Citation: Chen, Xie, et al. "Symmetry protected topological orders and the group cohomology of their symmetry group." Physical Review B 87.15 (2013): 155114.
Web: https://arxiv.org/abs/1106.4772
Tags: Physical, SPT/SETs, Topological-insulators, Foundational
This paper gives a good account of symmetry protected topological order, which has since been disputed. The main result is a classification of symmetry protected topological order in terms of group cohomology. Namely, d-dimensional SPTs protected by symmetry group G are enumerated by elements of the cohomology group H^(1+d)(G,U(1)). The controversy about this result comes from the fact that there exist SPT phases that don't fall into the cohomological classification, such as
> Burnell, Fiona J., et al. "Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order." Physical Review B 90.24 (2014): 245122.
There have since been lots of work to try to find the true classification of SPTs, for example the following:
> Kapustin, Anton. "Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology." arXiv preprint arXiv:1403.1467 (2014).
> Kapustin, Anton, et al. "Fermionic symmetry protected topological phases and cobordisms." Journal of High Energy Physics 2015.12 (2015): 1-21.
> Sopenko, Nikita. "An index for two-dimensional SPT states." Journal of Mathematical Physics 62.11 (2021).
The main conjecture from these works was that there is a classification of SPTs in terms of a certain "cobordism group". This was proved mathematically in 2019:
> Yonekura, Kazuya. "On the cobordism classification of symmetry protected topological phases." Communications in Mathematical Physics 368.3 (2019): 1121-1173.
However, this does not mark the end of the story. The fantastic paper
> Barkeshli, Maissam, et al. "Classification of (2+ 1) D invertible fermionic topological phases with symmetry." Physical Review B 105.23 (2022): 235143.
proves a classification result for fermionic SPT phases. Additionally, they shed some light on the situation with the status of the SPT classification: "Over the past several years, a range of techniques have been developed to characterize and classify SPT or invertible phases. Currently, the most comprehensive approach, which is believed to be applicable to all symmetry types and in general dimensions, is to assume that invertible topological phases of matter are described by deformation classes of invertible topological quantum field theories (TQFTs) [7], which are TQFTs whose path integrals have unit magnitude on every closed manifold. Invertible TQFTs, in turn, can be given an abstract classification in terms of bordism theory, although the mathematical results are fully proven only in cases where there is no thermal Hall effect [5-8]. While this approach is believed to be complete, it has two major drawbacks: first, the computations required to carry out the classification for any particular symmetry group G require difficult spectral sequence computations needing significant technical expertise and, as such, have been carried out only in a few cases [6,7,14,15]. Second, this approach is far-removed from the physical properties of the system. This obscures the physical distinction between different invertible phases, and also removes us from the setting of topological phases of matter, which rely on the notion of a gapped Hamiltonian acting on a many-body Hilbert space. A more direct approach to topological phases of matter, in terms of operator algebras, is also under development, although the results are so far less comprehensive than the TQFT approach [16-19]."
All this being said, the fact that H^(1+d)(G,U(1)) should serve as basis for the classification is intuitive. In the classical world, SPTs protected by a symmetry group G correspond to ordered media with order parameter in K(G,1). When continuity and homotopy equivalence are enforced energetically, one arrives at Dijkgraaf-Witten theory. When Dijkgraaf-Witten TQFTs are generalized to higher dimensional spaces, one finds that they are parameterized by elements of the cohomology group H^(1+d)(G,U(1)):
> Freed, Daniel S., et al. "Topological quantum field theories from compact Lie groups." arXiv preprint arXiv:0905.0731 (2009).
All that this paper is saying is that the (SPT)/Dijkgraaf-Witten theory connection given by gauging really does work.