# Tag Info

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In order: Because the term "gauge symmetry" pre-dates QFT. It was coined by Weyl, in an attempt to extend general relativity. In setting up GR, one could start with the idea that one cannot compare tangent vectors at different spacetime points without specifying a parallel transport/connection; Weyl tried to extend this to include size, thus the name ...

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The (big) difference between a gauge theory and a theory with only rigid symmetry is precisely expressed by the Noether first and second theorems: While in the case of a rigid symmetry, the currents corresponding to the group generators are conserved only as a consequence of the equations of motion. This is called that they are conserved "on-shell", in the ...

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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 ...

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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) ...

13

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 ...

11

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 ...

10

I think you need to define what you mean by a "topological state of matter", since the term is used in several inequivalent ways. For example the toric code that you mention, is a very different kind of topological phase than topological insulators. Actually one might argue that all topological insulators (maybe except the Integer Quantum Hall, class A in ...

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

In a "fully defined" TQFT the spaces of states are necessarily finite dimensional. This follows simply from the fact that the correlators assigned to the cap and the cup cobordism (the "2-point functions") equip the space of states with the structure of a dualizable object in the corresponding monoidal category of vector spaces, which are precisely the ...

8

1) Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc? Gauge symmetry is a local symmetry in CLASSICAL field theory. This may be why people call gauge symmetry a local symmetry. But we know that our world is quantum. In quantum system, gauge symmetry is not a symmetry, in the sense ...

8

First, the full paper is here: http://citeseer.ist.psu.edu/viewdoc/download;jsessionid=807BE383780883ACB4CAB8BD48E8C90B?doi=10.1.1.128.1806&rep=rep1&type=pdf Second, the paper has 150 citations. See all this information at INSPIRE (updated SPIRES): http://inspirebeta.net/record/278923?ln=en Third, the text between 3.4 and 3.5 looks ...

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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 ...

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

Goals one wants to achieve with those two theories are similar. We know that superstring theory is a potential theory of everything. One may want to ask what is the difference between the string-net-liquid approach and the superstring approach? Our understanding of the superstring theory has been evolving. According to an early understanding of the ...

7

RVB states were first coined in 1938 by Pauling in the context of organic materials and they were later extended to metals. Anderson revived the interest in this concept in 1973 when he claimed that they explained the Mott insulators. (Mott, not Matt and not Motl, which is a shame because I was born in 1973.) He wrote a new important paper in 1987 in which ...

7

It's not the making as opposed to verifying of topological superconductors that is difficult experimentally. One of the most useful techniques in identifying topological properties of a material is Angle-Resolved Photoemission Spectroscopy (ARPES). ARPES can independently image the bulk and surface modes of a 3-D solid with very good energy and momentum ...

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Recently, it is realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states. The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher ...

6

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

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 ...

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Here are some comments on the points: 1) I(nteger)QHE occurs due to the presence of Landau levels Yes 2) IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone IQHE is an example of topological order, although ...

5

Dijkgraaf and Witten used $\mathcal H^3[G,U(1)]$ to define CS theory for gauge group $G$. Recently, group cohomology has found applications in condensed matter physics. It may classify the so called "symmetry protected topological phases" of interacting bosons: The $d$-dimensional symmetry protected topological phases of interacting bosons with symmetry ...

5

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 ...

5

This is a very good question. Let me give a little back ground first. For a long time, physicists thought all different phases of matter are described by symmetry breaking. As a result, all continuous phase transitions between those symmetry breaking phases involve a change of symmetry. Now we know that there are new kind of phases of matter beyond ...

5

This is a very good question. First let me clarify a point. So far long range entanglement is only defined for gapped quantum systems. The gapless systems seems always "long range entangled". So the notion is useless. Do RVB states have long range entanglements? I think the string idea that you mentioned is a very good idea: A string liquid leads to long ...

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(1) What is topological degeneracy in strongly correlated systems such as FQH? From Wiki, http://en.wikipedia.org/wiki/Topological_degeneracy : Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system become degenerate in the large system size limit, and that such a degeneracy cannot be lifted ...

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Here is an explanation that's purely quantum. A charged quantum particle in a magnetic field is subject to Landau quantization. Taking the magnetic field in the $z$ direction, we can choose the Landau gauge for the vector potential: $$\mathbf{A} = B x \hat{y} ~~ \Rightarrow ~~ \mathbf{B} = B \hat{z}.$$ The Hamiltonian in the coordinates $xy$, ignoring (for ...

4

A short answer: The soliton in bosonic field theory can be fermionic because the model secretly contains massive fermions at high energies. This is because in order to define an bosonic field theory, we need to non-perturbatively regulate the field theory. So let us put the bosonic field theory on a lattice to non-perturbatively regulate the theory ...

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