AbstractWe study the minimality properties of a new type of “soft” theta functions. For a lattice $$L\subset {\mathbb {R}}^d$$
L
⊂
R
d
, an L-periodic distribution of mass $$\mu _L$$
μ
L
, and another mass $$\nu _z$$
ν
z
centered at $$z\in {\mathbb {R}}^d$$
z
∈
R
d
, we define, for all scaling parameters $$\alpha >0$$
α
>
0
, the translated lattice theta function $$\theta _{\mu _L+\nu _z}(\alpha )$$
θ
μ
L
+
ν
z
(
α
)
as the Gaussian interaction energy between $$\nu _z$$
ν
z
and $$\mu _L$$
μ
L
. We show that any strict local or global minimality result that is true in the point case $$\mu =\nu =\delta _0$$
μ
=
ν
=
δ
0
also holds for $$L\mapsto \theta _{\mu _L+\nu _0}(\alpha )$$
L
↦
θ
μ
L
+
ν
0
(
α
)
and $$z\mapsto \theta _{\mu _L+\nu _z}(\alpha )$$
z
↦
θ
μ
L
+
ν
z
(
α
)
when the measures are radially symmetric with respect to the points of $$L\cup \{z\}$$
L
∪
{
z
}
and sufficiently rescaled around them (i.e., at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies, and an approximation argument. Furthermore, for the honeycomb lattice $${\mathsf {H}}$$
H
, the center of any primitive honeycomb is shown to minimize $$z\mapsto \theta _{\mu _{{\mathsf {H}}}+\nu _z}(\alpha )$$
z
↦
θ
μ
H
+
ν
z
(
α
)
, and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centered-cubic, and face-centered-cubic lattices.
Lattice Green functions of the higher-dimensional face-centered cubic lattices
