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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]

Intrinsic topological order survives without a protecting symmetry and is characterized by features such as topology-dependent ground-state degeneracy and anyonic excitations. One-dimensional bosonic gapped systems have no intrinsic topological order, although they can have symmetry-protected topological order.

The Affleck--Kennedy--Lieb--Tasaki state places two virtual spin-one-half degrees of freedom at every site, puts neighboring virtual spins into singlets, and projects the two virtual spins at each site onto their symmetric spin-one triplet. Its local tensor is equivalently proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$ in the spin-one Cartesian basis, and its Affleck--Kennedy--Lieb--Tasaki parent Hamiltonian is

The one-dimensional cluster state is the simultaneous $+1$ eigenstate of the commuting stabilizersOn a periodic chain it is obtained by applying a controlled-$Z$ gate on every neighboring pair of the product state $|+{⟩}^{\otimes N}$.

After suitable blocking, the two states have the same nontrivial projective virtual symmetry of a matrix product state for the protecting group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. The two virtual symmetry generators can be represented by anticommuting Pauli matrices, so they realize the nontrivial projective class in group cohomology. Consequently the AKLT and cluster states can be connected by a symmetry-preserving gapped path, or equivalently by a symmetry-preserving finite-depth local circuit: this is the Symmetry-protected equivalence of the Affleck--Kennedy--Lieb--Tasaki state and cluster state.

Both states therefore exhibit one-dimensional symmetry-protected topological order. On an open chain their nontrivial virtual representation produces protected edge degrees of freedom and a characteristic degeneracy in the entanglement spectrum of a matrix product state. They do not have intrinsic topological order: if the protecting symmetry is discarded, either state can be connected to a product state by a finite-depth local circuit.

Apply the theorem to a symmetric injective MPS. The physically transformed tensor $A_g^i={\sum }_{j}U(g{)}_{ij}{A}^j$ generates the same state, so the theorem forcesApplying $g$ and then $h$ shows that $V(g)V(h)$ and $V(gh)$ implement the same gauge transformation. Injectivity makes that gauge unique up to a scalar, hence $V(g)V(h)=\omega (g,h)V(gh)$. Associativity gives the two-cocycle equation for $\omega$.

A continuous symmetry-preserving gapped path changes the tensor and $V(g)$ continuously but cannot change the discrete cohomology class $[\omega ]$ without losing injectivity, breaking the symmetry, or closing the gap. Thus the fundamental theorem of matrix product states turns the virtual projective representation into the invariant classifying one-dimensional symmetry-protected topological phases. With broken symmetry, one first records the permuted ground-state sectors and then applies the same argument to their unbroken subgroup.

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.

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.

Two local Hamiltonian operators describe the same gapped phase when an allowed continuous path $H(s)$, $0\le s\le 1$, connects them while retaining a nonzero bulk spectral gap. The path must keep locality bounds under control. For a family of finite systems with an isolated low-energy sector, a standard sufficient condition is a lower boundindependent of system size $L$ and path parameter $s$. A positive gap in each small finite system alone does not establish a gapped thermodynamic limit.

For sufficiently regular paths of local interactions, quasi-adiabatic continuation transports the low-energy subspace by a quasi-local evolution. Lieb-Robinson bounds control its spatial tails. The resulting equivalence concerns locality-preserving transport, not arbitrary unitary operators on a finite Hilbert space. ^[ref]

The permitted paths define what is being classified. In a symmetry-protected topological phase, they must preserve the specified symmetry. Allowing symmetry-breaking terms can join phases that were distinct under that restriction; the Haldane phase is an important example of why the symmetry condition matters.