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In this paper, we prove that any translation and $SU_2(\IC)$-invariant purestate of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, latticesymmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, isfinitely correlated and its two-point spatial correlation function decaysexponentially whenever $d$ is an odd integer. In particular, the Heisenbergiso-spin anti-ferromagnetic integer spin model admits unique low temperaturelimiting ground state and its spatial correlation function decaysexponentially. The unique low temperature limiting ground state of theHamiltonian is determined by the unique solution to Clebsch-Gordoninter-twinning isometry between two representations of $SU_2(\IC)$.

$SU_2(\mathbb{C})$ symmetry in quantum spin chain ground states and Haldane's conjecture