Equivalence of the modified Villain formulation and the dual Hamiltonian method in the duality of the XY-plaquette model

Regarding the duality of the XY-plaquette model in the $2+1d$ system, We compared and discussed the equivalence and difference between the modified Villain formulation, recently introduced by Gorantla et al. to study duality in exotic field theories such as fracton theory, and the standard Villain formulation and dual Hamiltonian method we introduced in our previous paper to study duality between Josephson junctions and quantum phase slips in the $1+1d$ system.