Topological Phase Transitions in Finite-size Periodically Driven Translationally Invariant Systems*

In the thermodynamic limit, the Chern number of a translationally invariant system cannot change under unitary time evolutions that are smooth in momentum space. But the Bott index, a real-space counterpart of the Chern number, has been shown to change in periodically driven systems with open boundary conditions. Using the Bott index, we show that, in finite-size translationally invariant systems, a Fermi sea under a periodic drive that is turned on slowly can acquire a nontrivial topology. This can happen provided that the gap-closing points in the thermodynamic limit are absent in the discrete Brillouin zone of the finite system, which allows the topological charge enclosed by the system to "leak out". Hence a periodic drive can be used to dynamically prepare topologically nontrivial states starting from trivial ones in finite-size systems, that are either translationally invariant or with open boundary conditions.