Ordering of hard rectangles in strong confinement

Using transfer operator and fundamental measure theories, we examine the structural and thermodynamic properties of hard rectangles confined between two parallel hard walls. The side lengths of the rectangle ($L$ and $D$, $L>D$) and the pore width ($H$) are chosen such that maximum two layers are allowed to form in planar order ($L$ is parallel to the wall), while only one in homeotropic order ($D$ is parallel to the wall). We observe three different structures: (i) a low density fluid phase with parallel alignment to the wall, (ii) an intermediate and high density fluid phase with two layers and planar ordering and (iii) a dense single fluid layer with homeotropic ordering. The appearance of these phases and the change in the ordering direction with density is a consequence of the varying close packing structures with $L$ and $H$. Interestingly, even three different structures can be observed with increasing density if $L$ is close to $H$.