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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'' ...


29

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)...


28

The Berezinskii-Kosterlitz-Thouless (BKT) scenario is one of the most beautiful transitions that is ubiquitous in 2d systems (though it can also occur in higher dimensions for particular kinds of models) that surprisingly requires non-perturbative effects (i.e. topological defects) to be realized. To understand all the fuss (and the nobel) around this ...


25

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 ...


21

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 ...


21

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 ...


21

Yes, a topological defect is a discontinuity that cannot be removed. Let me explain by giving an example similar to a liquid crystal. Fish in a pond I have a two-dimensional pond and I would like to fill it with fish. These fish are longer than they are wide, so when they are densely packed in the pool, they like to point in the same direction as their ...


20

As you have mentioned, topological insulators (TI) are "topological" because they can not be smoothly connected to trivial band insulators without closing the band gap (and without breaking certain symmetry). Simply generalize this to the many-body case, we may say that the topologically ordered states are called "topological" because they can not be ...


20

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 ...


19

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 ...


16

Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?" (1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-...


16

This is a very a general question, I think I could provide some insight but it will certainly need to be elaborated on by someone with this specific expertise. The $\mathbb{Z}_k$ para fermions arise in several statistical mechanics models. They are both interesting and subtle because their exchange statistics depend on their positions (in one-dimension). It ...


15

A quote from http://en.wikipedia.org/wiki/Symmetry_protected_topological_order : The SPT order (for both frermionic and bosonic systems) has the following defining properties: Distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry. However, they all can be ...


14

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 ...


14

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 ...


14

In theories with spontaneous symmetry breaking, the phase transition can usually be characterized by a local order parameter $\Delta(x)$, which is not invariant under the relevant symmetry group $G$ of the Hamiltonian. The expectation value of this field has to be zero outside the ordered phase $\langle\Delta(x)\rangle = 0$, but non-zero in the phase $\...


14

Let us consider a quantum phase transition (at T=0) from an ordered phase to a disordered phase, driven by the quantum fluctuations of the order parameter. We like to ask if the disordered phase has topological order or not. The importance of the topological defects in phase transitions have been emphasized by Kosterlitz and Thouless, who shared 2016 Nobel ...


12

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 ...


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The answer is Yes. See A physical understanding of fractionalization http://arxiv.org/abs/hep-th/0302201 Quantum order from string-net condensations and origin of light and massless fermions, Xiao-Gang Wen; Spin-1/2 and Fermi statistics from qubits http://arxiv.org/abs/hep-th/0507118 Quantum ether: photons and electrons from a rotor model, ...


11

All the non-trivalent graph can be obtain from the trivalent graph by combining some vertices together. For example, the Z2 string-net model is defined on a honeycomb lattice, which is a trivalent lattice. But if you combine the two sites in each unit cell together, and consider the whole unit cell as your "site", then the honeycomb lattice simply becomes ...


11

A symmetry-protected topological phase has a certain symmetry. Any Hamiltonian in this phase can be adiabatically deformed (i.e. without closing the gap) into a Hamiltonian whose ground state is a product state, but the symmetry must be explicitly broken during the deformation process and then restored at the end. As a visually analogy, there is a "wall" ...


10

The reason that people get away with ignoring category theory and homotopy theory in physics so much is that physics is already so rich locally and in perturbative approximations. But a general fact is that all global and non-perturbative effects, hence everything that concerns the full story, is fairly intractable without the toolbox of higher categroy/...


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_{...


10

(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is roughly the same as classifying phase structure of topologically ordered states. Some topologically ordered states are described by a group and can be related to a gauge theory. Some other topologically ordered states are not related to gauge theory. (2) One way to classify "Phase ...


10

At low temperature the spin-spin correlation function of the 2d O(2) model can be computed in perturbation theory, see, for example chapter 33 of Zinn-Justin, QFT and Critical Phenomena. The answer is $$ \langle e^{i\theta(0)}e^{-i\theta(r)}\rangle \sim r^{-t/(2\pi)} $$ where $t$ is the reduced temperature, and I have written the spin vector in terms of the ...


10

Although it's true, as Norbert Schuch pointed out, that the same state can be the ground state of Hamiltonians with and without gap, in general this behavior seems rather fine-tuned. I would expect that, for a generic perturbation to the gapless Hamiltonians he discussed, a gap would open up. For this reason, physicists tend to ignore this subtlety and treat ...


9

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 ...


9

The 'topological' in topological order can refer to: The fact that the ground state degeneracy is sensitive to the the topology of the manifold (as mentioned by Motl). The low energy, effective theory is a Topological Field Theory. The low energy excitations are anyons which obey a generalized form of exchange statistics. This steps into the realm of knot ...


9

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 ...


9

I don't think the provided comment gives the right answer. Topological insulators is the bigger group and Chern insulator are a subgroup of that. This means that every Chern insulator is a topological insulator, but not every topological insulator is a Chern insulator. Can maybe someone confirm that this is indeed true? In general a topological insulator is ...


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