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17

The realization of non-Abelian statistics in condensed matter systems was first proposed in the following two papers. G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991) Zhenghan Wang and I wrote a review article to explain FQH state (include non-Abelian FQH state) to mathematicians, which include the explanations ...


17

Majorana fermions are fermions which are their own antiparticles. As a result, they only have half the degrees of freedom as a regular Dirac electron. One physical interpretation, at least for Majorana fermion quasiparticles in condensed matter systems, is that they can be thought of a superposition of an electron and hole state. Only Majorana bound states ...


14

The distinction between "ordinary" and topological charges comes from the fact that the conservation of the ordinary charges is a consequence of the Noether's theorem, i.e., when the system under consideration possesses a symmetry, then according to the Noether's theorem, the corresponding charge is conserved. Topological charges, on the other hand, do ...


13

This was originally a comment on Joe's excellent answer, but it got too long. I'm trying to address the question of what φ ⊕ φ means. Suppose you look at the equation φ ⊗ φ ⊗ φ = φ ⊕ φ ⊕ I. What this says is that when you fuse three φ particles, there are two different ways of producing φ, and one way of producing I. The two ways are (a) and (b) ...


11

I put an extra answer, since I believe the first Jeremy's question is still unanswered. The previous answer is clear, pedagogical and correct. The discussion is really interesting, too. Thanks to Nanophys and Heidar for this. To answer directly Jeremy's question: you can ALWAYS construct a representation of your favorite fermions modes in term of Majorana's ...


10

There is a very nice set of lecture notes on the subject by Jiannis Pachos here. (see specifically section 1.3 on fusion and braiding properties). As regards the first question, the tensor product and direct product are basically different ways of divvying up the Hilbert space (see John Baez's illuminating discussion here). When you have a relation like ...


10

Majorana fermions as defined in http://en.wikipedia.org/wiki/Majorana_fermion are really fermions, as its name indicates. So Majorana fermion really have Fermi statistics. It is not proper to say Majorana fermions obey non-abelian statistics, since fermion always obey Fermi statistics by definition. The thing that people said to have non-Abelian statistics ...


9

The thing is that the operation "exchange of two particles" has to be defined properly. What is the meaning of $P$ ? We can imagine the operator $P$ is not physical (in the sense that is does not correspond to a physically possible operation). For instance, $P\psi(x_1,x_2)=\lambda\psi(x_2,x_1)$ in the sense that it only exchange the argument of the ...


9

Local quasiparticle excitations and topological quasiparticle excitations To understand and classify anyonic quasiparticles in topologically ordered states, such as FQH states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations. First let us define the notion of ``particle-like'' ...


9

The point is that the anyons are not electronic states at all. As you've rightly noted, the electrons are fermions, and there's nothing that's going to make them forget that, but very rarely are condensed matter systems so simple that electrons are appropriate degrees of freedom to work with. Instead, the fractional quantum Hall states conjectured to give ...


9

There is no such thing as "Abelian anyonic commutation relations", in the sense that the "Abelian anyonic commutation relations" that you write down does not describe Abelian anyons. So the starting point of the question is not valid. Also anyons do not have a Fock space description. The standard many-body text books stress on Fock space too much, which ...


8

The simple objects in the braided fusion category correspond to the possible particle types. In the simplest important example there are two particle types 1 and $\phi$. (Well, 1 is the vacuum so it's a slightly odd sort of particle type.) The non-simple objects don't have any intrinsic physical meaning, $\phi \oplus \phi$ just means any system "that can ...


8

Let me quote Phys. Rev. B 83, 115132 (2011) The one-dimensional representations of $S_n$ correspond to bosons and fermions. One might have hoped that higher-dimensional representations of $S_n$ would give rise to interesting 3D analogues of non-Abelian anyons. However, this is not the case, as shown in Ref. 18,19: any higher dimensional representation of ...


8

How to obtain this braiding matrix from Non-Abelian Chern-Simon theory? To obtain braiding matrix $U^{ab}$ for particle $a$ and $b$, we first need to know the dimension of the matrix. However, the dimension of the matrix for Non-Abelian Chern-Simon theory is NOT determined by $a$ and $b$ alone. Say if we put four particles $a,b,c,d$ on a sphere, the ...


