# Tag Info

16

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 ...

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 ...

7

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 ...

7

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 ...

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 ...

7

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 ...

5

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 ...

5

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 ...

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 ...

4

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 ...

4

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. ...

4

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.

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 ...

3

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). For gapped states, fractional charge in 1D is due to translation symmetry breaking, while fractional charge in 2D and higher is due to topological order (ie long-range entanglement). See A physical ...

3

Topological insulator, by definition, cannot exist in magnetic field. This is because the topological insulator is NOT topological. A topological insulator is a material with time reversal symmetry and particle number conservation. Without time-reversal symmetry, topological insulators cannot exist, since they become the same as trivial band insulators. So a ...

2

The so-called, lyotropic liquid crystals exhibit several topological transitions. The topology of a real space changes during such transitions. The most famous of them is the transition into the so-called, sponge phase. But there are also more simply ones. For example, lipid vesicles are known to transform into a string of beads (which still are ...

2

As for the second question (are the Majorana difficult to realize in lab ?) the answer is obviously yes, and for the same reason that we have no idea what to look for ! (NB: of course there are some predictions about the experimental signature of the Majorana, but no smoking gun experiment.

2

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 ...

2

The Majorana bound state inside a vortex of a topological superconductor is, indeed, not a chiral edge state. It does not follow that the topological superconductor does not have a chiral edge state. It does! Solve, for example, the BdG equations for a p+ip superconductor with open boundary conditions, and you'll see it. The existence of edge modes is ...

2

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 ...

2

Well, the answer is yes and no. The band inversion between the $s$-like (conduction) band $\Gamma_6$ and $p$-like (valence) band $\Gamma_8$ in HgTe is primarily responsible for its topologically nontrivial band structure. The bulk band structure of HgTe with (right) and without (left) spin-orbit coupling is shown in the figure below. There are a total of ...

2

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 ...

1

Fractional excitations are understood to be generic in 1D. An example with a "symmetry presreved" state (whatever that is supposed to mean in 1D) is the simple Luttinger liquid. The Luttinger liquid exhibits charge-fractionalization in to spin charge separation. This was first shown here, I believe.

1

For the vacuum/HgTe you'd have 2D TI somewhere on surface of HgTe which is not very convenient for study of this state. In addition, surface defects will make the result hardly usable even for experiments, don't even think of applications. In CdTe/HgTe/CdTe QW you have a good control on the properties of the system, it has a perfect lattice, etc.

1

You are just wrong. 1) The time reversal symmetry you are speaking of is not the time reversal symmetry which is considered when topological isulators are discussed. In the latter case just no magnetic field (both external and internal) is enough. In that case effective Hamiltonian of the system allows for the time inversion symmetry. Formally you may ...

1

For a time reversal invariant bloch hamiltonian (such as in a $\mathbb{Z}_2$ topological insulator) the Chern number is always zero. The topological invariant $\nu = 0,1$ classifies the insulator as trivial or topological. This can be found by counting the number of times the surface energy bands intersect the Fermi energy mod 2 as you mentioned above. ...

1

The short answer is that in topological insulators, spin-orbit coupling allows for the creation of a [topological] class of insulating bands where time reversal symmetry is unbroken with topologically protected edge states at the boundaries of the sample. These edge states are similar in spirit to those which arise in the integer quantum Hall effect but in ...

1

Thank you for you answer. But there might be different interpretations of the word "topology." In the TI case, "topological" means universal, independent of material properties while when it comes to the concept of "topological orders," "topology" implies robustness. After all, it is merely a name whose meaning varies from person to person. Yet, we can ...

1

From what I understand, "quantum phases" or "Berry-Pancharatnam phases" are examples of "topological phases". See Y Ben-Aryeh 2004 J. Opt. B: Quantum Semiclass. Opt. 6 R1, "Berry and Pancharatnam Topological Phases of Atomic and Optical Systems", http://arxiv.org/abs/quant-ph/0402003 . Under this definition of the terms, the simplest example of topological ...

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