Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

(1) As we know, we have theories of second quantization for both bosons and fermions. That is, let $W_N$ be the $N$ identical particle Hilbert space of bosons or fermions, then the "many particle" Hilbert space $V$ would be $V=W_0\oplus W_1\oplus W_2\oplus W_3\oplus...$ , and further we can define creation and annihilation operators which satisfy commutation(anticommutation) relations for bonsons(fermions).

So my first question is, do we also have a "second quantization theory" for anyons like bosons and fermions?

(2) Generally speaking, anyons can only happen in 2D. Is this conclusion based on the assumption that the particles are point-like?

In Kitaev's toric code model, the quasiparticles are point-like due to the local operators in the Hamiltonian. My question is, in 3D case, whether there exists a simple model whose Hamiltonian contains local operators and spatially extended operators, so that it has both poit-like quasiparticles(say, $\mathbf{e}$) and knot-like quasiparticles(say, $\mathbf{m}$), then the $\mathbf{e}$ and $\mathbf{m}$ particles have nontrivial mutual statistics in 3D?

share|improve this question
2  
Regarding (1), anyon means that the wavefunction describing a two- or many-anyon state picks up a phase other than a multiple of $\pi$ upon exchange of two particles. To define a Fock space (i.e. many-particle Hilbert space), one needs to attach exclusion statistics to the constituent particles, which will act as a generalized Pauli principle. If you do that, then yes, you can devise a second-quantized theory of anyons, because the occupation-number basis would be well defined. –  delete000 Apr 23 '13 at 3:37
1  
@ delete000 Thanks a lot. Do you mean the exclusion statistics is common to anyons(except bosons) like fermions? Do you have some related articles ? And do physicists now have a well developed theory of second quantization for anyons? –  K-boy Apr 23 '13 at 11:43
2  
Each type of anyons will have a different exclusion principle, which can be something between the Pauli principle (1 particle per state) and bosons (any number of particles per state). You may find this article useful: iopscience.iop.org/0305-4470/27/11/009 –  delete000 Apr 23 '13 at 13:32
1  
Regarding your last point, there are indeed generalizations of Kitaev's toric code model with pointlike and stringlike quasiparticles. –  Peter Shor Apr 27 '13 at 10:56
3  
This paper of Beni Yoshida proves theorems about a broad class of 2- and 3-dimensional generalizations of Kitaev's toric code. You may also want to look at the citations for earlier papers about these models. There are also a few papers about more general codes (corresponding to non-abelian anyons), but I can't easily locate these right now. –  Peter Shor Apr 27 '13 at 20:09

1 Answer 1

No, “this conclusion” is based on the topological properties of rotation groups. Namely, for any $n > 2$ $\mathrm{Spin}(n)$ is the universal cover of $\mathrm{SO}(n)$, whereas for $n = 2$ it is not. That’s why in $n > 2$ any thing has to be controlled by a representation of the Spin group, whereas in $n = 2$ it has not.

share|improve this answer
    
I'm sorry that I just saw your answer, thank you. I can not understand your explanation immediately, but I just found a related article. –  K-boy Sep 19 at 8:26

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.