Formation of Magnetic Microphases in ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathbf{O}}_6$

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

Figure 1

(a) Exchange couplings between the Co II ions: The FM coupling $J_1$ (a thick solid line) and the AFM couplings $J_2$ (thin solid lines) and $J_3$ (dash lines). The lines within the layers are projections of the interchain couplings. (b) The lattice projected on the $ab$ plane. Each dot represents a chain.

Figure 2

The upper panels show the specific heat, while the lower panels show $|{\sigma }_{\mathit{q}=0,\omega =0}^z{|}^2$ for $J_2=J_3=0.1|J_1|$ and $\Gamma =0.3|J_1|$. OBCs (PBCs) are imposed in the $c$ direction in (a) and (c) [(b) and (d)]. The arrows indicate a tiny anomaly in the low-$T$ regime. The insets provide enlarged views.

Figure 3

$S(\mathit{q})$ at $T=1.3|J_1|$ for $J_2=J_3=0.1|J_1|$, $\Gamma =0.3|J_1|$, $L=16$ ($N_{\mathrm{layer}}=480$), and OBCs along the $c$ axis. The vertical lines in (a) indicate $q_3=2\pi /3$ and $4\pi /3$, and the inset shows an enlarged view around $q_3=2\pi /3$. The wave vector is varied as (a) $\mathit{q}=(0,0,q_3)$ and (b) $\mathit{q}=(q_1,0,Q_3)$. Error bars are smaller than the symbol size. The line is a guide to the eye.

Figure 4

$⟨m_0{m}_l⟩$ for the same $J_2$, $J_3$, $\Gamma$, $T$, and $L$ as in Fig. 3. Open boundary conditions are imposed in the $c$ direction, and the center layer of the simulated lattice is chosen as $l=0$.

Figure 5

$q^{\prime }(T)$ obtained from the MF theory ($J_2=J_3=0.1|J_1|$). The inset shows an enlarged view.

Figure 6

Magnetization curve obtained by using a relaxation process described in the text, for $J_2=J_3=0.1|J_1|$, $\Gamma =0.3|J_1|$, $L=12$ ($N_{\mathrm{layer}}=360$), and OBCs in the $c$ direction.