Magnonic Floquet Quantum Spin Hall Insulator in Bilayer Collinear Antiferromagnets

We study irradiated two-dimensional insulating bilayer honeycomb ferromagnets and antiferromagnets coupled antiferromagnetically with a zero net magnetization. The former is realized in the recently synthesized bilayer honeycomb chromium triiodide CrI$_{\bf 3}$. In both systems, we show that circularly-polarized electric field breaks time-reversal symmetry and induces a dynamical Dzyaloshinskii-Moriya interaction in each honeycomb layer. However, the resulting bilayer antiferromagnetic system still preserves a combination of time-reversal and space-inversion ($\mathcal{PT}$) symmetry. We show that the magnon topology of the bilayer antiferromagnetic system is characterized by a $\pmb{\mathbb{Z}_2}$ Floquet topological invariant. Therefore, the system realizes a magnonic Floquet quantum spin Hall insulator with spin filtered magnon edge states. This leads to a non-vanishing Floquet magnon spin Nernst effect, whereas the Floquet magnon thermal Hall effect vanishes due to $\mathcal{PT}$ symmetry. We study the rich $\pmb{\mathbb{Z}_2}$ Floquet topological magnon phase diagram of the system as a function of the light amplitudes and polarizations. We further discuss the great impact of the results on future experimental realizations.