Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Classical discrete time crystals

Abstract

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: the discrete time crystal (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator–prey models to Faraday waves and a.c.-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially long timescales. Power-law correlations are present along a first-order line, which terminates at a critical point. We analyse the transition by mapping it to the locked-to-sliding transition of a d.c.-driven charge density wave. Finally, building upon results from the field of probabilistic cellular automata, we conjecture the existence of classical time crystals with true long-range order, where time-translation symmetry is broken out to infinite times.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Period-doubled dynamics ‘boil’ out of a uniform initial state.
Fig. 2: Diagnostics and phase diagram of an activated classical discrete time crystal.
Fig. 3: Parametric resonance of a single nonlinear pendulum.
Fig. 4: Characterizing the CDTC phase transition by measuring the rate of phase slips, \(v=\left\langle \dot{\tilde{\theta }}\right\rangle\), as a function of damping F and temperature T.
Fig. 5: Competition between period-doubled and undoubled dynamics near the putative first-order transition.
Fig. 6: The presence of power-law correlations at the first-order CDTC transition.
Fig. 7: Probing the CDTC using the stroboscopic spectral function.

Data availability

The data represented in Figs. 2b–d, 4a–d, 6b and 7b are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. 1.

    Van der Pol, B. & Van Der Mark, J. Frequency demultiplication. Nature 120, 363–364 (1927).

    ADS  Google Scholar 

  2. 2.

    May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).

    ADS  MATH  Google Scholar 

  3. 3.

    Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

    ADS  MATH  Google Scholar 

  4. 4.

    Brown, S. E., Mozurkewich, G. & Grüner, G. Subharmonic Shapiro steps and devil’s-staircase behavior in driven charge-density-wave systems. Phys. Rev. Lett. 52, 2277–2280 (1984).

    ADS  Google Scholar 

  5. 5.

    Parlitz, U., Junge, L. & Kocarev, L. Subharmonic entrainment of unstable period orbits and generalized synchronization. Phys. Rev. Lett. 79, 3158–3161 (1997).

    ADS  Google Scholar 

  6. 6.

    Rasband, S. N. Chaotic Dynamics of Nonlinear Systems (Dover Publications, 2015).

  7. 7.

    Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2014).

  8. 8.

    Linsay, P. S. Period doubling and chaotic behavior in a driven anharmonic oscillator. Phys. Rev. Lett. 47, 1349–1352 (1981).

    ADS  Google Scholar 

  9. 9.

    Kaneko, K. Period-doubling of kink–antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice: towards a prelude of a ‘field theory of chaos’. Prog. Theor. Phys. 72, 480–486 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  10. 10.

    Jackson, E. A. & Hübler, A. Periodic entrainment of chaotic logistic map dynamics. Physica D 44, 407–420 (1990).

    ADS  MathSciNet  MATH  Google Scholar 

  11. 11.

    Van der Pol, B. LXXXVIII. On “relaxation-oscillations”. Lond. Edinb. Dubl. Phil. Mag. 2, 978–992 (1926).

    Google Scholar 

  12. 12.

    Kuramoto, Y. Self-entrainment of a population of coupled non-linear oscillators. In Int. Symposium on Mathematical Problems in Theoretical Physics 420–422 (Springer, 1975).

  13. 13.

    Gupta, S., Campa, A. & Ruffo, S. Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise. Phys. Rev. E 89, 022123 (2014).

    ADS  Google Scholar 

  14. 14.

    Brown, S. E., Mozurkewich, G. & Grüner, G. Harmonic and subharmonic Shapiro steps in orthorhombic TaS3. Solid State Commun. 54, 23–26 (1985).

    ADS  Google Scholar 

  15. 15.

    Tua, P. & Ruvalds, J. Dynamics of driven charge-density waves: subharmonic Shapiro steps with devil’s staircase structure. Solid State Commun. 54, 471–474 (1985).

    ADS  Google Scholar 

  16. 16.

    Sherwin, M. & Zettl, A. Complete charge density-wave mode locking and freeze-out of fluctuations in NbSe3. Phys. Rev. B 32, 5536–5539 (1985).

    ADS  Google Scholar 

  17. 17.

    Wiesenfeld, K. & Satija, I. Noise tolerance of frequency-locked dynamics. Phys. Rev. B 36, 2483–2492 (1987).

    ADS  Google Scholar 

  18. 18.

    Bhattacharya, S., Stokes, J. P., Higgins, M. J. & Klemm, R. A. Temporal coherence in the sliding charge-density-wave condensate. Phys. Rev. Lett. 59, 1849–1852 (1987).

    ADS  Google Scholar 

  19. 19.

