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 a maximum of two layers is allowed to form when the long sides of the rectangles are parallel to the wall, while only one layer is possible in case the rectangles are perpendicular to the wall. We observe three different structures: (i) at low density, the rectangles align mainly parallel to the wall, (ii) at intermediate or high density, two fluid layers form in which the rectangles are parallel to the wall, and (iii) a dense single fluid layer with rectangles aligned mainly perpendicular to the wall. The transition between these structures is smooth without any non-analytic behaviour in the thermodynamic quantities; however, the fraction of particles perpendicular (or parallel) to the wall can exhibit a relatively sudden change if L is close to H. In this case, interestingly, even three different structures can be observed with increasing density.