Ferromagnetic order in dipolar systems with anisotropy: application to magnetic nanoparticle supracrystals

Single domain magnetic nanoparticles (MNP) interacting through dipolar interactions (DDI) in addition to the magnetocrystalline energy may present a low temperature ferromagnetic (SFM) or spin glass (SSG) phase according to the underlying structure and the degree of order of the assembly. We study, from Monte Carlo simulations in the framework of the effective one-spin or macrospin models, the case of a monodisperse assembly of single domain MNP fixed on the sites of a perfect lattice with fcc symmetry and randomly distributed easy axes. We limit ourselves to the case of a low anisotropy, namely the onset of the disappearance of the dipolar long-range ferromagnetic (FM) phase obtained in the absence of anisotropy due to the disorder introduced by the latter.Comment: 10 pages, 7 figure

Reducibility of nilpotent commuting varieties

Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\N_n)$ under certain condition. Then we prove that $C_r(\N_n)$ is reducible for all $n, r\ge 4$. The ingredients of this result are also useful for getting a new lower bound of the dimensions of $C_r(\N_n)$ and $C_r(\mathfrak{gl}_n)$. Finally, we investigate values of $n$ for which the variety $C_3(\N_n)$ of nilpotent commuting triples is reducible.Comment: 8 page