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.

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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.

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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.

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All authors contributed extensively to all aspects of this work.

Correspondence to Norman Y. Yao.

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Supplementary information

Supplementary Information

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

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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.

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

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