On exact solutions to three-dimensional Yukawa lattice model and BCC to FCC phase transitions

We determine the global minimizers of the three-dimensional Yukawa lattice energy at every fixed density. At unit covolume, BCC is the unique minimizer below a transition parameter ${\kappa }_{\ast }\approx 1.7179218684699$, FCC is the unique minimizer above it, and exactly these two lattice classes minimize at the transition. Mathematically, denote the Yukawa potential and the energy per particle under this potential by $$v_{\kappa }(r)=\frac{e^{-\kappa r}}{r},E_{\kappa }(L)=\frac{1}{2}\sum _{v\in L\setminus \{0\}}\frac{e^{-\kappa \mathrm{\mid }v\mathrm{\mid }}}{\mathrm{\mid }v\mathrm{\mid }}\mathrm{.}$$ Then, $$\arg \underset{\mathrm{\mid }L\mathrm{\mid }=1}{\min }{E}_{\kappa }(L)=\{\begin{array}{llll}\text{BCC lattice},& 0<\kappa <{\kappa }_{\ast },\text{BCC or FCC lattice},& \kappa ={\kappa }_{\ast },\text{FCC lattice},& \kappa >{\kappa }_{\ast }\mathrm{.}\end{array}$$ As a consequence, the zero-screening comparison recovers the unique BCC minimum for the renormalized Coulomb energy.