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In the papers I review they first start to talk about topologically ordered phases of matter. Their standard example of it is FQHE. Than they give another set examples which are quantum spin liquids, they also say that topological order is not enough to characterise the nature of QSL you also need to consider symmetry which gives rise to SETS. Now, are FQHE, is a subset of SETS or are they just topologically ordered phases which are not SETS.

Also, do all of the topologically ordered phases of matter consists of FQHE+QSL. Are these only possible topologically ordered phases of matter?

I ask these because there are several different papers of classifying SETs in 3d, and they seem to answer different questions which is not very clear for me.

For example https://arxiv.org/abs/1212.0835 this paper discusses the classification of SETs in 3D.

Also this paper classifies QSL in 3D https://arxiv.org/abs/1710.00743

However what different questions do they exactly answer? The frist one supposed to classify all of the possible QSL in 3d so why there is a need of second paper?

Also how complete is that second paper?

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Topological order has no symmetry. A classification of 3+1D topological orders for bosonic systems is given in arXiv:1704.04221 and arXiv:1801.08530

FQH states are examples of topological order if we ignore the electron conservation (the U(1) symmetry). But with the U(1) symmetry FQH states are examples of SETs. The two mentioned papers are about SETs with symmetry, not about topological orders without symmetry. A complete classification of 2+1D SETs are given in arXiv:1602.05946 for bosonic and fermionic systems and in arXiv:1410.4540 for bosonic systems.

Two special cases of the results in arXiv:1602.05946 also give rise to classifications 2+1D topological orders (ie without symmetry) for bosonic and fermionic systems.

QSL are very general, and they can realize all bosonnic topological orders in all dimensions (which are of infinite types).

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  • $\begingroup$ Thank for the answer I am a big fan of you, the FQHE does not need symmetry to have a topological order i.e anyonic excitations, GSD etc, but the anyonic excitations are actually the fractionalization of symmetry U(1) the anyons have fractional charge. So symmetry still plays a role. This is a little bit confusing to me. Also if topological order is not tied to symmetry what so ever, how it is classified, I mean in usual A&Z classification we have classifications respect to symmetry and dimensions, so in topological order what the classification is respect to. $\endgroup$ – physshyp Feb 4 at 15:21
  • $\begingroup$ A&Z classification deals with SPT (symmetry protected trivial) orders, which has nothing to do with topological order. Anyon by definition has fractional statistics which does not need symmetry. In the references given in my answer, topological orders are classified by category theory which does not need symmetry. $\endgroup$ – Xiao-Gang Wen Feb 5 at 19:40
  • $\begingroup$ I see, are topological orders in 3d also classified by category theory? $\endgroup$ – physshyp Feb 6 at 0:56
  • $\begingroup$ Yes. By certain 2-categories. $\endgroup$ – Xiao-Gang Wen Mar 13 at 0:36

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