Topological insulator are materials formed by a insulator bulk and metallic surfaces with topological origin. These materials may be important for developments in Quantum Computing and Spintronics.

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Simple models that exhibit topological phase transitions

There are a number of physical systems with phases described by topologically protected invariants (fractional quantum Hall, topological insulators) but what are the simplest mathematical models that ...
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How Fundamental is Spin-Orbit Coupling to Topological Insulators?

I'm well aware this is a very active area of research so the best answer one can give to this question may be incomplete. Topological states in condensed matter are well-known, even if not always ...
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Why are there chiral edge states in the quantum hall effect?

The most popular explanation for the existence of chiral edge states is probably the following: in a magnetic field, electrons move in cyclotron orbits, and such such cyclotron orbits ensure electrons ...
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What is the mathematical reason for topological edge states?

There are many free fermion systems that possess topological edge/boundary states. Examples include quantum Hall insulators and topological insulators. No matter chiral or non-chiral, 2D or 3D, ...
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Graphene and Klein bottle?

I am trying to understand graphene as a topological insulator. The spin orbital interaction in graphene is very small (~10mK?). But if we consider that, then graphene should be a topological ...
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Do topological superconductors exhibit symmetry-enriched topological order?

Gapped Hamiltonians with a ground-state having long-range entanglement (LRE), are said to have topological order (TO), while if the ground state is short-range entangled (SRE) they are in the trivial ...
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Reconciling topological insulators and topological order

We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic ...
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Is there a way to directly observe the spin texture of the surface states of topological insulators?

Is there a way to directly, here I means in real space, observe the interaction of the surface states of 3D topological insulators with defects (dopings and adatoms)? How to observe the spin texture ...
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When can we take the Brillouin zone to be a sphere?

When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen ...
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Is edge state of topological insulator really robust?

I am a little confused! Some people are arguing that the gapless edge state of Topological insulator is robust as long as the time reversal symmetry is not broken,while other people say that it is ...
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Topological insulators: why K-theory classification rather than homotopy classification?

I am reading a 2009 paper by Kitaev on K-theory classification of topological insulators. In the 4th page, 1st paragraph in the section "Classification principles", he says, Continuous ...
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Hall conductivity from Kubo: Bulk or edge?

Using the Kubo formula, Thouless, Kohmoto, Nightingale, and den Nijs (TKNN, PRL 49 405-408 (1982)), proved that upon summing all the contributions of the filled states of an insulator, the Hall ...
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Where does the Berry phase of $\pi$ come from in a topological insulator?

The Berry connection and the Berry phase should be related. Now for a topological insulator (TI) (or to be more precise, for a quantum spin hall state, but I think the Chern parities are calculated in ...
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Topological band structure, difference between a sphere and a donut

Kohmoto from TKNN(Thouless-Kohmoto-Nightingale-deNijs) who described the topology of the integer quantum hall effect always stressed the importance of the 2D Brillouin zone being a donut due to ...
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How metallic surfaces states can emerge in topological insulators?

Topological insulators are materials known to have bulk insulator and metallic surface states. But, what is the origin of these metallic surface states? And how the topology of band could help the ...
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Intuition on topologically nontrivial 2D-band structures?

I want to get more intuition on topologically nontrivial band structures. There's this popular 2D two-band model for a topological insulator where $H=\sum_{k}h(\boldsymbol{k})$ (see Qi, Hughes, and ...
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Are there topological non-trivial states in zero dimension?

The periodic table of topological insulators and superconductors suggests that there can be topological non-trivial phases in zero dimension in non-interacting system with certain symmetries. A 0D ...
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How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)

This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it ...
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Trivial and Non-trivial topology of band structure

I don't understand the meaning of the expression "trivial topology" or "non-trivial topology" for an electronic band structure. Does anybody have a good explanation?
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Counterexamples to the bulk-boundary correspondence (topological insulators)

In the literature on topological insulators and superconductors the 'bulk-boundary correspondence' features quite heavily. One version of this conjecture says roughly: "At an interface between two ...
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What's the meaning of “topological” in “symmetry protected topological phase”

I am trying to understand the symmetry protected topological phase. Most papers only explain the symmetry but none of them explain the meaning of "topological". My question is : What's the meaning ...
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How are topological invariants constructed?

I've seen several different definitions for what are called topological invariants, for instance in the context of Majorana unpaired modes, by Kitaev: http://arxiv.org/abs/cond-mat/0010440 ...
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Topology of Fermi surface

In The universe in a Helium droplet, Grigory Volovik relates the stability of a fermi surface to topology of a Green function. There he gives the example of a Fermi gas and says that the Green ...
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Descent equation and anomaly polynomial

I am just reading Ryu, Moore and Ludwig's paper on classifications of topological insulators and quantum anomaly. They are trying to relate the quantum anomaly as a signal of the presence of a ...
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What makes a superconductor topological?

I have read a fair bit about topological insulators and proximity induced Majorana bound states when placing a superconductor in proximity to a topological insulator. I've also read a bit about ...
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Whis is the difference between charge fractionalization in 1D and 2D?

Both 1D Polyacetelene and 2D fractional quantum Hall state can support fractional excitations. But as I can see, there are some differences: the ground state of Polyacetelene breaks translational ...
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Why are Topological Superconductors hard to make?

