symmetry protected trivial order

Symmetry-protected trivial order

Symmetry-protected trivial order


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)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)]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

Classification for free fermion SPT phases

Early discovery articles

  • Michael Levin, Zheng-Cheng Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120.
  • Yuan-Ming Lu, Ashvin Vishwanath, Theory and classification of interacting ‘integer’ topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86, 125119 (2012), arXiv:1205.3156.
  • Davide Gaiotto, Theo Johnson-Freyd, Symmetry protected topological phases and generalized cohomology, arxiv/1712.07950

Conference and seminar cycles

Last revised on February 15, 2021 at 01:14:20. See the history of this page for a list of all contributions to it.