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We consider the minimizing problem for energy functionals with two types ofcompeting particles and completely monotone potential on a lattice. We provethat the minima of sum of two completely monotone functions among lattices islocated exactly on a special curve which is part of the boundary of thefundamental region. We also establish a universal result for square latticebeing the optimal in certain interval, which is surprising. Our resultestablishes the hexagonal-rhombic-square-rectangular transition lattice shapesin many physical and biological system (such as Bose-Einstein condensates andtwo-component Ginzburg-Landau systems). It turns out, our results also apply tolocating the minimizers of sum of two Eisenstein series, which is new in numbertheory.

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials