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 features!

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

123 submissions , 104 unreviewed
3,547 questions , 1,198 unanswered
4,552 answers , 19,366 comments
1,470 users with positive rep
411 active unimported users
More ...

Fractionalization and the structure of spin rotation group?

+ 4 like - 0 dislike
35 views

As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the quasiparticals must be created or annihilated by pair.

On the other hand, consider the groups $SU(2)$ and $SO(3)$, they are the rotation groups for half-integer and integer spins, respectively. And we know that $SU(2)/\mathbb{Z}_2=SO(3)$, which means that each element in $SO(3)$ can be viewed as one pair $(U,-U)$, where $U\in SU(2)$ (otherwise put: the coset $\left\{U, -U\right\} \subset SU(2)$ in the quotient group $SU(2)/\mathbb{Z}_2$ is our element in $SO(3)$).

So I wonder that whether is there any underlying connection between the pair nature of quasiparticals in topological phase in physics side and the pair structure relating $SU(2)$ and $SO(3)$ in mathematics side?

Thank you very much.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user K-boy
asked Aug 8, 2013 in Theoretical Physics by Kai Li (975 points) [ no revision ]
Dear K-boy: I added a phrase about the coset $\left\{U, -U\right\}$ - just another way to say the "one pair in SU(2)" thing, which some readers might find a little obtuse. Group theorists will likely know what you are talking about, but spelling it out might make it clearer for the rest of us. Delete it if you feel it changes the sense, but it's nice if a wider audience can understand the question, even if they can't answer it.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance
The phenomenon you mentioned is called "symmetry fractionalization" (arxiv.org/abs/1012.4470), i.e. the symmetry of the system is $SO(3)$, but the quasiparticle carries fractionalized symmetry (or projective representation) $SU(2)$. This phenomenon belongs to the symmetry enriched topological (SET) order, classified (partly) by the group cohomology $H^2(SO(3),\mathbb{Z}_2)=\mathbb{Z}_2$, meaning that the non-trivial quasiparticles can be trivialized in pairs.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user Everett You
@ Everett You, thanks for your comment. What about the AKLT phase for spin-1 chain? It's a symmetry protected topological phase, and is it also a SET phase?

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user K-boy
@K-boy Yes, the physics is very much the same as AKLT chain (and AKLT chain is a SPT not a SET phase). But when you talk about "fractionalization", you mean something is broken apart and its pieces must be DECONFINED. In the AKLT chain, the spin-1/2 objects are confined (as 1+1D gauge theory always confining) to the end of the chain and can not move freely in the system. Moreover, if you close the AKLT chain, then no spin-1/2 excitations actually exist in the bulk. So you seem to break spin-1 into spin-1/2's, but they then confined back to spin-1, and we should not call it a fractionalization.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user Everett You
@K-boy So "symmetry fractionalization" actually means you can not only break spin-1 into spin-1/2's, but also the spin-1/2 excitations are defined in the bulk. Only in this case, we call it a "successful" fractionalization. And such senario can only be achieved in 2+1D or higher dimensions. In 2+1D, Yao, Fu and Qi (arxiv.org/abs/1012.4470) constructed a AKLT loop liquid state with exact solvable models. In that state, the deconfined spin-1/2 excitations arise in the spin-1 system, and the $\mathbb{Z}_2$ topo. order coexists with the fractionalization. So it is actually a SET phase.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user Everett You
@K-boy In fact, all "successful" fractionalization must come along with topological order, otherwise you can not gauge away the unphysical degrees of freedom arise in the fractionalization. Then it is not hard to understand why symmetry fractionalization is actually SET.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user Everett You
@ Everett You, I see, thank you very much.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user K-boy

1 Answer

+ 4 like - 0 dislike

The group of rotations of an $N$-dimensional space is $SO(N)$. Being a symmetry of nature, classical systems transform according to representations of $SO(N)$.

Quantum mechanics, on the other hand, allows systems which transform according to the universal covering groups of classical symmetries. This is the reason why we get in three dimensional quantum theory representations of $SU(2)$ which are not true representations of $SO(3)$, (the half integer spin representations). More generally, we have, in quantum theory, representations of $Spin(N) = SO(N) \ltimes \mathbb{Z}_2$.

However in the case of a two spatial dimensions, $SO(2) \cong U(1)$, and the universal covering of $U(1)$ is not $Spin(2)$ but rather $\mathbb{R}$.

In contrast to $SO(2)$ or $U(1)$ which allow discrete values of the two dimensional spin: $ u = e^{i n \theta}$ $n \in \mathbb{Z}$, $0\le \theta <2 \pi$, the universal covering $\mathbb{R}$ allows a continuum of spin values.

This is the basic reason of the fractionalization of spin in two dimensions.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user David Bar Moshe
answered Aug 8, 2013 by David Bar Moshe (3,505 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$\hbar$y$\varnothing$icsOverflow
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
...