Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Topology and Majorana bound states

+ 5 like - 0 dislike
800 views

I'm working at the moment on Majorana Bound states and their topological properties. Now I have a question about it. The Altland-Zirnbauer symmetry classes says us how many topological different ground states the system have just due to the presence or absence of symmetries and the spatial dimension.

Further, in one-dimensional systems with only particle-hole symmetrie (class D) emerges Majorana Bound states at the ends. I can this obtain by solving the Bogoliubov-de Gennes equation.

My question is there a "topological" method to obtain whether Majorana bound states emerges in the system in general? The Altland-Zirnbauer classes are the "Can Be"-condition, but at the present to prove the presence of Majorana bound states is to solve the Bogoliubov-de Genes equations for a special system and not in general.

This post imported from StackExchange Physics at 2014-03-06 21:59 (UCT), posted by SE-user Lars Milz
asked Dec 10, 2013 in Theoretical Physics by Lars Milz (45 points) [ no revision ]

1 Answer

+ 1 like - 0 dislike

You misunderstood the classification I believe. Let's take an example.

In class D and 1D, the classification tells you there are two possible vacua (you understood this apparently). This is the famous $\mathbb{Z}_{2}$ ensemble in the classification. Next the classification tells you also that: at the boundary between the two gapped vacua, a Majorana mode emerges.

More precisely, an evanescent, localised mode emerges at the boundary between the two vacua, since they are gapped. For superconductors this emergent mode is a Majorana one, thanks to the particle-hole symmetry.

The construction is topological in essence: you map the problem of your differential equation to a group language, you recognise some properties (the Cartan class for instance, above the D one) which allow you to classify your problem (using equivalence relation, the more complicated part being to choose relevant criterion).Then these classes have additional properties, like the boundary term discussed above. To be a little bit more clear, it is topological because you want to understand how the local solutions to your problem are gluing to some other local ones in order to make global ones. Think about the Möbius stripe that is locally differentiable, but not globally (check out fiber bundle also, this is the mathematical object describing the property you're looking for). For the class D in 1D, you can find the two solutions in both vacua, but you can not glue them continuously without making a Majorana, continuity here means making a global continuous solution, or wave-function.

Nevertheless, the topological property tells you that something strange exists, it will never give you the wave-function. You can think about group classifications in quantum mechanics: the groups of your molecule/lattice-cell tell you which interaction terms exist, they never give you their strength (or the energy band splitting if you prefer to see the problem that way). A microscopic calculation is always required to get microscopic details. Now, there are some tricks: since the characteristic length inside a superconductor is the coherence length $\xi=\hbar v_{F}/\Delta$ ($\Delta$ gap parameter, $v_{F}$ Fermi velocity), you can think as a decaying wave-function as $\Psi_{\text{Maj.}}\sim e^{-x/\xi}$ at the $x=0$ interface. This is cheating because this is not a wave-function, but it nevertheless gives the correct estimate for the real wave function which should be something like $\Psi_{\text{Maj.}}\sim e^{-x/\xi}\sin k_{F}x$ for instance ($k_{F}$ Fermi wave-vector).

Next, some limitations of the classification:

  • it works for non-interacting systems only (the Coulomb interaction is not taken into account for instance)
  • it works for pure clean system only (sometimes this issue is not trivial: for (so-called topological) $p$-wave superconductor the superconductivity itself is destroyed by disorder; so perhaps the topological properties are conserved below the gap, but the gap vanishes due to disorder...)

so in practise it can never be applied to condensed matter problem. Some people infer the topological property to avoid discussing these points. I think these points are the most relevant one, though.

This post imported from StackExchange Physics at 2014-03-06 21:59 (UCT), posted by SE-user Oaoa
answered Mar 5, 2014 by Oaoa (50 points) [ no revision ]
Some simple expressions for the wave function of the Majorana mode at the interface between a trivial vacuum (without particle) and the $p$-wave superconducting (then non-trivial) vacuum are given in arxiv.org/abs/1306.2519 in the quasi-classical approximation for a single interface and a $p$-wave wire of finite length, check the appendix of the paper.

This post imported from StackExchange Physics at 2014-03-06 21:59 (UCT), posted by SE-user Oaoa
Thanks for the answer.

This post imported from StackExchange Physics at 2014-03-06 21:59 (UCT), posted by SE-user Lars Milz

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOv$\varnothing$rflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...