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.

Table of contents
- Projective virtual symmetry of a matrix product state Symmetry-protected topological phase
  - Protected edge state of a symmetry-protected topological phase Projective virtual symmetry of a matrix product state
- Cluster state Symmetry-protected topological phase

Projective virtual symmetry of a matrix product state

 0  0

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.

Protected edge state of a symmetry-protected topological phase

 0  0

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.