Spontaneous $SU_2(\mathbb{C})$ symmetry breaking in the ground states of quantum spin chain

In this paper, we have proved that there exists no translation invariant pure state of $\mathbb{M}=\otimes_{k \in \mathbb{Z}}\!M^{(k)}_d(\mathbb{C})$ that is real, lattice symmetric with a certain twist and $SU_2(\mathbb{C})$ invariant for any even integer $d \ge 2$. In particular, this result also says that the Heisenberg iso-spin anti-ferromagnetic model with ${1 \over 2}$-odd integer spin degrees of freedom does not admit a unique ground state.