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Symmetry Protected Topological order (SPT order) is a new kind of order in zero-temperature states of matter that have a symmetry and a finite energy gap. 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.

Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states with a symmetry. Since short-range entangled states have only trivial topological orders, we may also refer the SPT order as Symmetry Protected Trivial order.

Characteristic properties of SPT order

1. The boundary of a SPT state is either gapless or degenerate, regardless how we cut the sample to form the boundary. A gapped non-degenerate boundary is impossible for a non-trivial SPT state. If the boundary is a gapped degenerate state, the degeneracy may be caused by spontaneous symmetry breaking and/or (intrinsic) topological order.
2. Monodromy defects in non-trivial 2+1D SPT states carry non-trival statistics and fractional quantum numbers of the symmetry group. Monodromy defects are created by twisting the boundary condition along a cut by a symmetry transformation. The ends of such cut are the monodromy defects. For example, 2+1D bosonic Z_n SPT states are classified by a Z_n integer m. One can show that n identical elementary monodromy defects in a Z_n SPT state labeled by m will carry a total Z_n quantum number 2m which is not a multiple of n.
3. 2+1D bosonic U(1) SPT states have a Hall conductance that is quantized as an even integer. 2+1D bosonic SO(3) SPT states have a quantized spin Hall conductance.

Relation between SPT order and (intrinsic) topological order

SPT states are short-range entangled while topologically ordered states are long-range entangled. Both intrinsic topological order and SPT order can some times have protected gapless boundary excitations. 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 intrinsic topological order has emergent fractional charge, emergent fractional statistics, and emergent gauge theory. In contrast, SPT order has no emergent fractional charge/fractional statistics for finite-energy excitations, nor emergent gauge theory (due to its short-ranged entanglements). 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.

Examples of SPT order

The first example of SPT order is the Haldane phase of spin-1 chain. It is a SPT phase protected by SO(3) spin rotation symmetry. 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.

On the other hand, fractionalquantum Hall states are not SPT states. They are states with (intrinsic) topological order and long-range entanglements.

Group cohomology theory for SPT phases

Using the notion of quantum entanglement, one obtains the following general picture of gapped phases at zero temperature. All gapped zero-temperature phases can be divided into two classes: long-range entangled phases (ie phases with intrinsic topological order) and short-range entangled phases (ie phases with no intrinsic topological order). All short-range entangled phases can be further divided into three classes: symmetry-breaking phases, SPT phases, and their mix (symmetry breaking order and SPT order can appear together).

It is well known that symmetry-breaking orders are described by group theory. Rencently, it was shown that the bosonic SPT orders are described by group cohomology theory: d+1D SPT states with symmetry G are labeled by the elements in group cohomology class $H^{d+1}[G,U(1)]$.

From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected 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 ($Z_2^T$ = time-reversal-symmetry group)

symm. group	1+1D	2+1D	3+1D	4+1D	note
$U(1)⋊Z_2^T$	$Z_2$	$Z_2$	$Z_2^2$	$Z\times Z_2$	bosonic topological insulator
$Z_2^T$	$Z_2$	$0$	$Z_2$	$0$	bosonic topological superconductor
$Z_n$	$0$	$Z_n$	$0$	$Z_n$
$U(1)$	$0$	$Z$	$0$	$Z$	2+1D: quantum Hall effect
$SO(3)$	$Z_2$	$Z$	$0$	$Z_2$	1+1D:Haldane phase; 2+1D: spin Hall effect
$SO(3)\times Z_2^T$	$Z_2^2$	$Z_2$	$Z_2^3$	$Z_2^2$
$D_{2h}$	$Z_2^4$	$Z_2^6$	$Z_2^9$	$Z_2^{12}$

Just like group theory can give us 230 crystal structures in 3+1D, group cohomology theory can give us various SPT phases in any dimensions with any on-site symmetry groups.

On the other hand, the fermionic SPT orders are described by group super-cohomology theory. So the group (super-)cohomology theory may allow us to classify all SPT orders even for interacting systems, which include interacting topological insulator/superconductor.

A complete classification of 1D gapped quantum phases (with interactions)

Using the notions of quantum entanglement and SPT order, one can obtain a complete classification of all 1D gapped quantum phases.

First, it is shown that there is no (intrinsic) topological order in 1D (ie all 1D gapped states are short-range entangled). Thus, if the Hamiltonians have no symmetry, all their 1D gapped quantum states belong to one phase—the phase of trivial product states. On the other hand, if the Hamiltonians do have a symmetry, their 1D gapped quantum states are either symmetry-breaking phases, SPT phases, and their mix.

Such an understanding allows one to classify all 1D gapped quantum phases: All 1D gapped phases are classified by the following three mathematical objects: $(G_H,G_{\mathrm{\Psi }},H^2[G_{\mathrm{\Psi }},U(1)])$ , where $G_H$ is the symmetry group of the Hamiltonian, $G_{\mathrm{\Psi }}$ the symmetry group of the ground states, and $H^2[G_{\mathrm{\Psi }},U(1)]$ the second group cohomology class of $G_{\mathrm{\Psi }}$. (Note that $H^2[G,U(1)]$ classifies the projective representations of $G$.) If there is no symmetry breaking (ie $G_{\mathrm{\Psi }}=G_H$), the 1D gapped phases are classified by the projective representations of symmetry group $G_H$.

See also

• AKLT Model
• Topological insulator
• Quantum spin Hall effect
• Topological order

References

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