A prismatic approach to $(\varphi, \hat G)$-modules and $F$-crystals

We give a new construction of $(\varphi, \hat G)$-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic $F$-crystals in finite locally free $\mathcal{O}_\Delta$-modules over $(\mathcal{O}_K)_\Delta$ and the category of lattices in crystalline representations of $G_K$, where $K$ is a complete discretely valued field of mixed characteristic with perfect residue field. We also generalize this result to semi-stable representations using the absolute logarithmic prismatic site defined by Koshikawa.