Topological insulators (TI) have already been made in lab. Topological superconductors (TSC), being close cousins of TI, seem harder to make. Why is that? It seems that materials in connection with ...
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“Topological” notions in physics

I've been trying to make sense recently of the usage of 'topological' in various fields of physics, and get sort of an intuition for what this means in context. This all boils down to my main question ...
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Experimental evidence for non-abelian anyons?

Since non-abelian anyons have become quite fashionable from the point of view of theory. I would like to know, whether there has actually been experimental confirmation of such objects. If you could ...
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A simple conjecture on the Chern number of a 2-level Hamiltonian $H(\mathbf{k})$?

For example, let's consider a quadratic fermionic Hamiltonian on a 2D lattice with translation symmetry, and assume that the Fourier transformed Hamiltonian is described by a $2\times2$ Hermitian ...
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Chern insulator vs topological insulator

What is the basic distinction between a Chern Insulator and a Topological Insulator? Right now I know that a Chern Insulator has "topologically non-trivial band structure" and that a Topological ...
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Jordan Wigner Transformation in 1d Majorana chain

So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for ...
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How can one localize the massless fermions in Dirac materials?

I noticed that finite electric potential cannot localize the low energy excitations in a graphene sheet. Is it possible to localize the massless fermions in the surface band of topological insulators ...
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Measurement of topological spin

How do you measure the topological spin of an anyon? So how could an experimental setup look like? Is topological spin an observable at all?
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Chiral edge state as topological properity of bulk state

As far as I know, quantum hall effect and quantum spin hall effect has chiral edge state. Chiral edge state is usually closely related with delocalization, since back scattering is forbidden. However, ...
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Is band-inversion a 'necessary and sufficient' condition for Topological Insulators?

According to my naive understanding of topological insulators, an inverted band strucure in the bulk (inverted with respect to the vaccum/trivial insulator surrounding it) implies the existence of a ...
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A question on the doped Kitaev-Heisenberg model?

Recently, some groups have studied the effects of doping the Kitaev model on honeycomb lattice(e.g.,http://arxiv.org/abs/1109.6681 and http://arxiv.org/abs/1109.4155) and their calculations show the ...
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Superconductivity and time-reversal symmetry

Let us consider a system of a 1D edge of a 2D topological insulator in proximity to an s-wave superconductor. The system is described by the Hamiltonian: $$ H =\frac{1}{2} \int \mathrm{d}x \ ...
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Equivalence classes of mappings from $T^{2}$ to an arbitrary space $X$

I was reading the paper "Homotopy and quantization in condensed matter physics", by J.E Avron et al. ( http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51). There they have classified the ...
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Effective theory of topological insulator in coulomb impurity

I am trying to solve for the Haldane model with a coulomb impurity at one site in the effective theory approach and look for some topology in the solutions of the wave functions. The Hamiltonian near ...
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How to perform stroboscopic measurements for Floquet topological insulators?

Floquet topological insulators (arXiv:1008.1792, arXiv:1211.5623) have attracted much research interests in condensed matter physics. The goal is to realize topological insulators from trivial ...
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What is the physics behind “Bulk-edge correspondence”?

There is a frequently mentioned concept in the field of topological insulator called "bulk-edge correspondence" or "bulk-boundary correspondence", which basically gives the relationship between the ...
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What are the prerequisites to study topological quantum computation/topological phases of the matter? [closed]

I am an undergraduate student and I would like to approach the subject of topological order with focus on topological quantum computation, I know (very) little QFT and basic algebraic topology (if ...
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Significance of Dirac cones in condensed matter physics

In condensed matter physics, Dirac cones can be found in graphene, topological insulators, cuprates, and iron-pnictides. This means that electrons behave as massless particles near the Dirac points. ...
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Defects in 3+1 TFTs/2+1 CFTs

I would like to know of good pedagogic references to learn about the notion of "defects" in TFTs and CFTs. I am specially interested in 3+1 TFTs (.and probably about their relation to 2+1 CFTs..) In ...
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1D topological insulator

This question is inspired by another one about the simplest model of topological insulator, where 4tnemele showed a nice two band model in the answer. I read that and am wondering if we and push that ...
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How to distinguish between a topological state from from a non-topological one?

How to distinguish between a topological state from from a non-topological one? Is there any standard procedure for identifying the topological features of a given hamiltonian? In general what are the ...
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What happens when a bare 3d topological insulator is subject to a magnetic field?

Effective field theory of 3d topological insulators (TI) predict some novel electromagnetic effects. Unfortunately it require a gapped surface which is hard to achieve experimentally. Then I have two ...
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What symmetry class does 1D spinless $p$-wave superconductor belongs to?

$Z_{2}$ topological invariant exist for Kitaev model. What symmetries does it conserve? And to what symmetry class it belongs to? The hamiltonian for kitaev model can be written as $$ H=\sum_k ...
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Topological Insulators: is HgX a special case?

I got confused by reading this article: Francois Virot, Roland Hayn, Manuel Richter, and Jeroen van den Brink. “Engineering topological surface-states: HgS, HgSe and HgTe.” arXiv preprint ...