    Falo, F., Floría, L. M., Martínez, P. J. & Mazo, J. J. Unlocking mechanism in the ac dynamics of the Frenkel–Kontorova model. Phys. Rev. B 48, 7434–7437 (1993).

    ADS  Google Scholar 

  20. 20.

    Balents, L. & Fisher, M. P. Temporal order in dirty driven periodic media. Phys. Rev. Lett. 75, 4270–4273 (1995).

    ADS  Google Scholar 

  21. 21.

    Tekić, J., He, D. & Hu, B. Noise effects in the ac-driven Frenkel-Kontorova model. Phys. Rev. E 79, 036604 (2009).

    ADS  Google Scholar 

  22. 22.

    Tekić, J. et al. The ac driven Frenkel-Kontorova Model (Vinča Nuclear Institute, 2016).

  23. 23.

    Lee, H. C. et al. Subharmonic Shapiro steps in Josephson-junction arrays. Phys. Rev. B 44, 921–924 (1991).

    ADS  Google Scholar 

  24. 24.

    Yu, W., Harris, E. B., Hebboul, S. E., Garland, J. C. & Stroud, D. Fractional Shapiro steps in ladder Josephson arrays. Phys. Rev. B 45, 12624–12627 (1992).

    ADS  Google Scholar 

  25. 25.

    Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).

    ADS  Google Scholar 

  26. 26.

    Shapere, A. & Wilczek, F. Classical time crystals. Phys. Rev. Lett. 109, 160402 (2012).

    ADS  Google Scholar 

  27. 27.

    Bruno, P. Impossibility of spontaneously rotating time crystals: a no-go theorem. Phys. Rev. Lett. 111, 070402 (2013).

    ADS  Google Scholar 

  28. 28.

    Nozières, P. Time crystals: can diamagnetic currents drive a charge density wave into rotation? Europhys. Lett. 103, 57008 (2013).

    ADS  Google Scholar 

  29. 29.

    Volovik, G. E. On the broken time translation symmetry in macroscopic systems: precessing states and off-diagonal long-range order. JETP Lett. 98, 491–495 (2013).

    ADS  Google Scholar 

  30. 30.

    Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A 91, 033617 (2015).

    ADS  Google Scholar 

  31. 31.

    Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Phys. Rev. Lett. 114, 251603 (2015).

    ADS  MathSciNet  Google Scholar 

  32. 32.

    Heo, M.-S. et al. Ideal mean-field transition in a modulated cold atom system. Phys. Rev. E 82, 031134 (2010).

    ADS  Google Scholar 

  33. 33.

    Citro, R. et al. Dynamical stability of a many-body Kapitza pendulum. Ann. Phys. 360, 694–710 (2015).

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Chandran, A. & Sondhi, S. L. Interaction-stabilized steady states in the driven O(N) model. Phys. Rev. B 93, 174305 (2016).

    ADS  Google Scholar 

  35. 35.

    Liggett, T. M. Interacting Particle Systems Vol. 276 (Springer, 2012).

  36. 36.

    Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).

    ADS  Google Scholar 

  37. 37.

    Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).

    ADS  Google Scholar 

  38. 38.

    von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal long-range order in Floquet systems. Phys. Rev. B 94, 085112 (2016).

    ADS  Google Scholar 

  39. 39.

    Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality and realizations. Phys. Rev. Lett. 118, 030401 (2017).

    ADS  MathSciNet  Google Scholar 

  40. 40.

    Khemani, V., vonKeyserlingk, C. W. & Sondhi, S. L. Defining time crystals via representation theory. Phys. Rev. B 96, 115127 (2017).

    ADS  Google Scholar 

  41. 41.

    Lazarides, A. & Moessner, R. Fate of a discrete time crystal in an open system. Phys. Rev. B 95, 195135 (2017).

    ADS  Google Scholar 

  42. 42.

    Iemini, F. et al. Boundary time crystals. Phys. Rev. Lett. 121, 035301 (2018).

    ADS  Google Scholar 

  43. 43.

    Else, D. V., Monroe, C., Nayak, C. & Yao, N. Y. Discrete time crystals. Preprint at https://arxiv.org/pdf/1905.13232.pdf (2019).

  44. 44.

    Yao, N. Y. & Nayak, C. Time crystals in periodically driven systems. Phys. Today 71, 40–47 (2018).

    Google Scholar 

  45. 45.

    Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

    ADS  Google Scholar 

  46. 46.

    Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).

    ADS  Google Scholar 

  47. 47.

    Rovny, J., Blum, R. L. & Barrett, S. E. Observation of discrete-time-crystal signatures in an ordered dipolar many-body system. Phys. Rev. Lett. 120, 180603 (2018).

