The recent discovery of skyrmions in Cu$_2$OSeO$_3$ has established a new platform to create and manipulate skyrmionic spin textures. We use high-field electron spin resonance (ESR) spectroscopy combining a terahertz free electron laser and pulsed ma
gnetic fields up to 64 T to probe and quantify its microscopic spin-spin interactions. Besides providing direct access to the long-wavelength Goldstone mode, this technique probes also the high-energy part of the excitation spectrum which is inaccessible by standard low-frequency ESR. Fitting the behavior of the observed modes in magnetic field to a theoretical framework establishes experimentally that the fundamental magnetic building blocks of this skyrmionic magnet are rigid, highly entangled and weakly coupled tetrahedra.
We present magnetodielectric measurements in single crystals of the cubic spin-1/2 compound Cu$_2$OSeO$_3$. A magnetic field-induced electric polarization ($vec{P}$) and a finite magnetocapacitance (MC) is observed at the onset of the magnetically or
dered state ($T_c = 59$ K). Both $vec{P}$ and MC are explored in considerable detail as a function of temperature (T), applied field $vec{H}_a$, and relative field orientations with respect to the crystallographic axes. The magnetodielectric data show a number of anomalies which signal magnetic phase transitions, and allow to map out the phase diagram of the system in the $H_a$-T plane. Below the 3up-1down collinear ferrimagnetic phase, we find two additional magnetic phases. We demonstrate that these are related to the field-driven evolution of a long-period helical phase, which is stabilized by the chiral Dzyalozinskii-Moriya term $D vec{M} cdot(bs{ abla}timesvec{M})$ that is present in this non-centrosymmetric compound. We also present a phenomenological Landau-Ginzburg theory for the ME$_H$ effect, which is in excellent agreement with experimental data, and shows three novel features: (i) the polarization $vec{P}$ has a uniform as well as a long-wavelength spatial component that is given by the pitch of the magnetic helices, (ii) the uniform component of $vec{P}$ points along the vector $(H^yH^z, H^zH^x, H^xH^y)$, and (iii) its strength is proportional to $eta_parallel^2-eta_perp^2/2$, where $eta_parallel$ is the longitudinal and $eta_perp$ is the transverse (and spiraling) component of the magnetic ordering. Hence, the field dependence of P provides a clear signature of the evolution of a conical helix under a magnetic field. A similar phenomenological theory is discussed for the MC.
