  We study the excitonic states in CaB6 in terms of the Ginzburg-Landau theory.
By minimizing the free energy and by comparing with experimental results, we
identify two possible ground states with exciton condensation. They both break
time-reversal and inversion symmetries. This leads to various magnetic and
optical properties. As for magnetic properties, it is expected to be an
antiferromagnet, and its spin structure is predicted. It will exhibit the
magnetoelectric effect, and observed novel ferromagnetism in doped samples and
in thin-film and powder samples can arise from this effect. Interesting optical
phenomena such as the nonreciprocal optical effect and the second harmonic
generation are predicted. Their measurement for CaB6 will clarify whether
exciton condensation occurs or not and which of the two states is realized.
