Hard rigid rods on Husimi lattices

We study the thermodynamic behavior of hard rigid rods of size $k$ (i.e., $k$-mers) on four- and six-coordinated Husimi lattices (HLs), respectively built with squares (square HL) and triangles (triangular HL). In both lattices, dimers ($k=2$) and trimers ($k=3$) only present a isotropic phase, whereas a isotropic-nematic transition is observed for $k \ge 4$. In the square HL, this transition is continuous and occurs at a critical \textit{monomer} activity which displays a nonmonotonic variation with $k$, while the critical \textit{rod} activity and density are always decreasing functions of $k$. The isotropic-nematic transition is discontinuous in the triangular HL, but the $k$-dependence of the coexistence activities and density is analogous to that found for the square case. No transition from the nematic to a high-density disordered phase is found in these HLs. In general, this scenario is very similar to that already observed for rods on the Bethe lattice, though the critical parameters obtained here are in most cases closer to those reported in the literature for the square and triangular lattices. The entropy per site of fully-packed rods is also investigated in detail in the triangular HL, where its value for dimers differs by only 0.7\% from the exact result for the triangular lattice.