Thank you alot. In fact my Goal is to understand Kitaev model. But i am afraid if i should have some basics before start reading. I know Quantum Mechanics, second quantization... But i am wondering if i need further things.DeathbyGreen said:A great place to start is a review by Hasan and Kane on topological insulators:
https://arxiv.org/pdf/1002.3895.pdf
A more theory based introduction can be found here:
https://arxiv.org/pdf/1608.03395.pdf
If you're looking for interesting introductions, the 1D spinless fermion chain with p-wave superconductivity by Kitaev is a good model to see the dynamics. A good professor to look at did some notes here:
https://arxiv.org/pdf/1206.1736.pdf
And you can look at all the references those papers cite to continue down the rabbit hole.
shiraz said:Thank you alot. In fact my Goal is to understand Kitaev model. But i am afraid if i should have some basics before start reading. I know Quantum Mechanics, second quantization... But i am wondering if i need further things.
Thank you a lot for your help
Really Thank you. I will do that sure. Good luck in your researchDeathbyGreen said:No problem! As long as you have a background with some second quantization you should be fine. If you really want to understand the kitaev model, I would start with the original paper:
https://arxiv.org/abs/cond-mat/0010440
Then, maybe work through a little project:
1. start with the real space Hamiltonian and Fourier transform into momentum space, using periodic boundary conditions in x and y; perform a Bogoliubov transformation and diagonalize to get an expression for the dispersion relation. Then try to find:
- The spectrum E(k)
- The density of states D(E)
- The wavefunctions (eigenvectors)
3. Now make the chain finite (use the real space model) and solve numerically (simple MATLAB eig(H) type function will do the trick). You should find
- The spectrum E(k)
- The density of states D(E)
- The wavefunctions
4. Reflect on the comparison between the two cases.
5. Also take a look at the review
https://arxiv.org/pdf/1202.1293.pdf
and use it's suggestions to calculate the Chern number, which will give you some insight into topology. The best way to learn this stuff is to really push through the math!
A topological superconductor is a type of material that exhibits both superconducting and topological properties. This means that it can conduct electricity without resistance and also have unique surface properties that are protected from external disturbances.
Traditional superconductors are characterized by the formation of Cooper pairs, which are two electrons bound together to carry electrical current without resistance. In topological superconductors, these Cooper pairs also have a topological property called non-Abelian statistics, which could potentially be used for quantum computing.
Topological superconductors have potential applications in quantum computing, as mentioned before, as well as in creating qubits for quantum information storage and processing. They could also be used in creating more efficient power transmission lines and high-speed electronic devices.
An introductory textbook on topological superconductors should cover the basics of superconductivity and topology, as well as their intersection. It should also include information on current research and potential applications. Visual aids, such as diagrams and illustrations, can also be helpful in understanding the concepts.
Yes, there are several online resources available for learning about topological superconductors, including lecture notes and videos from universities, scientific articles, and online courses. Some recommended textbooks include "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and "Topological Superconductivity" by B. Andrei Bernevig and Taylor Hughes.