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 ...

  What is the "physical" or "experimental" meaning of fusion rule / multiplicity $N_{ab}^c$?

+ 7 like - 0 dislike
1918 views

In 2-dimensional topologically ordered system with anyon excitations, we write the fusion of a anyon $a$ with another anyon $b$ outputting several possible outcomes, say anyons $c$. And there is a mysterious integer weight in the front of $c$, called the fusion rule or the fusion multiplicity

$N_{ab}^c$: $$a \times b = \sum_c N_{ab}^c c$$

  • question 1: What are the physical meanings or physical observables for $N_{ab}^c$?
  • question 2: does $N_{ab}^c$ say anything about the probability of having the anyon $c$ by fusing $a$ and $b$? Say, if $a \times b = N_{ab}^{c_1} c_1+ N_{ab}^{c_2} c_2$, then do $N_{ab}^{c_1}$ and $N_{ab}^{c_2}$ represent the probability weight of having a output anyon $c_1$ or $c_2$? Please try to answer this question from a experimental quantum physicist's perspective. Make the answer down to the earth. Thanks. :-)
asked Dec 25, 2014 in Theoretical Physics by RKKY (325 points) [ revision history ]
Kitaev and Preskill have a nice application of these topics in computing the topological entanglement entropy of a ground state whose effective TQFT is known. http://arxiv.org/abs/hep-th/0510092

2 Answers

+ 6 like - 0 dislike

Question 1: Before understanding "What is the physical or experimental meaning of fusion rule/multiplicity $N^c_{ab}$?", one needs to undertstand what is topological excitation and what is the fusion space. The section II of our paper http://arxiv.org/abs/1311.1784 has an explaination of all those notions.

Question 2: The setup that leads to the probability is not specified, and the question is hard to answer. I guess in one setup, the probability should be proportional to $N^c_{ab}d_c$ where $d_c$ is the quantum dimension of $c$. 

answered Dec 25, 2014 by Xiao-Gang Wen (3,485 points) [ revision history ]
edited Aug 8, 2015 by Xiao-Gang Wen
+ 4 like - 0 dislike

For question 2, consider a setup as follows: from the vacuum create a pair of $a$ and $\bar{a}$, and a pair of $b$ and $\bar{b}$. Then fuse $a$ and $b$ and measure the resulting topological charge. The probability of finding $c$ is $\frac{N_{ab}^c d_c}{d_ad_b}$. I think this is the setup Professor Wen had in mind in his answer.

answered Aug 9, 2015 by Meng (550 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$\hbar$ysics$\varnothing$verflow
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
...