On minima of difference of theta functions and application to hexagonal crystallization

Abstract

Let z = x+iy ∈ H := {z = x+iy ∈ C : y > 0} and θ(α; z) = (m,n)∈Z 2 e −α π y |mz+n| 2 be the theta function associated with the lattice L = Z ⊕ zZ. In this paper we consider the following minimization problem of difference of two theta functions min H θ(α; z) − βθ(2α; z)where α ≥ 1 and β ∈ (−∞, +∞). We prove that there is a critical value βc = √ 2 (independent of α) such that if β ≤ βc, the minimizer is 1 2 + i √ 32 (up to translation and rotation) which corresponds to the hexagonal lattice, and if β > βc, the minimizer does not exist. Our result partially answers some questions raised in [7,8,10,11] and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.