# Topological phase transitions in finite-size periodically driven translationally invariant systems

## TL;DR

This paper demonstrates that in finite-size translationally invariant systems, periodic driving can induce topological phase transitions by changing the Bott index, even though the Chern number remains invariant in the thermodynamic limit.

## Contribution

It proves the equivalence of the Bott index and Chern number in the thermodynamic limit and shows how finite-size effects enable topological changes under periodic drive.

## Key findings

- Bott index and Chern number are identical in the thermodynamic limit.
- Periodic drive can change the topology of a finite system's Fermi sea.
- Topologically nontrivial states can be dynamically prepared from trivial states in finite systems.

## Abstract

It is known that, in the thermodynamic limit, the Chern number of a translationally invariant system cannot change under unitary time evolutions that are smooth in momentum space. Yet a real-space counterpart of the Chern number, the Bott index, has been shown to change in periodically driven systems with open boundary conditions. Here we prove that the Bott index and the Chern number are identical in translationally invariant systems in the thermodynamic limit. 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 different topology from that of the initial state. This can happen provided that the gap-closing points in the thermodynamic limit are absent in the discrete Brillouin zone of the finite system. Hence, in such systems, a periodic drive can be used to dynamically prepare topologically nontrivial states starting from topologically trivial ones.