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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Is there a method for differentiating fractional quantum Hall states aside from finding Chern numbers?

+ 7 like - 0 dislike
222 views

The ground state for a quantum Hall system on a torus with fractional filling factor can be classified by the Chern number, which is why the Hall conductance is quantized. Is there another method or classification one can use to distinguish states?

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user reallygoodname
asked Nov 23, 2010 in Theoretical Physics by reallygoodname (35 points) [ no revision ]
A more clear version of this question: Different quantum Hall states can be characterized by their different Hall conductances. Is there other characterizations that one can use to distinguish different quantum Hall states?

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user Xiao-Gang Wen
I propose to change the title of this question to "Is there a method for differentiating fractional quantum Hall states aside from measuring the Hall conductance?"

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user Xiao-Gang Wen

2 Answers

+ 3 like - 0 dislike

In your title, you ask about fractional QHE but the Chern number description as far as I know holds for integer QHE. I don't know that much about FQHE, but let me say a bit about IQHE which I understand a bit better.

The famous paper of TKNN established that the Hall conductivity at integer filling is proportional to a topological invariant associated to the band structure of 2 dimensional Hamiltonians. This topological invariant, the Chern number, is an integer which tells us how the band structure "twists" over the Brillouin zone (this is the "torus" in your question) (more formally, the Chern number classifies the complex vector bundle associated to the band Hamiltonian). Remember for now that this is a property of the band structure, which comes out of a description where electron-electron interactions are neglected, i.e. we are not dealing with a "strongly-correlated system".

Side comment: considering a sort of $Z_2$ equivariant (roughly, time-reversal invariant) version of this topological invariant led to the current extremely hot topic of 3D topological insulators, as initiated by Fu and Kane (among others).

The fractional QHE does not admit a single-particle description -- that is, you can't understand the properties from band theory, as in IQHE -- it is a phase of electron behavior which results from interactions. Thus, I don't think the Chern number description carries over easily.

The FQHE admits a Chern-Simons Landau-Ginzburg description, which I've been reading a bit about in this 1992 review by Shou-Cheng Zhang. The Chern-Simons term in this field theory should not be confused with Chern numbers! (I'm not sure if that's what you're doing in the question, but I want to make that clear.) The notions are related mathematically, but I believe the physics here is distinct.

If you were just asking about IQHE, the insight of TKNN that the IQHE states are classified by a topological invariant probably rules out other independent descriptions. I might be confused as to your intent, but it seems unlikely that there could be a useful description of the IQHE states which doesn't use topology (topological invariant = stable to perturbations, which leads to the amazing plateaus, after all), and the topological situation is really pretty well understood at this point in time.

Please let me know if anything is unclear or if I've said something wrong. I too am just a learner in this field.

I might come back and add some stuff about FQHE if I ever get around to understanding it better.

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user j.c.
answered Nov 24, 2010 by j.c. (260 points) [ no revision ]
From what I understand, Haldane extended the work of TKNN to the FQHE, where the generalized boundary conditions are the adiabatic parameters which deform the hamiltonian. When the ground state is gapped over the full range of the parameters then the state can be classified by a Chern number. I think. I'm still getting my head around this.

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user reallygoodname
Thanks for the comment - do you have a reference for the paper by Haldane, as you know, he's written many :)

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user j.c.
Phys. Rev. B 33, 3844–3850 (1986),Impurity effect, degeneracy, and topological invariant in the quantum Hall effect. is the paper that I'm working from

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user reallygoodname
Thanks, I found that one after I asked you, but I wasn't able to read it carefully yet. My main source for this kind of stuff is actually Xiao-Gang Wen's book and I have been home for Thanksgiving so I didn't get to compare. What I will probably do is ask one of my friends who knows much more about this and get back to you.

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user j.c.
+ 3 like - 0 dislike

Ground states of two-dimensional gapped systems typically obey an entanglement area law, namely, if you compute the entropy $S_A = -\mathrm{Tr}(\rho_A \log_2 \rho_A)$ of the reduced density operator $\rho_A$ on a subsystem $A$, where $A$ is far from any system boundaries, has a smooth boundary, is simply connected, and contractible, then $$ S_A = c |\partial A| - \gamma + \dots $$ where $|\partial A|$ denotes the number of subsystems on the boundary of the region $A$, and $c$ is a constant of proportionality. Here, the conditions "far" and "smooth" can be thought of as relative to the natural length scale of the problem, which is the correlation length, or roughly the inverse of the spectral gap of the Hamiltonian.

The leading order correction to this scaling behavior, $\gamma$, is a topological term which counts the (log of the) number of superselection sectors of the low-energy effective TQFT. (More precisely, the log of the total quantum dimension.) It is a universal term, as far as anyone knows. Different theories with different values of $\gamma$ cannot be mapped to one another by local transformations, so this is a suitable basis for a classification of topological states.

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user Steve Flammia
answered Jun 4, 2012 by Steve Flammia (110 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:
p$\hbar$ysicsO$\varnothing$erflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...