Floquet engineering via periodic light fields has emerged as a powerful method to control topological phase transitions in quantum materials. However, Floquet-induced two-dimensional (2D) weak topological insulator phases, characterized by nontrivial 2D Zak phases, remain unexplored. In this work, we theoretically investigate two distinct models using Floquet theory: the extended Qi-Wu-Zhang model and Bernevig-Hughes-Zhang model with $d$-wave altermagnetism. For the extended Qi-Wu-Zhang model, we demonstrate that the polarized light can induce a 2D weak topological insulator phase characterized by an emergent 2D Zak phase. In the $d$-wave altermagnetic Bernevig-Hughes-Zhang model, we show that the linearly polarized light can drive more intricate topological phase transitions between weak topological insulator, strong topological insulator, and normal insulator. Furthermore, we find a distinct Floquet-induced topological phase, where one spin component exhibits a floating edge state characterized by the 2D Zak phase and the other spin component hosts a chiral edge state characterized by the Chern number.

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• Edge states
• Topological insulators
• Altermagnets
• Floquet systems
• Topological materials