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.

The $S=1$ Heisenberg antiferromagnet chain realizes the Haldane phase: a gapped symmetry-protected topological phase with effective spin one-half degrees of freedom at open ends. Time-reversal symmetry is one possible protecting symmetry. Integer spin alone does not imply the same protected class: odd- and even-integer chains differ. The broader gap distinction is the Haldane conjecture.^[ref]

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.

On an open chain, a nontrivial virtual projective representation appears as edge degrees of freedom. Symmetry-preserving local perturbations cannot remove their projective class without a bulk phase transition or symmetry breaking.

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.