Microscopic approach to orientational order of domain walls

We develop a fully microscopic, statistical mechanics approach to study phase transitions in Ising systems with competing interactions at different scales. Our aim is to consider orientational and positional order parameters in a unified framework. In this paper, we consider two-dimensional stripe-forming systems, where nematic, smectic, and crystal phases are possible. We introduce a nematic order parameter in a lattice, which measures orientational order of interfaces. We develop a mean-field approach that leads to a free energy, which is a function of both the magnetization (density) and the orientational (nematic) order parameters. Self-consistent equations for the order parameters are obtained and the solutions are described for a particular system, the dipolar frustrated Ising ferromagnet. We show that this system has an Ising-nematic phase at low temperatures in the square lattice, where positional order (staggered magnetization) is zero. At lower temperatures, a crystal-stripe phase may appear. In the continuum limit, the present approach connects to a Ginsburg-Landau theory, which has an isotropic-nematic phase transition with breaking of a continuous symmetry.