Simulation: tweaking the nanowire¶
We have two choices for your coding assignments of this week. Consider the task complete when you finish one of the two.
This is especially true since both of the assignments constitute a complete paper :)
Tilted magnetic field¶
Explore what happens when we change one of the important knobs of the nanowire model: the external magnetic field. We studied what happens when is pointing along the direction. However, what happens when the magnetic field is tilted?
Generalize the Hamiltonian of the nanowire to the case of a magnetic field with three components . How do the new terms look like?
Go into the nanowire notebook. Modify the nanowire_chain function to include the magnetic field pointing in a general direction.
Plot the band structure for different field directions, and compare to the original case of having only . What changes?
Compare your results with what you find over here:
From to ¶
Now let’s switch to the signatures of Majoranas. The code for these is in the signatures notebook.
How does the -periodic Josephson effect disappear? We argued that we cannot just remove a single crossing. Also, periodicity isn’t a continuous variable and cannot just change. So what is happening?
Study the spectrum of a superconducting ring as a function of magnetic field, as you make a transition between the trivial and the topological regimes.
What do you see? Compare your results with the paper below.
Review assignment¶
As we mentioned, there are really hundreds of papers that use the models and concepts that we used in the lecture.
Here is a small selection of the ones that you may find interesting.
Mourik et al. (2012)¶
Hint: Welcome to the real world.
Wimmer et al. (2011)¶
Hint: Majorana conductance with many modes.
Alicea et al. (2010)¶
Hint: To play a nice melody, you just need a keyboard. This paper first showed how Majoranas in wire networks can be moved around
Lutchyn et al. (2010)¶
Hint: Real nanowires are more complicated.
Bonus: Find your own paper to review¶
Do you know of another paper that fits into the topics of this week, and you think is good? Then you can get bonus points by reviewing that paper instead!
- Osca, J., Ruiz, D., & Serra, L. (2014). Effects of tilting the magnetic field in 1D Majorana nanowires. 10.48550/ARXIV.1403.4464
- Pientka, F., Romito, A., Duckheim, M., Oreg, Y., & von Oppen, F. (2012). Signatures of topological phase transitions in mesoscopic superconducting rings. 10.48550/ARXIV.1210.3237
- Mourik, V., Zuo, K., Frolov, S. M., Plissard, S. R., Bakkers, E. P. A. M., & Kouwenhoven, L. P. (2012). Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. 10.48550/ARXIV.1204.2792
- Wimmer, M., Akhmerov, A. R., Dahlhaus, J. P., & Beenakker, C. W. J. (2011). Quantum point contact as a probe of a topological superconductor. 10.48550/ARXIV.1101.5795
- Alicea, J., Oreg, Y., Refael, G., von Oppen, F., & Fisher, M. P. A. (2010). Non-Abelian statistics and topological quantum information processing in 1D wire networks. 10.48550/ARXIV.1006.4395
- Lutchyn, R. M., Stanescu, T., & Sarma, S. D. (2010). Search for Majorana fermions in multiband semiconducting nanowires. 10.48550/ARXIV.1008.0629