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Let z = x + i y ∈ H : = { z = x + i y ∈ C : y > 0 } and θ (α ; z) = ∑ (m , n) ∈ Z 2 e - α π y | m z + n | 2 be the theta function associated with the lattice L = 1 Im (z) Z ⊕ z Z . In this paper we consider the following minimization problem of difference of two theta functions min H θ (α ; z) - β θ (2 α ; z) where α ≥ 1 and β ∈ (- ∞ , + ∞) . We prove that there is a critical value β c = 2 (independent of α ) such that if β ≤ β c , the minimizer is 1 2 + i 3 2 (up to translation and rotation) which corresponds to the hexagonal lattice, and if β > β c , the minimizer does not exist. Our result partially answers some questions raised in Bétermin (SIAM J Math Anal 48(5):3236–269, 2016), Bétermin (Nonlinearity 31(9):3973–4005, 2018), Bétermin et al. (Models Methods Appl Sci 31(2):293–325, 2021) and Bétermin and Petrache (Anal Math Phys 9(4):2033–2073, 2019) and gives a new proof in the hexagonal crystallization among lattices under Yukawa potential. [ABSTRACT FROM AUTHOR]