    ADS  MathSciNet  Google Scholar 

  48. 48.

    Rovny, J., Blum, R. L. & Barrett, S. E. 31P NMR study of discrete time-crystalline signatures in an ordered crystal of ammonium dihydrogen phosphate. Phys. Rev. B 97, 184301 (2018).

    ADS  Google Scholar 

  49. 49.

    Nyquist, H. Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110–113 (1928).

    ADS  Google Scholar 

  50. 50.

    Gács, P. Reliable computation with cellular automata. J. Comput. Syst. Sci. 32, 15–78 (1986).

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Gács, P. Reliable cellular automata with self-organization. J. Stat. Phys. 103, 45–267 (2001).

    ADS  MathSciNet  MATH  Google Scholar 

  52. 52.

    Gray, L. F. A reader’s guide to Gacs’s ‘positive rates’ paper. J. Stat. Phys. 103, 1–44 (2001).

    ADS  MATH  Google Scholar 

  53. 53.

    Toom, A. L. Nonergodic multidimensional system of automata. Problemy Peredachi Informatsii 10, 70–79 (1974).

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Toom, A. Unstable multicomponent systems. Problemy Peredachi Informatsii 12, 78–84 (1976).

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Toom, A. Stable and attractive trajectories in multicomponent systems. Adv. Probab. 6, 549–575 (1980).

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Bennett, C. H., Grinstein, G., He, Y., Jayaprakash, C. & Mukamel, D. Stability of temporally periodic states of classical many-body systems. Phys. Rev. A 41, 1932–1935 (1990).

    ADS  MathSciNet  Google Scholar 

  57. 57.

    Landau, L. D. & Lifshitz, E. M. Statistical Physics Vol. 5: Course of Theoretical Physics (Pergamon Press, 1969).

  58. 58.

    Wolfram, S. Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983).

    ADS  MathSciNet  MATH  Google Scholar 

  59. 59.

    Frenkel, J. & Kontorova, T. On the theory of plastic deformation and twinning. Izv. Akad. Nauk Fiz. 1, 137–149 (1939).

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Braun, O. M. & Kivshar, Y. S. The Frenkel–Kontorova Model: Concepts, Methods, and Applications (Springer Science & Business Media, 2013).

  61. 61.

    Zounes, R. S. & Rand, R. H. Subharmonic resonance in the non-linear Mathieu equation. Int. J. Non-Linear Mech. 37, 43–73 (2002).

    ADS  MathSciNet  MATH  Google Scholar 

  62. 62.

    Brizard, A. J. Jacobi zeta function and action-angle coordinates for the pendulum. Commun. Nonlinear Sci. Numer. Simul. 18, 511–518 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  63. 63.

    Stewart, W. Current–voltage characteristics of Josephson junctions. Appl. Phys. Lett. 12, 277–280 (1968).

    ADS  Google Scholar 

  64. 64.

    McCumber, D. E. Effect of ac impedance on dc voltage–current characteristics of superconductor weak-link junctions. J. Appl. Phys. 39, 3113–3118 (1968).

    ADS  Google Scholar 

  65. 65.

    Else, D. V., Bauer, B. & Nayak, C. Prethermal phases of matter protected by time-translation symmetry. Phys. Rev. X 7, 011026 (2017).

    Google Scholar 

  66. 66.

    Hänggi, P., Talkner, P. & Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990).

    ADS  MathSciNet  Google Scholar 

  67. 67.

    Büttiker, M., Harris, E. P. & Landauer, R. Thermal activation in extremely underdamped Josephson-junction circuits. Phys. Rev. B 28, 1268–1275 (1983).

    ADS  Google Scholar 

  68. 68.

    Dykman, M. I., Maloney, C. M., Smelyanskiy, V. N. & Silverstein, M. Fluctuational phase-flip transitions in parametrically driven oscillators. Phys. Rev. E 57, 5202–5212 (1998).

    ADS  Google Scholar 

  69. 69.

    Purcell, E. M. & Pound, R. V. A nuclear spin system at negative temperature. Phys. Rev. 81, 279 (1951).

    ADS  Google Scholar 

  70. 70.

    Braun, S. et al. Negative absolute temperature for motional degrees of freedom. Science 339, 52–55 (2013).

    ADS  Google Scholar 

  71. 71.

    Nakagawa, M., Tsuji, N., Kawakami, N. & Ueda, M. Negative-temperature quantum magnetism in open dissipative systems. Preprint at https://arxiv.org/abs/1904.00154 (2019).

  72. 72.

    Kapitza, P. L. A pendulum with oscillating suspension. Uspekhi Fizicheskikh Nauk 44, 7–20 (1951).

