Questions tagged [topological-insulators]

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

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3answers
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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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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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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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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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Is there an algebraic approach for the topological boundary (defect) 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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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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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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1answer
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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 deformation, ...
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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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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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Derivation of Kubo Formula for Hall Conductance

I am trying to derive the result of the TKNN formula but am experiencing difficulty in deriving the Kubo formula. The Kubo formula used in the TKNN paper is, $$ \sigma_{xy}= \frac{ie^2}{\hbar} \sum_{E^...
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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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Why bulk states in quantum hall effect do not contribute to electric conductivity

Most reviews and textbooks explain quantum hall effect as insulating bulk states and conducting edge states, as is shown in the following picture. My question is: why bulk states are insulating in ...
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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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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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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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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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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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$\mathbb Z_2$ or $\mathbb Z$ invariant for the Su-Schrieffer-Heeger (SSH) model

I am trying to understand topological insulators and topological invariant. The Su-Schrieffer-Heeger (SHH) model is often invoked as a protoypical topological insulator in 1D that carries localized ...
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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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Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The Berry phase accumulated on a path can be described by a matrix when we look at adiabatic time evolution with a Hamiltonian with degenerate energy levels. The Berry phase matrix is given by $$ \...
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1answer
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$Z_2$ topological insulator: odd vs. even number of edge state pairs

I am having trouble in understanding why in $Z_2$ topological insulators odd number of Kramers' pairs on one edge are protected by time reversal symmetry against elastic backscattering while even ...
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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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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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Book recommendations - Topological Insulators for dummies

Is there a pedagogical explanation of what is a topological insulator for those that do not even know what the Berry phase is but have a basic understanding of quantum mechanics and solid state ...
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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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1answer
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Berry phase in 1D materials

The Berry phase $\phi_B$ is the phase that an eigenstate acquires after its momentum vector goes around a circle at constant energy around the Dirac point. It is defined as $\phi_B = -i \int \langle\...
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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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Topological insulators - Surface states have a phase?

When I look at the circle of the Dirac cone around the Dirac point of, let's say, $Bi_2Se_3$, then the electron winds around and it is true that it goes from momentum $-k$ and spin-up to $+k$ and spin-...
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2answers
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Would HgTe be a topological insulator?

In "Quantum Spin Hall Insulator State in HgTe Quantum Wells", researchers observed a 2D topological insulator by sandwiching HgTe between CdTe. Is the CdTe really necessary? Would Vacuum/HgTe/Vacuum ...
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Chiral anomaly in Weyl semimetal

In the presence of electromagnetic fields $E$ and $B$, four current is not conserved in a Weyl semimetal i.e. $\partial_{\mu} j^{\mu}\propto E\cdot B \neq 0$. There are some proofs in the literature ...
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Why is the Berry curvature odd under time reversal?

The question is: Why is the Berry curvature, defined as $$\mathcal{F}=-\mathrm i\, \epsilon_{ij}\, \left\langle\partial_{ki}u_{n}(k)\mid \partial_{kj}u_{n}(k) \right\rangle ,$$ odd if I apply time ...
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1answer
694 views

How to show that Chern number gives the amount of edge states?

When talking about topological insulator and talking about bulk-edge correspondence, it seems to be widely accepted conclusion that the band Chern number (winding number) is equal to, when the ...
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Why is Brillouin zone a torus?

I am told that Brillouin Zone is a torus, which plays an important role in defining chern number and other topological invariants. The argument is that Bloch wave function $\psi_k$ is periodical in ...
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Band structure of topological insulators

The above figure is Rashba-split free electron-like surface state in a projected bulk band gap. The below figure is the band structure of a topological insulators. x axis is the wave vector, y axis is ...
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1answer
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Topological invariant in 1D

In 2D, with state $\psi(k_x, k_y)$, it is common to calculate measure of topology of material: 1 - Calculate Berry connection $a = -i <\psi | \partial_{\boldsymbol{k}} | \psi>$. 2 - Calculate ...
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1answer
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Does time reversal symmetry hold for (kitaev model) 1D spinless $p-$ wave superconductor?

The hamiltonian 1D spinlesss p wave superconductor can be written as $$ H=\sum_k \phi_k^\dagger \begin{pmatrix} \xi(k) & 2i\Delta \sin(k)\\ -2i\Delta \sin(k ) & -\xi(k)\end{pmatrix}\phi_k $$ ...
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1answer
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What is “trivial” about the trivial topological superconducting phase?

Once more I am stuck on my favorite word: "trivial". I am reading a bunch of stuff about topological superconductors at the moment and people keep talking about having to distinguish between the ...
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1answer
536 views

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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1answer
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Relation between topological insulators and breaking of time reversal symmetry

Whenever one talks about topological insulators, the breaking of time reversal symmetry is always mentioned. Is there an intuitive reason as to why one need time reversal symmetry to be broken in ...
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1answer
196 views

Resources on non-Abelian gauge transformations and etc

I am currently studying topological insulators and occasionally the concept of non-Abelian phases and non-Abelian gauge transformations comes up in related literature. However, my understanding of ...
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1answer
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Questions about Berry Phase

I'm learning about the Berry Phase from the original paper, and from the TIFR Infosys Lectures The Quantum Hall Effect by David Tong (2016). I have some questions regarding the original derivation of ...
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1answer
485 views

How can there be negative energy gaps sandwiching a topological insulator?

At the 37:37 instant of the timeline in this youtube video Stanford University theoretical physicist Shoucheng Zhang presents a graph of "theoretical prediction of the first topological insulator". ...
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How to visualize multi-dimensions in topological periodic table?

This is a question for those who are familiar with topological periodic table. The first row and right side columns represents dimensions of topological materials in periodic table. I know that for ...