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From my basic understanding of Time Crystals, we need a periodically driven Hamiltonian, say $H(t)$ with period $T$, $H(t)= H(T+t)$ and a local order parameter $O(t)$ with a period $nT$ satisfying rigidity and the periodic behavior should persist in the thermodynamic limit (Ref: https://arxiv.org/pdf/1908.11339.pdf). Regarding rigidity, it is said that $O(t)$ should have a period of $nT$ without the Hamiltonian being fine-tuned, (a) does this imply and (b) if so, is it always necessary that the Hamiltonian strength should be time-independent? In most of the examples considered, the Hamiltonian is of the form $H_{0} = J_{1}H_{1} + J_{2}H_{2}$ where the $H_{1}$ is driven for $t_{1}$ and then $H_{2}$ is driven from $t_{1}$ to $t_{1} + t_{2} = T$, where $J_{1}$ and $J_{2}$ are time independent.

  1. Is it also possible to choose $J_{1}$ and $J_{2}$ to be time dependent with the constraint that the total Hamiltonian is periodic?

  2. How strong is the condition of locality on the order parameter $O(t)$ which breaks the time translational symmetry? There has been a reference to non-local order parameter in Appendix A of https://arxiv.org/pdf/1612.08758.pdf, so it would be helpful to know how strict this condition this?

  3. Also, it would be helpful to know a permissible error on the periodicity, i.e., if $O(t + nT) = O(t) + \epsilon$, in the sense the limits of $\epsilon$, implying how much of $\epsilon$ is too much?

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