Ising Magnetism and Ferroelectricity in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Ca</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>CoMnO</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math>

The origin of both the Ising chain magnetism and ferroelectricity in Ca3CoMnO6 is studied by ab initio electronic structure calculations and x-ray absorption spectroscopy. We find that Ca3CoMnO6 has alternate trigonal prismatic Co2+ and octahedral Mn4+ sites in the spin chain. Both the Co2+ and Mn4+ are in the high-spin state. In addition, the Co2+ has a huge orbital moment of 1.7micro_{B} which is responsible for the significant Ising magnetism. The centrosymmetric crystal structure known so far is calculated to be unstable with respect to exchange striction in the experimentally observed upward arrow upward arrow downward arrow downward arrow antiferromagnetic structure for the Ising chain. The calculated inequivalence of the Co-Mn distances accounts for the ferroelectricity.