  When a Heisenberg honeycomb ferromagnet is irradiated by high frequency
circularly polarized light, the underlying uncharged magnons acquire a time
dependent Aharonov Casher phase, which makes it a Floquet topological magnon
insulator. In this context, we investigate the many body interaction effects of
Floquet magnons in laser irradiated Heisenberg honeycomb ferromagnets with
ocontaining Dzyaloshinskii Moriya interaction under the application of
circularly polarized off resonant light. We demonstrate that the quantum
ferromagnet systems periodically laser driven exhibits temperature driven
topological phase transitions due to Floquet magnon magnon interactions. The
thermal Hall effect of Floquet magnons serves as a prominent signature for
detecting these many body effects near the critical point, enabling
experimental investigation into this phenomenon. Our study complements the lack
of previous theoretical works that the topological phase transition of the
Floquet magnon under the linear spin wave approximation is only tunable by the
light field. Our study presents a novel approach for constructing Floquet
topological phases in periodically driven quantum magnet systems that goes
beyond the limitations of the linear spin wave theory. We provide numerical
results based on the well known van der Waals quantum magnet CrX3 (X=F, Cl, Br,
and I), calling for experimental implementation.
