  We propose a class of projected BCS wave functions and derive their parent
spin Hamiltonians. These wave functions can be formulated as infinite Matrix
Product States constructed by chiral correlators of Majorana fermions. In 1D,
the spin Hamiltonians can be viewed as SO(n) generalizations of Haldane-Shastry
models. We numerically compute the spin-spin correlation functions and Renyi
entropies for n=5 and 6. Together with the results for n=3 and 4, we conclude
that these states are critical and their low-energy effective theory is the
SO(n)_1 Wess-Zumino-Witten model. In 2D, we show that the projected BCS states
are chiral spin liquids, which support non-Abelian anyons for odd n and Abelian
anyons for even n.
