Phonon-induced Floquet topological phases protected by space-time symmetries

For systems with spatial and nonspatial symmetries, the topological classification depends not only on these symmetries but also on the commutation/anticommutation relations between spatial and nonspatial symmetries. The coexistence of spatial and nonspatial symmetries together with appropriate commutation/anticommutation relations between them can give rise to crystalline and higher-order topological phases, which host gapless boundary modes. Alternatively, space-time symmetries in a Floquet system can take the role of spatial symmetries in deciding the topological classification. Promoting a spatial symmetry to a space-time symmetry can alter the commutation relations, which in turn can modify the topological properties of the system. We show how a coherently excited phonon mode can be used to promote a spatial symmetry with which the static system is always trivial to a space-time symmetry which supports a nontrivial Floquet topological phase. We demonstrate this effect by considering two systems: The first is a second-order topological superconductor, and the second is a first-order crystalline topological insulator. In both these cases, a coherently excited phonon mode is responsible for promoting the reflection symmetry to a time-glide symmetry. This newly introduced symmetry allows the previously trivial system to host gapless modes. In the first case, these are protected corner modes, while in the second case, these are gapless edge modes.