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 ...

  Realization of Witten-type topological quantum field theory in condensed matter physics

+ 7 like - 0 dislike
2049 views

It is well-known that some exotic phases in condensed matter physics are described by Schwarz-type TQFTs, such as Chern-Simons theory of quantum Hall states. My question is whether there are condensed matter systems that can realize Witten-type TQFTs?

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Tengen
asked Nov 6, 2012 in Theoretical Physics by Tengen (105 points) [ no revision ]
Wikipedia seems to only give one example of what a Witten-type TQFT is, and that is the WZW model which appears many places in condensed matter physics. I always thought that Witten type TQFT's are cohomological field theories, where there exist an operator $Q^2=0$ (so you have susy or BRST structure) and so on. I haven't seen any of these type theories in condensed matter physics

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Heidar
The type of TQFT's that Atiyah's axioms capture (where its a functor between certain appropriate categories) are the ones that usually appear in condensed matter context (at least in toy models). According to wikipedia, these include the Schwarz-type TQFT's but it is not clear that they describe the Witten-types fully.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Heidar
@Heidar: That wikipedia article is wrong. The WZW model is not a topological field theory. It's conformal, but that's not the same thing.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user user1504
@user1504 This is also the first time I have seen the WZW being called a topological field theory. The theory is gapless and has a lot of dynamical degrees of freedom, so its quite hard for me to see in what sense it could be topological.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Heidar
It's simply not a topological field theory, and especially not one of Witten class. Whoever edited that wikipedia article didn't know what they were talking about. The WZW model isn't supersymmetric; it doesn't have a nilpotent supercharge. (It's intimately related to Chern-Simons theory, but that's a different story, and not relevant here.)

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user user1504
It will be very helpful to explain what is "Witten-type TQFTs"? At least to indicate if the theories have an energy gap, if the theories have supersymmetries, in which space-time dimensions, what are their Hamiltonian or Lagrangian, etc

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Xiao-Gang Wen

2 Answers

+ 4 like - 0 dislike

The TQFTs that Witten introduced are those obtained by a topological twist of a supersymmetric field theory. This includes notably the A-model and the B-model TQFTs.

Despite what seems to be suggested in the comments here and on Wikipedia, these are also "Schwarz type" (come from the Poisson sigma-model) and they do have a desciption in terms of functorial TQFT if only one allows what are called (infinity,1)-functors: they are "TCFTs" (i.e. non-compact 2d homotopy TQFTs).

Now, under homological Mirror symmetry these are related to other TCFTs known as Landau-Ginzburg models. And these do have applications in solid state physics.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Urs Schreiber
answered Jul 5, 2013 by Urs Schreiber (6,095 points) [ no revision ]
I very much appreciate this answer because it actually defines what the question is asking about. But these Landau-Ginzburg models appear how in solid state? As fixed points of disordered systems in two dimensions?

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user BebopButUnsteady
Well, the Landau-Ginzburg theory is an old model for superconductivity and the behaviour of its potential term is what gives the Landau-Ginzburg model its name. But yeah, otherwise the relation is not super-close, I agree.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user Urs Schreiber
That's just a linguistic coincidence then... These models are by necessity supersymmetric and conformal (and even dimensional)? - the only place in solid state where I know of supersymmetric methods is in disordered systems/random matrices. If you don't know of a particular example, then that is fine, I just wanted to clarify.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user BebopButUnsteady
+ 0 like - 0 dislike

I don't think there are any Witten-type TQFTs which are directly relevant to condensed matter physics. Witten-type TQFTs are very strange beasts: they violate spin-statistics, they aren't unitary, etc. It'd be pretty tricky to find a physical system you could model with one in the usual way.

There are some indirect connections between Chern-Simons theory and Gromov-Witten theory, but that's all I can think of.

This post imported from StackExchange Physics at 2014-04-05 04:33 (UCT), posted by SE-user user1504
answered Nov 6, 2012 by user1504 (1,110 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$\hbar$ysi$\varnothing$sOverflow
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
...