On a variational model for the continuous mechanics exhibiting hexagonal to square Phase Transitions

Inspired by Conti and Zanzotto \cite{Conti2004A}, we reformulate a simple variational model for reconstructive phase transitions in crystals arising in continuum mechanics in the framework of Landau's theory of phase transition(with slight modification). We provide and prove that this class of modular invariant functions admit exactly hexagonal-square lattices minimizers without passing through rhombic lattices, being the first rigorous result in this regard. Our result gives an affirmative answer to an open problem by in \cite{Conti2004A}. In addition, our result has independent interest from number theory.