Entangled tetrahedron ground state and excitations of the magnetoelectric skyrmion material ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

The strongly correlated cuprate ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ has been recently identified as the first insulating system exhibiting a skyrmion lattice phase. Using a microscopic multiboson theory for its magnetic ground state and excitations, we establish the presence of two distinct types of modes: a low-energy manifold that includes a gapless Goldstone mode and a set of weakly dispersive high-energy magnons. These spectral features are the most direct signatures of the fact that the essential magnetic building blocks of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ are not individual Cu spins, but rather weakly coupled ${\mathrm{Cu}}_4$ tetrahedra. Several of the calculated excitation energies are in excellent agreement with terahertz electron spin resonance, Raman, and far-infrared experiments, while the magnetoelectric effect determined within the present quantum-mechanical framework is also fully consistent with experiments, giving strong evidence in the entangled ${\mathrm{Cu}}_4$ tetrahedra picture of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. The predicted energy and momentum dependence of the dipole and quadrupole spin structure factors call for further experimental tests of this picture.

Figure 1

(a) Distorted pyrochlore structure of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. Solid (dashed) lines indicate the strong (weak) exchange couplings. For clarity, the longer-range $J_{\mathrm{O}..\mathrm{O}}^{\mathrm{AF}}$ coupling (see text) is not shown here. (b) The energy spectrum of an isolated strong tetrahedron [23].

Figure 2

(a), (b) Energy levels of ${\mathcal{H}}_0^{\left(t\right)}$ and ${\mathcal{H}}_{\mathrm{TMF}}^{\left(t\right)}$ (see text). (c), (d) Multiboson spectrum of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. The width of each line represents the matrix elements of the magnetic dipole (c) and quadrupole (d) operators. Dashed lines correspond to vanishing matrix elements due to selection rules. Yellow squares A–E in (d) indicate the five Raman lines reported in Ref. [21] (see text). (e) Brillouin-zone path taken in (c) and (d).

Figure 3

(a) The $\left[110\right]$ and (b) the $\left[001\right]$ components of the induced electric polarization in magnetic field rotating perpendicular to the $\left[\overline{1}10\right]$ axis. (c) $P_{\left[001\right]}$ in rotating field about the $\left[001\right]$ axis. (d), (e) The corresponding geometries, chosen according to the convention of Ref. [14].