Hard-Needle Elastomer in One Spatial Dimension

We perform exact statistical mechanics calculations for a system of elongated objects (hard needles) that are restricted to translate along a line and rotate within a plane, and that interact via both excluded-volume steric repulsion and harmonic elastic forces between neighbors. This system represents a one-dimensional model of a liquid crystal elastomer, and has a zero-tension critical point that we describe using the transfer-matrix method. In the absence of elastic interactions, we build on previous results by Kantor and Kardar, and find that the nematic order parameter Q decays linearly with tension sigma. In the presence of elastic interactions, the system exhibits a standard universal scaling form, with Q/|sigma| being a function of the rescaled elastic energy constant k/|sigma|(Delta), where Delta is a critical exponent equal to 2 for this model. At zero tension, simple scaling arguments lead to the asymptotic behavior Q similar to k(1/Delta), which does not depend on the equilibrium distance of the springs in this model.