Formation of Magnetic Microphases in Ca$_3$Co$_2$O$_6$

We study a frustrated quantum Ising model relevant for Ca$_3$Co$_2$O$_6$ that consists of a triangular lattice of weakly-coupled ferromagnetic (FM) chains. According to our quantum Monte Carlo simulations, the chains become FM and form a three-sublattice "up-up-down" structure for $T \leq T_\text{CI}$. In contrast, long-period spin-density-wave (SDW) {\it microphases} are stabilized along the chains for $T_\text{CI} < T < T_c$. Our mean field solutions reveal a quasi-continuous temperature dependence of the SDW wavelength, implying the existence of metastable states that explain the very slow dynamics observed in Ca$_3$Co$_2$O$_6$. We also discuss implications of microphases for the related multiferroic compounds Ca$_3$CoMnO$_6$ and Lu$_2$MnCoO$_6$.