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 $Q \sim k^{1/\Delta}$, which does not depend on the equilibrium distance of the springs in this model.