Symmetry-protected topological (SPT) order[1][2] is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap.
To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points).
[1] The SPT order has the following defining properties: (a) distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry.
The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.
Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states with a symmetry (by contrast: for long-range entanglement see topological order, which is not related to the famous EPR paradox).
The difference is subtle: the gapless boundary excitations in intrinsic topological order can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations that do not break the symmetry.
So the gapless boundary excitations in intrinsic topological order are topologically protected, while the gapless boundary excitations in SPT order are symmetry protected.
In contrast, a SPT order has no emergent fractional charge/fractional statistics for finite-energy excitations, nor emergent gauge theory (due to its short-range entanglement).
Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by modifying the Hamiltonian.
The first example of SPT order is the Haldane phase of odd-integer spin chain.
[10][11][12][13][14] It is a SPT phase protected by SO(3) spin rotation symmetry.
[1] Note that Haldane phases of even-integer-spin chain do not have SPT order.
A more well known example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.
They are states with (intrinsic) topological order and long-range entanglements.
Using the notion of quantum entanglement, one obtains the following general picture of gapped phases at zero temperature.
For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by group cohomology theory:[15][16] those (d+1)D SPT states with symmetry G are labeled by the elements in group cohomology class
For other (d+1)D SPT states[17] [18] [19] [20] with mixed gauge-gravity anomalous boundary, they can be described by
is the Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).
From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected by U(1) and time-reversal symmetry) and bosonic topological superconductors (the SPT states protected by time-reversal symmetry), as well as many other new SPT states protected by other symmetries.
A list of bosonic SPT states from group cohomology
On the other hand, the fermionic SPT orders are described by group super-cohomology theory.
[22] So the group (super-)cohomology theory allows us to construct many SPT orders even for interacting systems, which include interacting topological insulator/superconductor.
First, it is shown that there is no (intrinsic) topological order in 1D (ie all 1D gapped states are short-range entangled).
Such an understanding allows one to classify all 1D gapped quantum phases:[15][24][25][26][27] All 1D gapped phases are classified by the following three mathematical objects:
), the 1D gapped phases are classified by the projective representations of symmetry group