Ferronematics: integrality of the limiting interface in the strong coupling regime

We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics – composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features two order parameters: a reduced $\text{Q}$-tensor field $\text{Q}$, describing the nematic liquid crystal component, and a vector field $\text{M}$, accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between $\text{Q}$ and $\text{M}$. It has been shown in~\cite{CDS1,CDS2} that, as a small parameter $\text{eps}$ tends to zero, the energy of critical pairs $({\text{Q}}_{\text{eps}},{\text{M}}_{\text{eps}})$ concentrates on \emph{distinct} singular sets: a finite set of points for the ${\text{Q}}_{\text{eps}}$-component and the support of a \H^1-rectifiable varifold for the ${\text{M}}_{\text{eps}}$-component, with first variation supported on the singular set for the ${\text{Q}}_{\text{eps}}$-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a material constant, such a limiting varifold has integer multiplicity.