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Normalize the stated Pauli matrices asThe factor $3^{-1/2}$ changes only the overall normalization at fixed $N$. In the spin-one Cartesian basis, the associated periodic uniform matrix product state isThis is the Pauli-matrix representation of the Affleck--Kennedy--Lieb--Tasaki state.

The Pauli matrix multiplication lawshows that products on two neighboring sites span all of $M_2(\mathbb{C})$, so the tensor is an injective matrix product state after blocking two sites. More geometrically, its two-site image is the scalar plus antisymmetric subspace of $3\otimes 3$, namely the total-spin $J=0$ and $J=1$ sectors. By the Two-site support of the Pauli-matrix Affleck--Kennedy--Lieb--Tasaki tensor, the missing subspace is the five-dimensional symmetric traceless $J=2$ sector.

Let $P_{n,n+1}^{(2)}$ be the orthogonal projection onto that $J=2$ sector. The parent Hamiltonian of a matrix product state isEach term annihilates $|{\Psi }_N⟩$, so this is a frustration-free quantum Hamiltonian and the MPS is a ground state. Writing $x={\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$ and using the eigenvalues $-2,-1,1$ in the three total-spin sectors gives the explicit projectorThis is the Affleck--Kennedy--Lieb--Tasaki parent Hamiltonian. Injectivity implies that its periodic ground state is unique for every sufficiently long chain.

For a unique gapped ground state, one-dimensional bosonic phases protected by an on-site symmetry $G$ are classified by the cohomology classof a projective virtual symmetry of a matrix product state. After blocking finitely many sites, an injective matrix product state tensor satisfiesChanging the virtual gauge or rephasing $V(g)$ changes $\omega$ by a coboundary and leaves $[\omega ]$ fixed.

If symmetry breaking and degenerate ground states are allowed, the additional data are the unbroken subgroup $H\subseteq G$, the way $G$ permutes the broken-symmetry sectors, and a class in $H^2(H,U(1))$ within one sector. Acting with ${⨂}_i{U}_i(g)$ preserves a symmetric ground state up to phase in the unbroken case; in a broken phase it generally permutes the different ground states.