Noncollinear Ground State from a Four-Spin Chiral Exchange in a Tetrahedral Magnet

We propose a quartic chiral term $m_x{m}_y{m}_z\nabla ·\mathbf{m}$ for the energy density of a cubic ferromagnet with broken parity symmetry (point group $T_d$). We demonstrate that this interaction causes a phase transition from a collinear ferromagnetic state to a noncollinear magnetic cone ground state provided its strength exceeds the geometric mean of magnetic exchange and cubic anisotropy. The corresponding noncollinear ground state may also be additionally stabilized by an external magnetic field pointing along certain crystallographic directions. The four-spin chiral exchange does also manifest itself in peculiar magnon spectra and favors spin waves with the wave vector that is perpendicular to the average magnetization direction.

Figure 1

Schematic illustration of the magnetic cone state that minimize the energy of Eq. (6). The state wave vector is perpendicular to the average magnetization, $\mathbf{k}·\mathbf{n}=0$, that is characteristic for the four-spin chiral interaction $w_{4S}\propto m_x{m}_y{m}_z\nabla ·\mathbf{m}$.

Figure 2

The color plot is obtained by numerical minimization of the function $\mathcal{E}(\mathbf{k},\mathbf{n},\theta )$ of Eq. (6) and represents the value of $\sin \theta$ (the span of magnetic cone) at the global minimum, provided the external magnetic field is directed as $\tilde{\mathbf{H}}=\tilde{H}(0,1,1)/\sqrt{2}$. The noncollinear magnetic cone state (finite $\theta$ and $\mathbf{k}$) is realized for moderate values of $\tilde{K}$ and $\tilde{H}$. The upper left panel shows the horizontal cross section with $\tilde{H}=0$, while the lower left panel shows three vertical cross sections for $\tilde{K}=0.03$, 0.3, and 0.51. The angle $\theta$ smoothly deviates from zero across the lines $\tilde{K}=2-|\tilde{H}|$, which correspond to the second order phase transition. Noisy borders for $\tilde{K}\approx \pm 0.5$ correspond to the first order phase transition from collinear to a noncollinear state with a finite $\theta$. The corresponding jumps are also seen in the left panels.

Figure 3

Absolute value of the vector $\mathbf{v}$ that defines the asymmetry of the magnon dispersion $\delta {\omega }_{\mathbf{q}}=-16B\mathbf{v}·\mathbf{q}$ as the function of the magnetization direction $\mathbf{n}=(\cos \alpha \sin \beta ,\sin \alpha \sin \beta ,\cos \beta )$.