Hot answers tagged

55

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 "...


31

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 ...


23

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 systems, gauge symmetry is not a symmetry, in the sense ...


18

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 ...


18

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 ...


15

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 ...


14

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) ...


14

The 'topological' in topological order means 'robust against ANY local perturbations'. According to such a definition, topological insulator is not 'topological' since its properties are not robust against ANY local perturbations, such as the perturbation that break the U(1) and time reversal symmetry. So a more proper name for topological insulator is 'U(1)...


13

This is a heavy question, that contains many topics in it that are worthy of their own questions, so I'm not going to give a complete answer. I am relying mainly on this excellent review paper by Nayak, Simon, Stern, Freedman and Das Sarma. The first part can be skipped by anyone already familiar with anyons. Abelian and non-Abelian anyons Anyons are ...


12

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 ...


12

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 ...


11

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 ...


11

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 ...


11

In short, what makes a superconductor topological is the nontrivial band structure of the Bogoliubov quasiparticles. Generally one can classify non-interacting gapped fermion systems based on single-particle band structure (as well as symmetry), and the result is the so-called ten-fold way/periodic table. The topological superconductivity mentioned in the ...


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 ...


10

There are different categories of topological superconductors. I’m guessing that you are referring to the time-reversal invariant (class DIII) ones, in 2D or 3D. Yes, it is possible to distinguish the surface/edge states of 3D/2D topological superconductors from the bulk. I'm not talking about designing some intricate experimental technique to separate out ...


10

After stating the solution, I'll try to give some physical insights to the best of my knowledge and some more references. The dimension of the required state space is given by the Verlinde formula, having the following form for a general compact semisimple Lie group $G$ on a Riemann surface with genus $g$ corresponding to the level $k$: $$ \mathrm{dim} V_{...


9

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 ...


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

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 "topological" in "topological order" and the "topological" in "topological insulator" have different meanings. The 'topological' in topological order means 'robust against ANY local perturbations'. The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of ...


8

My short answer is No, they're not too useful, but let me discuss some details, including positive ones. Categories, especially derived categories, have been appropriate to describe D-brane charges - and not only charges - beyond the level accessible by homology and K-theory. See e.g. http://arxiv.org/abs/hep-th/0104200 However, I feel it is correct ...


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

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 ...


8

(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 ...


8

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 ...


8

The previous understanding of the quantum spin liquid as a ground state of spin systems with spin rotation symmetry is not only out-of-date but also misleading. In modern language, quantum spin liquids are classified as symmetry enriched topological(SET) states, which possess anyon excitations carrying fractionalized symmetry charges, meaning that the anyons ...



Only top voted, non community-wiki answers of a minimum length are eligible