As far as I know, the SPT orders(or SPT phases) are all gapped and protected by symmetry. However they are short range entangled, and the topological order phases are all long range entangled. So the SPT phases are "trivial" topological order. This confused me because it is called "topological". And why are the SPT phases worth studied since they are "trivial" ？ The name SPT and topological order are so similar. Do they have relations?
A symmetry-protected topological phase has a certain symmetry. Any Hamiltonian in this phase can be adiabatically deformed (i.e. without closing the gap) into a Hamiltonian whose ground state is a product state, but the symmetry must be explicitly broken during the deformation process and then restored at the end. As a visually analogy, there is a "wall" crossing the submanifold of parameter space that respects the symmetry, and the wall separates the SPT phase from the totally trivial phase with a product ground state. But if you are allowed to temporarily break the symmetry, then you can leave the submanifold and "jump over the wall" before ending up back in the submanifold and restoring the symmetry.
A Hamiltonian in a topologically ordered phase cannot be deformed into a Hamiltonian with a product ground state by any means whatsoever (without closing the gap). Here, the "wall" crosses the entire parameter space of all possible (local) perturbations (it's infinitely high and can't be jumped over). The phase does not need to have any symmetry. This is much stronger condition.
The two concepts are closely related mathematically as well. It turns out that topologically ordered states are much more exotic than SPT states. (E.g. they have "anyonic" excitations with neither bosonic nor fermionic exchange statistics, while SPT's do not. At least, not in the bulk - things get a little subtle at the boundary.) But if you mathematically "gauge" the symmetry that protects the SPT, then you get a theory that is morally very similar to a topologically ordered state. Also, both types of systems can usefully by classified using cohomology theory.
Topological order and symmetry-protected topological (or trivial, as some people would prefer) order are orders that cannot be classified by the conventional Landau symmetry breaking paradigm. People have found out that there are different phases of matter with the same symmetry (e.g. different fractional quantum Hall liquid with the same symmetry), which cannot be explained by Landau's theory.
The difference between these two would be:
Topological order could endure any perturbations (as long as it's not too big), while SPT order could only endure perturbations that respect the symmetry, which protects the state.
Topological order has long-range entanglement, while SPT has only short-range entanglement. This means that topological order is a global property of the system (e.g. toric code), while SPT is still a local property (e.g. AKLT chain).
Topological order could have fractional excitations (i.e. quasiparticles that have fractional statistics, but not necessarily though), while SPT cannot (though there might be symmetry fractionalization as the boundary, but that is not excitations).
The methods to classify those two orders are quite different. There exists no unified way to classify topological order (as far as I know), while for 1D SPT (for bosonic and spin systems), one can use group cohomology to classify different SPT states (there is the well-known 10-fold way for non-interacting fermions for any dimension too).
Examples: Topological order: fractional quantum Hall systems, spin liquids, toric code, Kitaev honeycomb model, etc. SPT: Haldane/AKLT chain, topological insulators (quantum spin Hall), topological superconductors, etc.
References: arXiv: 1610.03911