Symmetry-protected topological phase (SPT phase)

A symmetry-protected topological phase is short-range entangled but cannot be deformed to a product state without breaking a protecting symmetry or closing the gap.

An on-site symmetry of an injective MPS acts on virtual indices by matrices defined up to phase. Their projective cohomology class is the one-dimensional SPT invariant.

The one-dimensional cluster state is the common positive eigenstate of commuting stabilizers $Z_{j-1}{X}_j{Z}_{j+1}$ and realizes a nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT phase.