SPT Phases

Definition

A symmetry-protected topological (SPT) phase is a gapped phase with a unique symmetric ground state on any closed manifold — trivial in the bulk — but a protected boundary (degeneracy, gaplessness, or an anomalous edge realization) that the symmetry cannot remove. Unlike topological-order, an SPT is short-range entangled: it can be deformed to a product state by a finite-depth local-unitary circuit, but not by one that respects the symmetry.

Key facts:

• The edge realizes the symmetry anomalously — the same ‘t Hooft anomaly by inflow: an SPT in $d+1$ dimensions is the bulk whose boundary carries a $d$-dimensional anomaly (see t-hooft-anomaly).
• Bosonic SPTs with symmetry $G$ are largely classified by group cohomology $H^{d+1}(G,U(1))$ (Chen–Gu–Liu–Wen). The 1d case $H^2(G,U(1))$ is exactly projective edge representations.
• Gauging an SPT produces a Dijkgraaf–Witten topological gauge theory with action $\omega \in H^d(G,U(1))$; the two ${\mathbb{Z}}_2$ theories in 2+1d — toric-code and double semion — differ by the SPT cocycle and hence by anyon self-statistics.

The physical archetype is the Haldane/AKLT spin-1 chain: a symmetric bulk gap with protected spin-$\frac{1}{2}$ edge modes.

Role in Research

SPT phases are the short-range-entangled half of the topological-phases story in the generalized-symmetries course, and the bridge from anomalies to topological order. Via gauging they connect protected phases to the topologically ordered Dijkgraaf–Witten theories that Block 3 of the course diagonalizes, and via inflow they make anomalies concrete and finite-dimensional. The cluster-state SPT is also a resource for measurement-based quantum computation, a point of contact with condensed-matter-connections and quantum information.

Relations

• t-hooft-anomaly — the protected edge realizes a ‘t Hooft anomaly (inflow)
• topological-order — the long-range-entangled cousin; gauging an SPT yields topological order
• toric-code — the ${\mathbb{Z}}_2$ Dijkgraaf–Witten theory obtained by gauging the trivial vs nontrivial SPT
• higher-form-symmetries — higher-group and higher-form generalizations of SPT protection

Papers

Chen–Gu–Liu–Wen, arXiv:1106.4772 (group-cohomology classification); Dijkgraaf–Witten, Commun. Math. Phys. 129 (1990) 393; Haldane (1983), Affleck–Kennedy–Lieb–Tasaki (1987). See the course bibliography.

Notes

• “Protected” is the operative word: remove the symmetry and the SPT is trivial. This is the sharp contrast with intrinsic topological order, which is stable with no symmetry at all.
• The group-cohomology classification is not the whole story (beyond-cohomology / invertible-field-theory refinements exist), but it captures the cases the course uses.