Symmetry-protected topological phases
Some short-range-entangled phases become distinct only when a symmetry is enforced.
$$\text{SPT protection}=\text{symmetry}+\text{bulk gap}$$
What this means
Some short-range-entangled phases become distinct only when a symmetry is enforced.
Deep dive: mathematical and physical mechanism
Symmetry-protected topological (SPT) phases cannot be smoothly deformed to trivial product states while retaining the relevant symmetry and gap. They may host protected boundary states, but unlike intrinsic topological order they need not have fractional bulk anyons.
How it appears in quantum phenomena
SSH chains and time-reversal topological insulators are band-theory examples.
Assumptions, caveats and limitations
Breaking the protecting symmetry may remove the distinction without closing the gap.
Connect the ideas
For context, see the Interacting Topological Phases topic guide and the reading references.