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Floquet topological insulators (arXiv:1008.1792, arXiv:1211.5623) have attracted much research interests in condensed matter physics. The goal is to realize topological insulators from trivial insulators by applying periodic driving forces. The idea is based on the Floquet theory: given a periodic driven system described by the Hamiltonian $H(t)$ (with $H(t+T)=H(t)$), the time evolution operator $U$ over a period

$$U(t_0+T,t_0)=\mathcal{T}_t\exp\left(-\mathrm{i}\int_{t_0}^{t_0+T}\mathrm{d}t\,H(t)\right)$$

can be given as the stroboscopic time evolution generated by a time independent Floquet Hamiltonian $H_F$

$$U(t_0+T,t_0)=\exp(-\mathrm{i}H_F T).$$

It is possible to design the driving terms in $H(t)$ cleverly, such that $H_F$ looks like the Hamiltonian of a topological insulator.

Because $\exp(-\mathrm{i}H_F t)$ matches the actual time evolution operator only at $t=nT+t_0$ ($n\in\mathbb{Z}$), does that mean that the system is a topological insulator only at these discrete stroboscopic time instances? If so, how do we perform the measurement of the system stroboscopically? Are there any stroboscopic measurements in solid-state or cold-atom systems that have been proposed?

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No.

The whole time evolution $[t_0,t_0+T]\ni t\mapsto U(t_0+t,t_0)$ is the topological system and determines the topological class of the system, and $H_F$ itself does not suffice to determine this topology.

Indeed in papers by the authors you cite (Rudner, Lindner et al) there are examples of systems where $H_F$ induces trivial topology yet $[t_0,t_0+T]\ni t\mapsto U(t_0+t,t_0)$ is non-trivial.

What is true, however (and this I still don't quite fully understand) is that only if $H_F$ has a gap can we call the system topological and define an appropriate invariant. Note that $H_F$ having a gap does not imply anything about $[t_0,t_0+T]\ni t\mapsto H(t)$ being an insulator. Also note that $H_F$ having a gap is equivalnt to $[t_0,t_0+T]\ni t\mapsto U(t_0+t,t_0)$ having a gap at time $t=T$. So yes, a gap at a very opportune moment makes the system topological at all, but the topology is determined by the entire evolution map.

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