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Anticlinic order of long-range repulsive rod-like magnetic particles in 2D

In the field of liquid crystals, it is well-known that rod-like molecules interacting via long-range attractive interactions or short-range repulsive potentials can exhibit orientational order. In this work, we are interested in what would happen to systems of rod-like particles interacting via long-range repulsive potential. In our model, each particle consists of a number of point dipoles uniformly distributed along the particle length, with all dipoles pointing along the $z-$ axis, so that the rod-like particles repel each other when they lie in the $x-y$ plane. Dipoles from different particles interact via an $r^{-3}$potential, where $r$ is the distance between the dipoles. We have considered two model systems, each with $N$ particles in a unit cell with periodic boundary conditions. In the first, particle centers are fixed on a square or triangular lattice but they are free to rotate. In the second, particles are free to translate as well as in cells with variable shapes. Here they self-assemble to form configurations where the stress tensors are isotropic. Our numerical results show that at low temperatures the particles tend to form stripes with alternating orientations, resembling herringbone patterns or the anticlinic Sm-C$_{A}$ liquid crystal phase.