• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Topologically proctected twist in the wave function (TKNN invariant)

+ 1 like - 0 dislike

In the famous TKNN paper and subsequents the authors wisely argue that the transversal conductance in the Integer Quantum Hall Effect has a topological interpretation as the integral of the curvature of the Berry connection on the Brillouin zone which happens to coincide with the first Chern number for a $U(1)$ principal bundle on a manifold. In particoular Kohmoto justifies the quantization of the TKNN invariant with the following argument:

let's $\psi_{\mathbf{k}} ( \mathbf{x} )$ be the wave function in the Brillouin zone associated with $\mathbf{k}$ quasi-momentum. He choses a section of the bundle such that $\psi_{\mathbf{k}} ( \mathbf{x}_0 )$ at fixed $\mathbf{x}_0$ is real. This of course doesn't work at the points where the wavefunction nullify. So he encircles them and defines the connection $A$ smoothly inside and outside the circles, with different prescriptions. He then applies Stoke's theorem to this situation getting a sum of line integral of the difference between the connection inside and outside the circles encompassing the singularities:

\sigma_{xy} = \sum_{i} \int_{\partial S_i} ( \mathbf{A}_{out} - \mathbf{A}_{in} ) \cdot \mathbf{d l}  

The difference of the connections is due to the discontinous choice of the phase: 

\psi_{\mathbf{k}}^{out} = e^{i f(\mathbf{k})} \psi_{\mathbf{k}}^{in}

$e^{i f(\mathbf{k})}$ needs to be single valued but $f(\mathbf{k})$ needn't! $e^{if}$ is basically the transition function that defines the principal bundle. If the function $f$ is differentiable then we can write a gauge transformation for the potential $A$ which looks like:

\mathbf{A}_{out} = \mathbf{A}_{in} + \nabla f

then the total conductivity becomes:

\sigma_{xy} = \sum_{i} \int_{\partial S_i} ( \nabla f) \cdot \mathbf{d l}  

now the requirement for the phase difference to be well defined fixes each integral to be $2 \pi n$. 

I've alway forgetten constants in my derivations, so don't mind the extra $2 \pi$ that happears. Here we come to my questions:

Is the smoothness hypothesis on $f$ well motivated? I think so because one can smoothly choose the phases of the wave function in the inner and outer regions. Then the difference will be smooth. 

This arguments shows the quantization of the integral but don't show the stability under smooth variation of $\psi_{\mathbf{k}}$ which I expect indeed to be true. Can we see it with basic notions? Basically I want to understand how the twists in the wavefunction are protected against variations of the Hamiltonian.

Please, I don't know advanced maths and I cannot follow real mathematical general motivations one can find on topology books. I was looking for a more down to earth justification. 

Thanks in advance!

asked May 25, 2018 in Theoretical Physics by MrRobot (20 points) [ no revision ]

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:
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