# nLab symmetry protected trivial order

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# Symmetry-protected trivial order

## Idea

Symmetry Protected Trivial order (SPT order) (also known as Symmetry Protected Topological 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:

1. distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry.
2. 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.
Using the notion of topological order, we can say that SPT states are symmetric states with trivial topological order.

## 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 the $SO(3)$ spin rotation group symmetry. Another example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.

## Group cohomology theory for SPT phases

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

## SPT phases in free fermion systems

Free fermion system can also have non-trivial SPT phases, such as topological insulators and topological superconductors. Those free fermion SPT phases are classified by K-theory.

Related entries: TQFT, topological order, group cohomology, entanglement

#### Classification for bosonic SPT phases

Discussion via higher dimensional WZW models is in

• Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf