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~$\mathbf{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\mathbf{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\mathbf{Q}$ and~$\mathbf{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\varepsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~$\mathbf{Q}$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~$\mathbf{M}$-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component.