Microscopic model for spiral ordering along (110) on the MnSi lattice

We study an extended Heisenberg model on the MnSi lattice. In the cubic B20 crystal structure of MnSi, Mn atoms form lattices of corner-shared equilateral triangles. We find an ubiquitous spiral ordering along (110) for ${J}_{1}l0$, ${J}_{2}g0$, and ${J}_{3}g0$, where ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ are the first, second, and third nearest-neighbor Heisenberg interactions, respectively. While the spiral order along (110) is reasonably robust to the introduction of Dzyaloshinskii-Moriya (DM) and anisotropic magnetic (AM) interactions, the ferromagnetic state for small ${J}_{2}$ and ${J}_{3}$ is replaced by spiral order along either (100) or (111) depending on the strength of DM and AM interactions. We discuss the possible relevance of these results to the partially ordered state recently reported in MnSi.