Spin-Density-Wave Order and Slow Dynamics in Ca$_{3}$Co$_{2}$O$_{6}$

We study a frustrated quantum Ising model relevant for Ca$_{3}$Co$_{2}$O$_{6\, }$that comprises a triangular lattice of weakly coupled ferromagnetic chains$_{\, }$[Y. Kamiya and C. D. Batista, PRL \textbf{109}, 067204 (2012)]. Our quantum Monte Carlo simulation shows that the chains become ferromagnetic and form a three-sublattice ``up-up-down'' structure in the lowest temperature regime $T\le T_{CI}$ due to a quantum effect. In contrast, long-wavelength spin-density-wave (SDW) modulations along the chains are stabilized for $T_{CI}$\textless $T$\textless $T_{c}$ in agreement with recent experiments. We also discuss a simple mean-field theory revealing quasi-continuous change of the modulation periodicity as a function of $T$ and implying the existence of metastable states in the SDW phase, which explains the slow low-temperature dynamics that has been observed in Ca$_{3}$Co$_{2}$O$_{6}$. The closely related multiferroic materials Ca$_{3}$CoMnO$_{6}$ and Lu$_{2}$CoMnO$_{6\, }$will also be discussed.