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18

The short answer: graphene is a counterexample. The longer version: 1) You do not need to break the time reversal symmetry. 2) spin-orbit coupling does not break the time-reversal symmetry. 3) In graphene, there are two valleys and time inversion operator acting on the state from one valley transforms it into the sate in another valley. If you want to stay ...


14

I am still not sure what you precisely want to be a Klein Bottle, but let me make some comments that might help you clarify what exactly you want to know. (Warning: I am writing this while being very tired, people are invited to correct me.) First of all one must be careful to distinguish band structure of the bulk from band structure of a semi-infinite ...


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 see how that can be confusing. Unfortunately understanding how to reconcile these statements will require a lot of background. I will try to answer this as concisely as I can (hopefully) without relying on concepts that are too advanced. Well, topological insulators do not possess a so-called intrinsic topological order. It means that the bulk states of a ...


12

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 ...


11

Very loosely speaking the reasoning is this. Imagine a two band system in which the fermi sea has one filled band with Chern number $n$ and another system with $N$ filled bands but also with Chern number $n$. Physically they have the same topological properties (for example the same Hall conductance, edge states and so on), but cannot be deformed ...


9

Topological insulators are gapped states of free fermions with particle number conservation and time-reversal symmetry. According to the K-theory classification, there is no Topological insulator in 1D. However, 1D interacting fermions with time-reversal symmetry do have non-trivial symmetry protected topological phases if the particle number is conserved ...


8

Sorry this answer got too long. I have categorized it into three points. (1) I think the reason Kohmoto stresses the importance of the Brillouin zone being a torus $BZ = T^2$, is because he wants to say that BZ is compact and has no boundary. This is important because of the subtlety that makes everything work. The Hall conductance is given by $\sigma_{xy} ...


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 ...


7

One of the early triumphs of QM (through e.g. Kronig-Penney model) was the explanation of the insulating state of matter. Energy bands (and gaps) appear as the result of hybridization of many atomic orbitals, and for a specific filling you can end up with the top most pair of bands being either entirely filled (valence band) or entirely empty (conduction ...


7

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 ...


6

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 ...


6

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 ...


6

The topological spin $h$ of an anyon (a quasi-hole in a FQH state) is the exponent in the Green function of the quasi-hole along the edge of the FQH state [see eq.(61) in my review paper http://arxiv.org/abs/1203.3268 ], which can be measured by the I-V curve: $I\propto V^{4h-1}$ in the tunnelling experiments between FQH edges.


6

There is no proof of bulk-boundary correspondence for topological phases in general. In fact, topological phases like toric code model does not have gapless excitations on the boundary. For non-interacting fermion systems protected by internal symmetries (as in the "periodic table" classification), bulk-boundary correspondence holds. For non-interacting ...


6

The notion of fractional charge is not well defined in 1D Luttinger liquid (despite many papers say that the charge is fractionalized in 1D Luttinger liquid). In fact, it is hard to define fractional charge in any gapless state if the low energy excitations are not described by free quasiparticles. For gapped states, fractional charge in 1D is due to ...


6

Here is an algebraic approach to understand the edge state. Let us start from a generic Dirac Hamiltonian for the bulk fermions in the $d$-dimensional space. $$H=\sum_{i=1:d}\mathrm{i}\partial_i\alpha^i+m(x_i)\beta,$$ where $\alpha^i$ and $\beta$ are anti-commuting gamma matrices ($\{\alpha^i,\alpha^j\}=2\delta^{ij}$, $\{\alpha^i,\beta\}=0$, $\beta\beta=1$), ...


6

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 ...


6

I think I understand what you mean when you say that you're not satisfied with the “nontrivial bulk topology argument” when it comes to thinking about edge states. The Chern number (for time-reversal breaking) and $\mathbb{Z}_{2}$ invariant (for time-reversal symmetric) systems, as DaniH suggested, does indeed give you information about the edge states; the ...


6

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 ...


5

For the sake of simplicity, let us limit this discussion to non-interacting fermionic phases: e.g. band insulators with electron quasiparticles and fully gapped superconductors with Bogoliubov quasiparticle excitations. Classification of interacting and/or non-fermionic phases is still work in progress. Before outlining basic differences between topological ...


5

This won't completely answer your question, but maybe it will help. I remember when I was encountering the topic that I used to be completely bewildered by the discussion. I think two facts really helped me to understand things: We are working in a non-interacting picture of electrons. This means that we only need to consider a single-particle Hamiltonian. ...


5

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 ...


4

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 ...


4

A short answer: Why not? HQ states do not have time reversal symmetry. So the right moving excitations and left moving excitations may behave differently -- thus chiral. The edge states of most FQH states are very chiral, in the sense that even the numbers of left moving modes and right moving modes are different. Topological insulator and topological ...


4

As the first author of arXiv:1109.4155, my answer to this question is yes. The topological p-wave SC state is insensitive to the sign of the coupling. The argument provided in our paper is quite general, the time-reversal broken SC state is supported by the underlying topological order in the Kitaev spin liquid, as described by the particular spin-gauge ...


4

Why do you want to have an understanding of the gapless edge states without using bulk topology? If you allow me to use the bulk topology, an argument is that you can continuously move the edge and consider that as an adiabatic parameter which interpolate two systems. To be more precise, you can consider a sphere with part of it in one topological state A ...


4

The answer of David Aasen is correct, but let me add some comments which connect to your question of the relation of between the $\mathbb Z_2$ invariant $\nu$ and the first Chern-Number $C_1$. Such a relation does not exist unless you require some extra symmetry than the generic symmetries usually required in the classification of topological insulators ...


4

All three questions can be answered by first artificially separating the graphene sheet into two sheets: (a) first sheet with only spin up electrons, and (b) second sheet with only spin down electrons. This statement alone should partially answer your third question; for the sake of organization, however, I will repeat a summary of this paragraph (in the ...


4

The follow article directly address your question (which is a very good question): Reconciling topological insulators and topological order The article explains the different meaning of "topological" in "topological insulators" and "topological order".



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