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
5,082 questions , 2,232 unanswered
5,353 answers , 22,788 comments
1,470 users with positive rep
820 active unimported users
More ...

  Approximation scheme in Kitaev's paper "Unpaired Fermions in 1D Wires" (2000)?

+ 3 like - 0 dislike
828 views

Kitaev's paper Unpaired Majorana Fermions in 1D Quantum Wires (https://arxiv.org/abs/cond-mat/0010440) is famous as a promising experimental proposal for realizing topologically-robust zero-energy fermionic quasiparticles in standard superconductor-semiconductor heterostructures. However, after reading the paper many times through, the reader gets the impression that Kitaev does not give a rigorous mathematical origin of the "effective low-energy Hamiltonian"

$$ H_{eff} = tb'b''  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~(15)$$

which Kitaev writes down in equation (15). The assertion of this effective Hamiltonian is central to the arguments of the paper, and is ultimately used to characterize and show the existence of Majorana Zero Modes (MZM's) in the proposed system. This effective Hamiltonian is only approximate, because the exact solutions of the Bogoliubov-deGennes equations for (15) are not solutions of the Bogoliubov-deGennes equations for the full Hamiltonian. 

This confuses me, so I wonder: is there a standard approximation scheme which is being invoked here? Perhaps a Schrieffer-Wolff transformation?  My personal hunch: Kitaev's note regarding the physical intuition for $t$ as the "tunneling amplitude for a quasiparticle to tunnel across the chain" smells a lot like discrete WKB is being invoked here. Indeed, in the WKB method, the scaling of the energy gap between low-energy states (spatially separated by a large energy barrier) is proportional to the scaling of the overlap of their corresponding wavefunctions. However, I am not yet able to make this statement any more precise.   

asked Aug 6, 2017 in Theoretical Physics by David B Roberts (135 points) [ revision history ]
edited Aug 6, 2017 by David B Roberts

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$\hbar$ysicsOverf$\varnothing$ow
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
...