# Time Crystals: Rigidity, tuning of Hamiltonians

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?