Nematic Phase in two-dimensional frustrated systems with power law decaying interactions

We address the problem of orientational order in frustrated interaction systems as a function of the relative range of the competing interactions. We study a spin model Hamiltonian with short range ferromagnetic interaction competing with an antiferromagnetic component that decays as a power law of the distance between spins, $1/r^\alpha$. These systems may develop a nematic phase between the isotropic disordered and stripe phases. We evaluate the nematic order parameter using a self-consistent mean field calculation. Our main result indicates that the nematic phase exists, at mean-field level, provided $0<\alpha<4$. We analytically compute the nematic critical temperature and show that it increases with the range of the interaction, reaching its maximum near $\alpha\sim 0.5$. We also compute a corse-grained effective Hamiltonian for long wave-length fluctuations. For $0<\alpha<4$ the inverse susceptibility develops a set of continuous minima at wave vectors $|\vec k|=k_0(\alpha)$ which dictate the long distance physics of the system. For $\alpha\to 4$, $k_0\to 0$, making the competition between interactions ineffective for greater values of $\alpha$.