Symmetry protected topological order

Symmetry Protected Topological order (SPT order) 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). 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.
(b) however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation.

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). Since short-range entangled states have only trivial topological orders we may also refer the SPT order as Symmetry Protected "Trivial" order.

SPT states are short-range entangled while topologically ordered states are long-range entangled. Both intrinsic topological order, and also SPT order, can sometimes have protected gapless boundary excitations. 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.

We also know that an intrinsic topological order has emergent fractional charge, emergent fractional statistics, and emergent gauge theory. 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.

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