Affleck--Kennedy--Lieb--Tasaki state (AKLT state)

The Affleck--Kennedy--Lieb--Tasaki state is an injective spin-one MPS whose virtual degrees of freedom are spin one-half. Its ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry acts linearly on physical spins and projectively on the virtual spins.

In a Cartesian basis of the physical spin-one space, the AKLT MPS matrices are proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$. Products of two such matrices span the full virtual matrix algebra, and the two-site parent term projects onto total spin two.

Because ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{\epsilon }_{ijk}{\sigma }_k$, the two-site image of the Pauli-matrix AKLT tensor is the direct sum of the scalar and antisymmetric subspaces, which have total spin zero and one. Its orthogonal complement is the symmetric traceless total-spin-two subspace.

The AKLT parent Hamiltonian is $H={\sum }_j{P}_{j,j+1}^{(2)}$, where $P^{(2)}$ projects two neighboring spin-one particles onto total spin two. It is frustration free, and the periodic AKLT MPS is its unique ground state on a sufficiently long chain.

After suitable blocking, the AKLT and one-dimensional cluster states realize the same nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-protected topological phase. Their virtual symmetries carry the same nontrivial projective class and produce protected edge degrees of freedom.