The formation of gradient-driven singular structures of codimension one and two in 2D: 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 minimising) 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~$\Q$-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.

Joint work with Giacomo Canevari (UniVR) and Bianca Stroffolini (UniNA)