The compass-type anisotropy appears naturally in diverse physical contexts with strong spin-orbit coupling (SOC) such as transition metal oxides and cold atomic gases etc, and it has been receiving substantial attention. Motivated by recent studies and particularly recent experimental observations on helimagnet MnGe, we investigate the critical roles of this compass-type anisotropy in modulating various spin textures of chiral magnets with strong SOC, by Monte Carlo simulations based on a classical Heisenberg spin model with Dzyaloshinsky-Moriya interaction and compass anisotropy. A phase diagram with emergent spin orders in the space of compass anisotropy and out-of-plane magnetic field is presented. In this phase diagram, we propose that a hybrid super-crystal structure consisting of alternating half-skyrmion and half-anti-skyrmion is the possible zero-field ground state of MnGe. The simulated evolution of the spin structure driven by magnetic field is in good accordance with experimental observations on MnGe. Therefore, this Heisenberg spin model successfully captures the main physics responsible for the magnetic structures in MnGe, and the present work may also be instructive to research on the magnetic states in other systems with strong SOC.

A magnetic skyrmion lattice is a microstructure consisting of hexagonally aligned skyrmions. While a skyrmion as a topologically protected carrier of information promises a number of applications, an easily accessible probe of the skyrmion and skyrmion lattice at mesoscopic scale is of significance. It is known that neutron scattering, Lorentz transmission electron microscopy, and spin-resolved STM as effective probes of skyrmions have been established. In this work, we propose that the spatial contour of dielectric permittivity in a skyrmion lattice with ferromagnetic interaction and in-plane (xy) Dzyaloshinskii-Moriya (DM) interaction can be used to characterize the skyrmion lattice. The phase field and Monte Carlo simulations are employed to develop the one-to-one correspondence between the magnetic skyrmion lattice and dielectric dipole lattice, both exhibiting the hexagonal symmetry. Under excitation of in-plane electric field in the microwave range, the dielectric permittivity shows the dumbbell-like pattern with the axis perpendicular to the electric field, while it is circle-like for the electric field along the z-axis. The dependences of the spatial contour of dielectric permittivity on external magnetic field along the z-axis and dielectric frequency dispersion are discussed.