Magnetic structure and phase diagram in a spin-chain system: Ca$_3$Co$_2$O$_6$

The low-temperature structure of the frustrated spin-chain compound Ca$_3$Co$_2$O$_6$ is determined by the ground state of the 2D Ising model on the triangular lattice. At high-temperatures it transforms to the honeycomb magnetic structure. It is shown that the crossover between the two magnetic structures at 12 K arises from the entropy accumulated in the disordered chains. This interpretation is in an agreement with the experimental data. General rules for for the phase diagram of frustrated Ising chain compounds are formulated.