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 , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  A naive question on the Quantum Hall Effect(QHE) and the confinement in gauge theory?

+ 2 like - 0 dislike
1161 views

The non-interacting 2D lattice QH system is described by the Hamiltonian $H=\sum t_{ij}e^{iA_{ij}}c_i^\dagger c_j+H.c$

My confusion is:

Does this imply that the $2D$ lattice QHE is described by the $2+1$ compact lattice $U(1)$ gauge theory? If it is, then according to the general conclusion " the $2+1$ compact lattice $U(1)$ gauge theory is always confinement ", thus, does this mean that the low-energy quasiparticles of $2D$ lattice QHE are not the free electrons?

I'm confused on this point, thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
asked Oct 10, 2013 in Theoretical Physics by Kai Li (980 points) [ no revision ]
I'm a bit confused in so far as in the integer QH effect, the gauge field is a classical, non-dynamical degree of freedom, whereas in 2+1 lattice U(1) gauge theory, I would have guessed that the U(1) gauge field is dynamical and the two systems are thus fundamentally different (at least when you are referring to confinement/deconfinement transition I would presume that you are referring to a BKT like transition, which is definitely linked to the non-trivial topology U(1)/S1 of the dynamical degrees of freedom).

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Jascha Ulrich

1 Answer

+ 4 like - 0 dislike

If you integrate out the fermions in the quantum Hall system (or the Chern insulator), you will end up with an effective $U(1)$ gauge theory, with a Chern-Simons term. The Chern-Simons term is originated from the non-zero Chern number of the occupied fermion bands, and reflects the Hall response of the system. This Chern-Simons term makes a great difference. With out it, the $U(1)$ compact lattice gauge theory is known to be in the confinement phase. But with the Chern-Simons term, a dynamic mass of photon is generated, so that the gauge fluctuation is gap away from the low-energy spectrum, and the theory is deconfined! Therefore the fermions (or quasiparticles) in the quantum Hall system are free.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Everett You
answered Oct 10, 2013 by Everett You (785 points) [ no revision ]
@ Everett You Thanks a lot. So do you mean that as long as the gauge field is gapped, then the corresponding gauge theory is deconfined? And I read Wen's book of section 6.3.2 on page 265, as the title said,"The instanton effect in the (1 + 2)-dimensional U(1) gauge theory gives the gauge boson a finite energy gap and causes a confinement between the U(1) charges", so does this contradict to what you mean? I'm confused....

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
@K-boy The confinement gap and the Chern-Simons gap are different gaps. A gap simply means the system is not critical. You can not tell from the gap whether the gauge theory is confined or not. Due to the Chern-Simons term, the instanton event is accompanied by charge pumping, which causes energy. So the instanton effect is suppressed, and hence the confinement is relieved.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Everett You

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$\hbar$ys$\varnothing$csOverflow
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
...