Q-domains in Multiferroic CoCr$_2$O$_4$

In spinel CoCr$_2$O$_4$, the observed spin state at low temperature is, approximately, a ``ferrimagnetic spiral"\footnote{N. Menyuk et al., J. de Physique \textbf{25}, 528 (1964)}, given by\footnote {D. H. Lyons et al., Phys. Rev. \textbf{126},540 (1962)} $\mathbf{S}_{n\nu} =\sin\theta_\nu[\hat{x}\cos(\mathbf{Q} \cdot\mathbf{R}_{n\nu}+\gamma_\nu)+ \hat{y}\sin(\mathbf{Q} \cdot\mathbf{R}_{n\nu}+\gamma_\nu)] +\cos\theta_\nu\hat{z}$ . $\nu=1\cdots6$ goes over the six magnetic fcc sublattices, $\mathbf{R}_{n\nu}$ are the positions of the magnetic ions, $\hat{z}$ = [001] crystallographic direction, $\theta_\nu$ are the cone half-angles on which the spins lie, and $\gamma_\nu$ are the phases of the 6 conical spirals, all with wave vector $\mathbf{Q}$ in the [110] direction. This yields magnetization $\mathbf{M},^1$ and, via the Katsura et al mechanism\footnote{H. Katsura et al., Phys. Rev. Lett. \textbf {95}, 057205 (2005)}, electric polarization~$\mathbf{P} $.\footnote{Y. Yamasaki et al., Phys. Rev. Lett. \textbf{96}, 207204 (2006)} Equivalent $\mathbf{Q}$'s, e.g. $\pm\mathbf{Q} $, with associated $\mathbf {M}$'s and $\mathbf{P}$'s, are expected to give degenerate states, ``$\mathbf{Q-M-P}$ domains"; poling in electric and magnetic fields selects a single such domain. Reversal of magnetic field then leads to $\mathbf{P}$ reversal$^{4,}$\footnote{Y. J. Choi et al, submitted for publication} and $\mathbf{Q}$ reversal$^5$. But $\mathbf{Q}\rightarrow -\mathbf{Q}$ in the equation above does not appear to give a degenerate state. I show, via the Heisenberg model and the Generalized Luttinger-Tisza method used in the prediction of the spin state,$^2$ that $\gamma_\nu\rightarrow- \gamma_\nu$ on $\mathbf{Q}$ reversal, making manifest the $\mathbf{Q} \rightarrow-\mathbf {Q}$ degeneracy.