Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$

Abstract

We prove the existence of a nematic phase in a model of $l\times w$ hard rectangles on the square lattice with two allowed orientations and a large aspect ratio $k:=l/w$. The proof is based on a two-scale cluster expansion method developed previously by Disertori--Giuliani for hard rods in 2D and Disertori--Giuliani--Jauslin for hard plates in 3D. Our main contributions lie in explicitly tracking the constants and parameters and completing the arguments left implicit in these works. Hence, the proof produces a sufficient set of quantitative conditions from which estimates for the required aspect ratio can be extracted. A non-optimized evaluation of these conditions yields the bound $k\ge 10^{72}$. Although it vastly overshoots the numerical prediction, $k_{\min}=7$, our result appears to be the first rigorous estimate of the aspect ratio required for the formation of a nematic phase in hard rectangle systems.