Self-Teaching Topological Insulators/Phases/Symmetry

[TopologicalPassion]

Hey I am a recent graduate with a B.S. in physics and mathematics. The highest physics class I officially took is Quantum Mechanics. I am very interested in learning about topological insulators, topological phases, and topological symmetry, but when I look at papers in the field on Arxiv I can tell that there is a long road ahead. I would like to know the best way to move forward that is most efficient/fastest. If you can suggest sources/a path with lots of problem solving that would be ideal, I tried to get my way through without doing problems and just reading so I can go through the material faster but I realized how futile and silly that is.

 

[radium]

There are several review papers and also several books. The most widely known book is Topological Insulators and Superconductors by Bernevig and the best reviews can be found under arXiv:1008.202 and 1002.3895. There is also a book Field Theories of Condensed Matter physics which is a general book but has parts discussing topological insulators along with the quantum Hall effect and the Haldane model which are both very important to understand topolological insulators.

Also note that when people say topological insulator they usually are referring to “time reversal invariant (TRI) Z₂ topological insulators even though it is really a general term which just describes states gapped in the bulk with conducting surface states. To be more specific, the TRI topological insulator is a symmetry protected state, breaking time reversal gaps the edge states. There are other states like the FQHE which are actually topologically ordered state in that the state is protected against ANY local perturbation.

 

[TopologicalPassion]

There are several review papers and also several books. The most widely known book is Topological Insulators and Superconductors by Bernevig and the best reviews can be found under arXiv:1008.202 and 1002.3895. There is also a book Field Theories of Condensed Matter physics which is a general book but has parts discussing topological insulators along with the quantum Hall effect and the Haldane model which are both very important to understand topolological insulators.

Also note that when people say topological insulator they usually are referring to “time reversal invariant (TRI) Z₂ topological insulators even though it is really a general term which just describes states gapped in the bulk with conducting surface states. To be more specific, the TRI topological insulator is a symmetry protected state, breaking time reversal gaps the edge states. There are other states like the FQHE which are actually topologically ordered state in that the state is protected against ANY local perturbation.

I apologize for the long reply. The first one you gave me "arxiv:1008.202" doesn't lead to anything, perhaps you meant 1008.2026, which is a good review to look over very slowly. I took a look at the 1002.3895 and I like it so far; it looks like a great intro. It probably will take me some time to dissect it because I haven't taken a solid state physics course but I can learn as I go along. I got the book by Bernevig, it's good but some of the material is a bit advanced for me at this point so I will save it for a bit later. Since I took a gap year I am going through quantum mechanics material and doing as many problems as possible with the aim of mastering the material like the back of my hands. Is that something that you would advise?

I found the book by Field Theories of Condensed Matter physics to be very mathematically elegant and interesting, but again I'm going to have to keep an eye on it as I develop some of the prerequisites.