Mathematical theory for topological insulators

[taishizhiqiu]

I have been learning topological insulators recently, and I become more and more curious about the link between topological insulators and mathematical theory these days.

I know topological insulators have something to do with fiber bundles and K-theory. I have a relatively good background of undergraduate topology and recently I have read some introduction about fiber bundles and K-theory. What is missing in my mind is the link between math and physics. That is, what exactly do we regard as fiber bundles and classify them as trivial and non-trivial?

Can someone kindly give me the answer or some references?

 

[radium]

Look at the books by Fradkin, Bernevig/Hughes, and Wen.

When you talk about TIs or the IQHE/FQHE, the nontrivial topology is in the Bloch wave function. You are looking at the Berry phase of this wave function and the fact that the wf must be single valued gives you some quantization condition. For the QHE you get the chern number, for TIs you get a Z2 invariant. If the system is topologically nontrivial, you cannot define a wf that is valid globally. This is the same idea as the Dirac monopole or Aharanov Bohm effect

 

[taishizhiqiu]

Look at the books by Fradkin, Bernevig/Hughes, and Wen.

When you talk about TIs or the IQHE/FQHE, the nontrivial topology is in the Bloch wave function. You are looking at the Berry phase of this wave function and the fact that the wf must be single valued gives you some quantization condition. For the QHE you get the chern number, for TIs you get a Z2 invariant. If the system is topologically nontrivial, you cannot define a wf that is valid globally. This is the same idea as the Dirac monopole or Aharanov Bohm effect

Can you give me names of the books? I can't find them on google.

 

[radium]

Field theories of condensed matter, topological insulators and topological superconductors, and quantum field theory of many body systems: from the origin of sound to an origin of light and electrons. The first two have chapters explicitly on Z2 TIs and TSCs. The second has a chapter or two on the IQHE and FQHE and a chapter about topology in condensed matter. It does not explicitly discuss Z2 TIs but does discuss things you should know about them.

In general, the best way to learn about these states is to start from the integer QHE, then go to the Haldane model for graphene, then go to the KM model in graphene (it is good to read that paper and a few of the ones after that) then go to 3D TIs/FKM model. Then you can learn about TSCs (the mapping from the quantum ising model to the majorana chain is very important).

If you want to learn about the FQHE, I would save that for last, it is incredibly subtle.