The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~{II}: Refined structure of the energy-concentration set

In this paper, we continue our study, started in~\cite{CDS1}, of 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~$\text{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\text{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\text{Q}$ and~$\text{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\text{eps}$ tends to zero. While in~\cite{CDS1} we showed that the (rescaled) energy density for the~$\text{Q}$-component concentrates, to leading order, on a finite number of singular points, in this paper we prove the energy density for the~$\text{M}$-component concentrates along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\text{M}$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\text{Q}$-component. Crucial to our arguments will be the energy estimates and compactness results proved in~\cite{CDS1}.