 Solution

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.