Quantum Theory of Multiferroic Helimagnets: Collinear and Helical Phases

We study the quantum fluctuation in the cycloidal helical magnet in terms of the Schwinger boson approach. In sharp contrast to the classical fluctuation, the quantum fluctuation is collinear in nature which gives rise to the collinear spin density wave state slightly above the helical cycloidal state as the temperature is lowered. Physical properties such as the reduced elliptic ratio of the spiral, the neutron scattering and infrared absorption spectra are discussed from this viewpoint with the possible relevance to the quasi-one dimensional ${\mathrm{LiCu}}_2{\mathrm{O}}_2$ and ${\mathrm{LiCuVO}}_4$.

Figure 1

Schematic lattice structure and exchange interactions of the effective spin model. $J_1$ is ferromagnetic, while $J_2$ is antiferromagnetic. $xyz$-coordinates and $abc$-axes are also shown.

Figure 2

RVB order parameters ${\eta }_1$ and ${\eta }_2$, and the gap $\Delta$ of the $S=1/2$ 1D model with varying $J_1/J_2$ at zero temperature. Inset shows the temperature dependence of ${\eta }_1$ and ${\eta }_2$ at $J_1/J_2=0.5$. We use the unit $J_2=k_{\mathrm{B}}=1$. The transition temperature is given by $T_{\mathrm{RVB}}=J_2(S+1/2)/\ln (1+1/S)$.

Figure 3

(a) Schematic energy dispersion of $\gamma$-particles. Minima are at $\mathit{k}=\pm \mathit{Q}/2$. (b) The reorganization of the SB due to the collinear magnetic order. The origin of the momentum $\mathit{k}$ is shifted by $\pm \mathit{Q}/2$. (c) Goldstone and amplitude modes associated with the BEC in the helical phase.

Figure 4

Im ${ϵ}_{aa}(\omega )$ in the paramagnetic phase with $T=0.8$ (red/right), $T=0.35$ (green/middle), and the helical phase with $T=0$ (blue/left, $\tilde{\Delta }=\sqrt{2}\Delta (0)$). We take $J_1=0.25$ and $J_2=1$. The inset shows the enlargement of the low-frequency region. Singularities are smeared out by the interchain interaction as shown by the black lines. Behaviors near thresholds are also indicated. Closed-form expressions for $\mathrm{Im}{\epsilon }_{aa}(\omega )$ are provided in 32.