems: A mini review, Nonlinear Phenom. Complex Syst. 26, 209 (2023).
  65. A. Wehrl, On the relation between classical and quantum-mechanical entropy, Rep. Math. Phys. 16, 353 (1979).
  66. E. H. Lieb, Proof of an entropy conjecture of Wehrl, Commun. Math. Phys. 62, 35 (1978).
  67. S. Gnutzmann and K. Zyczkowski, Rényi-Wehrl entropies as measures of localization in phase space, J. Phys. A: Math. Gen. 34, 10123 (2001).
  68. W. T. Lu, S. Sridhar, and M. Zworski, Fractal Weyl laws for chaotic open systems, Phys. Rev. Lett. 91, 154101 (2003).
  69. E. G. Altmann and T. Tél, Poincaré recurrences and transient chaos in systems with leaks, Phys. Rev. E 79, 016204 (2009).

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Phys. Rev. E 114, 034222– Published 25 September, 2026
  • Received 6 January 2026
  • Revised 12 August 2026
  • Accepted 14 September 2026
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