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.
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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
- Differential and Algebraic Geometry
- Field Theories in Lower Dimensions