The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~Q-tensor for the liquid crystal component and a magnetisation vector field~M, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~Q and~M. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~Q-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~M-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the -component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i. e. ~the singular set for the~-component.