    ADS  Google Scholar 

  73. 73.

    Braun, O. M., Dauxois, T., Paliy, M. V. & Peyrard, M. Nonlinear mobility of the generalized Frenkel–Kontorova model. Phys. Rev. E 55, 3598–3612 (1997).

    ADS  Google Scholar 

  74. 74.

    Braun, O., Bishop, A. & Röder, J. Hysteresis in the underdamped driven Frenkel–Kontorova model. Phys. Rev. Lett. 79, 3692–3695 (1997).

    ADS  Google Scholar 

  75. 75.

    Zheng, Z., Hu, B. & Hu, G. Resonant steps and spatiotemporal dynamics in the damped dc-driven Frenkel-Kontorova chain. Phys. Rev. B 58, 5453–5461 (1998).

    ADS  Google Scholar 

  76. 76.

    Huse, D. A. & Fisher, D. S. Dynamics of droplet fluctuations in pure and random Ising systems. Phys. Rev. B 35, 6841–6846 (1987).

    ADS  Google Scholar 

  77. 77.

    Vanossi, A., Manini, N., Urbakh, M., Zapperi, S. & Tosatti, E. Colloquium: modeling friction: from nanoscale to mesoscale. Rev. Mod. Phys. 85, 529–552 (2013).

    ADS  Google Scholar 

  78. 78.

    Bylinskii, A., Gangloff, D. & Vuletić, V. Tuning friction atom-by-atom in an ion-crystal simulator. Science 348, 1115–1118 (2015).

    ADS  Google Scholar 

  79. 79.

    Masluk, N. A., Pop, I. M., Kamal, A., Minev, Z. K. & Devoret, M. H. Microwave characterization of Josephson junction arrays: implementing a low loss superinductance. Phys. Rev. Lett. 109, 137002 (2012).

    ADS  Google Scholar 

  80. 80.

    Welch, P. The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).

    Google Scholar 

  81. 81.

    Grinstein, G., Mukamel, D., Seidin, R. & Bennett, C. H. Temporally periodic phases and kinetic roughening. Phys. Rev. Lett. 70, 3607–3610 (1993).

    ADS  Google Scholar 

  82. 82.

    Rajak, A., Citro, R. & DallaTorre, E. G. Stability and pre-thermalization in chains of classical kicked rotors. J. Phys. A 51, 465001 (2018).

    MathSciNet  MATH  Google Scholar 

  83. 83.

    Mori, T. Floquet prethermalization in periodically driven classical spin systems. Phys. Rev. B 98, 104303 (2018).

    ADS  Google Scholar 

  84. 84.

    Abanin, D. A., De Roeck, W., Ho, W. W. & Huveneers, Fmc Effective Hamiltonians, prethermalization and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017).

    ADS  Google Scholar 

  85. 85.

    Martin, P. C., Siggia, E. D. & Rose, H. A. Statistical dynamics of classical systems. Phys. Rev. A 8, 423–437 (1973).

    ADS  Google Scholar 

  86. 86.

    Sieberer, L. M. & Altman, E. Topological defects in anisotropic driven open systems. Phys. Rev. Lett. 121, 085704 (2018).

    ADS  Google Scholar 

  87. 87.

    Witthaut, D. & Timme, M. Kuramoto dynamics in Hamiltonian systems. Phys. Rev. E 90, 032917 (2014).

    ADS  Google Scholar 

  88. 88.

    Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F. & Spigler, R. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–186 (2005).

    ADS  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the insights of and discussions with E. Altman, D. Huse, S. Gazit, R. Goldstein, L. Sieberer, S. Sondhi and B. Zhu. This work was supported, in part, by the DARPA DRINQS programme (D18AC00033), the David and Lucile Packard Foundation and the W. M. Keck Foundation. L.B. was supported by the NSF Materials Theory programme through grant DMR1506119.

Author information

Affiliations

Authors

Contributions

All authors contributed extensively to all aspects of this work.

Corresponding author

Correspondence to Norman Y. Yao.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Dmitry Abanin, Emanuele Dalla Torre and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary discussion and numerical simulations, Figs. 1 and 2 and references.

Source data

Source Data Fig. 2

Source data for Fig. 2b-d.

Source Data Fig. 4

Source data for Fig. 4a-d.

Source Data Fig. 6

Source data for Fig. 6b.

Source Data Fig. 7

Source data for Fig. 7b.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yao, N.Y., Nayak, C., Balents, L. et al. Classical discrete time crystals. Nat. Phys. 16, 438–447 (2020). https://doi.org/10.1038/s41567-019-0782-3

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing