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Is there a pedagogical explanation of what is a topological insulator for those that do not even know what the Berry phase is but have a basic understanding of quantum mechanics and solid state physics?

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    $\begingroup$ If you don't know what a Berry phase is but what to learn what it is, I recommend Bernevig and Hughes' book. It is not a fast explanation, but a very clear and comprehensive one. Quite some typos though. If you want something even faster, Qi and Zhang's review paper or Hasan and Kane's review paper. $\endgroup$ – pathintegral Aug 24 '18 at 15:19
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I think the clear answer is A Short Course on Topological Insulators. There's both a textbook and an arXiv version. I think it's notable for assuming the absolute minimum knowledge and building up from there. It teaches the adiabatic theorem, Berry phase, Chern number, etc all from scratch. It is also notable in being the only textbook on the subject I've seen that doesn't use the second-quantization formalism. The only background necessary is that you understand quantum mechanics, and have seen tight-binding models before.

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I did not start too long ago, but when I did, I used the following resources in the given order. I used the chapters about adiabatic evolution and Berry phase in Griffith's book (they are very short!), then used both this website 2 and this review 3 hand-in-hand for a basic exposition of topological insulators (maybe the section about the Kene-Mele model might be overkill for a very basic overview). These two reads gave me the basic idea, but I still had trouble thinking of things in the bigger picture physically. Then I started working through the book by Shun-Quin 4, along with other lecture slides from the internet (like 5, which has great figures and was written by Haldane, a pioneer in the area). Perhaps, if you are willing to spend the time, starting with Shun-Qing's book might be better for a comprehensive understanding; it starts with most things like Dirac dispersion relations that most other materials take for granted, and helped me fit topological insulators into the general context within the Hall effect.

  1. Introduction to Quantum Mechanics 2nd edition, by Griffiths
  2. Topology in Condensed Matter website
  3. An Introduction to Topological Insulators, by Fruchart and Carpentier
  4. Topological Insulators — Dirac Equation in Condensed Matters, by Shun-Qing Shen
  5. Topology and geometry on condensed matter physics, by Haldane
  6. A Short Course in Topological Insulators, by Asbóth, Oroszlány, and Pályi

Hope this helps!

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  • $\begingroup$ I found the resource that Jahan recommended only last week, and it was pretty good! $\endgroup$ – TribalChief Aug 24 '18 at 15:34
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A more mathematical approach aimed at physicists (and to really understand topological insulators, you do need quite advanced mathematics) is the book by Emil Prodan and Hermann Schulz-Baldes titled “Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics”. Strictly speaking, they only cover two of the 10 Altland-Zirnbauer classes, but they do explain and later define what K-theory is, how to derive and understand bulk-boundary correspondences and so forth. They also give a good account of the state of the art at the time of writing (circa 2015) when it comes to experiments.

Prodan also collaborates with experimentalists, so in my mind much of the book is written in a style that is amenable to physicists.

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On top of the excellent recommendations mentioned above, I'd recommend Berry Phases in Electronic Structure Theory by David Vanderbilt. The book is pedagogically written. It starts with simple explanations in terms of polarization in 1D systems and slowly builds up to introduce the concept of Berry Phase and finally talks about topological insulators.

Another great aspect about the book is that the book gives you simple tight-binding codes to reproduce some of the figures in the book using pythTB package. This was super useful to me in order to visualize the concepts mentioned in the book.

I hope this helps.

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