7

The dichotomy of bosonic and fermionic behaviour essentially arises because of the nature of the rotation group in dimensions greater than 2+1. The exchange of 2 particles (which determines the statistics) introduces the same phase that you get when you rotate a particle by 2pi - this really comes from the spin-statistics theorem that tells you that ...


6

If I remember correctly, isomorphism classes of simple objects correspond to different types of particles (which is assumed to be finite), furthermore more is structure is usually needed than a fusion category, for example braiding (which is the reason why anyons are so interesting). Let me be very concrete. A physically (and experimentally) relevant ...


6

I feel that I finally understand the physical meaning of composite (ie non-simple) objects like $\phi\oplus\phi$. It is explained in the section II of my paper with Tian Lan arxiv.org/abs/1311.1784 . We know that putting a few anyons (ie the objects in tensor category) on a Riemann surface may generate degenerate states (ie the fusion space of the objects ...


5

Not sure about the condensed matter context, but in general the answer is NO. For instance, quarks have fractional charge but are regular fermions.


5

Let me start from the beginning, also making explicit what I understand from user35388's answer. Consider a system of a couple of identical particles. Their states are pictured by normalized wavefunctions $\psi(x,y)$. However this representation is not one-to-one (this holds for every quantum system): states are wavefunctions up to phases. That is $\psi$ ...


5

The way Shankar addresses the problem (pg. 278) is by introducing an "Exchange Operator" $P_{1,2}$, which would swap your two particles as follows: $P_{1,2} |\xi_1, \xi_2 \rangle = |\xi_2, \xi_1 \rangle$ I like the operator notation because it makes it clear (to me, at least) that applying the operator twice is just the identity operator, since swapping ...


5

You are right. In a space-time with one time dimension and $D$ spatial dimensions, finding possible different statistics is equilalent to look at the fundamental group (first homotopy group) of $SO(D)$ For $D=1$, the fundamental group is trivial. For $D=2$, the fundamental group is $\mathbb{Z}$. For $D>=3$, the fundamental group is $\mathbb{Z}_{2}$ ...


4

The (unitary) "phase" factor for non-Abelian anyons satisfies the (non-Abelian) Knizhnik-Zamolodchikov equation: $$\big (\frac{\partial}{\partial z_{\alpha}} + \frac{1}{2\pi k} \sum_{\beta \neq \alpha} \frac{Q^a_{\alpha}Q^a_{\beta}}{z_{\alpha} - z_{\beta}}\big )U(z_1, ....,z_N) = 0 $$ Where $z_{\alpha}$ is the complex plane coordinate of the particle ...


4

Ady Stern, Anyons and the quantum Hall effect - a pedagogical review, arXiv:0711.4697 is a gentle introduction.


4

Both fractional/non-Abelian statistics and fractional charges come from the same origin: long-range entanglements. This is why fractional/non-Abelian statistics common for fractional charges. One way to realize long-range entanglements is through the string-net liquid phase of a pure bosonic model. The ends of strings in string-net liquid are non-local and ...


4

Please, let me first refer you to the original paper by Leinaas and Myrheim where the existence of anyonic statistics was first predicted before its actual discovery. All the ingredients of the understanding of the special properties of the two dimensional case already exist in this old paper, however, I'll try to cast it in a more modern terminology: By ...


4

My question is: What is the effect of braiding of , for example, anyon 1 with anyon 2 on the dimension? What is the effect of a twist? Short answer: None. Consider that for anyons $N_{ab}^c=N_{ba}^c$ and that twisting is really just a braiding with some special stuff. Longer answer: In order for this to make sense, we have to dig a little deeper and clear ...


3

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.


3

The best way to answer the question "How are anyons possible" is to use the "dynamical" path integral formalism, rather than the "static" wave function formalism. The permutation group action on the wave function is "static" in the sense that only initial and final states are specified. It will be ambiguous if there are more than one non-equivalent ways to ...


3

I don't think that is the case. A useful reference is: http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.80.1083. One way to approach a theory of anyons is to start by writing down a list of particle types along with their fusion rules. Once doing this one may obtain consistency equations from solving the hexagon and pentagon equations arising from ...



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