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 $T\le T_{\text{CI}}$. 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${}_6$.