(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?

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    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
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    @ 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? – Kai Li Apr 23 '13 at 11:43
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    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
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    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
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    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
up vote 1 down vote accepted

(1) For anyons to be created locally in a physical model they must be created in groups such that the local excitation is a boson or a fermion. However, the local excitation can fractionalize into anyonic parts which can propagate independently. In terms of second quantized operators the expectation is that the the local fermionic/bosonic degree of freedom can be written as a product of anyon creation/annihilation operators. This can be explicitly realized in exactly solvable models such as the Toric Code or the Kitaev Honeycomb model. So the answer to whether anyons have creation and annihilation operators is yes.

However, as pointed out by @delete000, we need knowledge of the exclusion statistics to characterize the Fock space of an anyon type. In exactly solvable models this is apparent if there is a direct algebraic fractionalization as just described. But, I don't think there is a complete understanding of exclusion statistics for an anyon given set of fractional quantum numbers so I cannot completely answer part (1) of your question, although there is a recent discussion for the special case of parafermions.

(2) As pointed out in the comments, there are Toric Code models whose quasiparticles are extended operators that realize non-trivial mutual statistics. One good example is the recent exactly soluble 3D models by Lin and Levin realizing realizing braiding between points and loops.

The particle-loop braiding picture is also important for the gapless phase of the 3D variants of the Kitaev Honeycomb model (although gaplessness makes it difficult to identify the anyon in the wavefunction, it exists at the operator level) where the confinement transition takes place at finite temperature on the 3D lattice because the loops need not only to exist, but be very large to lead to the cancellation between spinon paths through about around the loops [cite: 1309.1171 and 1507.01639]. This is unlike the 2D case where a point defect already does the job, making the 2D Kitaev spin liquid unstable to finite temperatures.

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.

  • 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. – Kai Li Sep 19 '14 at 8:26

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