Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of timecrystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time-crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete one dimensional model of a Hamiltonian chain. It is obtained by a reduction from the Q-ball Lagrangian that describes time dependent nontopological solitons. We show that a time crystal appears as a minimum energy domain wall configuration, along the chain.
F. Wilczek, Superfluidity and space-time translation symmetry breaking, Phys. Rev. Lett. 111 (2013) 250402.
A. Shapere and F. Wilczek, Regularizations of time-crystal dynamics, PNAS 116 (2019) 18772.
K. Sacha and J. Zakrzewski, Time crystals: a review, Rep. Prog. Phys. 81 (2018) 016401 [arXiv:1704.03735].
N.Y. Yao and C. Nayak, Time crystals in periodically driven systems, Phys. Today 71 (2018) 40.
V. Khemani, A. Lazarides, R. Moessner and S.L. Sondhi, Phase structure of driven quantum systems, Phys. Rev. Lett. 116 (2016) 250401.
D.V. Else and C. Nayak, Classification of topological phases in periodically driven interacting systems, Phys. Rev. B 93 (2016) 201103.
D.V. Else, B. Bauer and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117 (2016) 090402.
N.Y. Yao, A.C. Potter, I.-D. Potirniche and A. Vishwanath, Discrete time crystals: rigidity, criticality, and realizations, Phys. Rev. Lett. 118 (2017) 030401 [Erratum ibid. 118 (2017) 269901].
D.V. Else, C. Monroe, C. Nayak and N.Y. Yao, Discrete time crystals, arXiv:1905.13232.
J. Zhang et al., Observation of a discrete time crystal, Nature 543 (2017) 217.
S. Choi et al., Dynamics of interacting fermions under spin-orbit coupling in an optical lattice clock, Nature Phys. 543 (2017) 221.
S. Pal, N. Nishad, T.S. Mahesh and G.J. Sreejith, Temporal order in periodically driven spins in star-shaped clusters, Phys. Rev. Lett. 120 (2018) 180602.
J. Rovny, R.L. Blum, S.E. Barrett, Observation of discrete-time-crystal signatures in an ordered dipolar many-body system, Phys. Rev. Lett. 120 (2018) 180603.
J. Rovny, R.L. Blum and S.E. Barrett, 31 P NMR study of discrete time-crystalline signatures in an ordered crystal of ammonium dihydrogen phosphate, Phys. Rev. B 97 (2018) 184301.
J. Smits, L. Liao, H.T.C. Stoof and P. van der Straten, Observation of a space-time crystal in a superfluid quantum gas, Phys. Rev. Lett. 121 (2018) 185301.
J. Smits, L. Liao, H.T.C. Stoof and P. van der Straten, Dynamics of a space-time crystal in an atomic Bose-Einstein condensate, Phys. Rev. A 99 (2018) 013625.
K. Giergiel, A. Kosior, P. Hannaford and K. Sacha, Time crystals: analysis of experimental conditions, Phys. Rev. A 98 (2018) 013613.
J. Dai, A.J. Niemi, X. Peng and F. Wilczek, Truncated dynamics, ring molecules, and mechanical time crystals, Phys. Rev. A 99 (2019) 023425.
J.E. Marsden and T.S. Ratiu, Introduction to mechanics and symmetry a basic exposition of classical mechanical systems, second Edition, Springer, Germany (1999).
P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series volume 2, New York, U.S.A. (1964).
A. Wasserman, Equivariant differential topology, Topology 8 (1969) 127.
D.M. Austin and P.J. Braam, Morse-Bott theory and equivariant cohomology in The Floer memorial volume, H. Hofer et al. eds., Progress in Mathematics volume 133, Birkhäuser, Basel Switzerland (1995).
L. Nicolaescu, An invitation to morse theory, second edition, Springer, Germany (2011).
A. Alekseev and F. Petrov, A principle of variations in representation theory, in The orbit method in geometry and physics: in honor of A.A. Kirillov, C. Duval et al. eds., Progress in Mathematics volume 213, Birkhäuser, Boston U.S.A. (2000).
S.R. Coleman, Q balls, Nucl. Phys. B 262 (1985) 263 [Erratum ibid. 269 (1986) 744] [INSPIRE].
T.D. Lee and Y. Pang, Nontopological solitons, Phys. Rept, 221 (1992) 251.
R.H. Byrd, J.C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Progr. 89 (2000) 149.
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ArXiv ePrint: 2002.07023
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Alekseev, A., Dai, J. & Niemi, A.J. Provenance of classical Hamiltonian time crystals. J. High Energ. Phys. 2020, 35 (2020). https://doi.org/10.1007/JHEP08(2020)035