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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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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Bulk-boundary correspondence in topological insulators

I'm not an expert in the area; just recently I checked this paper because of my research. What puzzles me a lot is the so called bulk-boundary correspondence. Can anyone explain in succint terms what'...
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2D Chern Simons action by integrating out fermions

In Qi, Hughes, and Zhang's paper (https://arxiv.org/abs/0802.3537), they show how the Chern number appears as a coefficient of response function. Given the Hamiltonian (49) of a (2+1) or (4+1)D ...
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“Weak” and “Strong” topological insulators

For translationally invariant systems, we can define some topological invariant based on the translational symmetry, which is referred to "weak" topological invariant. For example, according to Kitaev'...
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The origin of $Z_2$ in topological insulators

In condensed matter physics, a study of topological insulators use the term $Z_2$ (defined as topology index) very frequently. What is the origin of $Z_2$? Why not $Z_3$ or $Z$? My understanding is ...
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Weyl Semimetal and Fermi Velocity

In a Weyl semimetal, the Nielson-Ninomiya theorem enforces the fact that number of positive and negative chirality Weyl points must be equal. Is there any restriction on the form of the Weyl points? ...
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Is topological surface state always tangential to bulk bands?

Think of a topologically nontrivial $D$-dimensional system. Its bulk bands form a $D+1$-dimensional manifold ($+1$ from energy). Its surface/edge bands form a $D$-dimensional one. Is the latter always ...
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Time reversal of Bloch Hamiltonian (at TRIM points)

I know that time-reversal symmetry requires the Bloch Hamiltonian $H(\textbf{k})$ to transform as: $$ \Theta H(\textbf{k}) \Theta^{-1}=H(-\textbf{k}) $$ where $\Theta$ is the time-reversal operator. ...
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Parity of Bloch states at TRIM points

There is an argument presented in Fu and Kane's paper on inversion symmetric topological insulator which I have not yet convinced myself. Just below Eq.(3.6), the authors said that because of ...
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Topological Classification of Band Insulators (in terms of Green's functions)

I am currently reading Topological Classification and Stability of Fermi Surfaces by Y. X. Zhao and Z. D. Wang (PRL 110, 240404 (2013)). They remark that the Green's function (along the complex ...
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Formula for the topological invariant for each of the symmetry classes

Is there a reference that systematically derives the topological invariant/winding number for all the ten symmetry classes in Altland and Zirnbauer's periodic table? For example, in this answer, there ...
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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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Linking phase of flux lines and excitation energy of monopole

I am reading this paper and on the left-hand side of pp.10 it states the following relation between linking phase and excitation energy of monopole: Now the $\theta = \pi$ term in the bulk implies ...
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Winding number of SSH model 3

SSH model can be written as $$H=-\sum_n\big[Jc_n^\dagger d_n + J'd_n^\dagger c_{n+1}\big]+h.c.$$ in Fourier space $$H(k)= \begin{bmatrix} c_k^\dagger && d_{k}^\dagger \end{bmatrix} \begin{...
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What does the Chern number physically represent?

In 2D the Chern number can be written as $$C_m=\frac 1{2\pi}\int_{BZ}\Omega_m(\mathbf k)\cdot d^2 \mathbf k$$ where we are integrating over the Brillouin zone. In 2D this is equivalent to finding ...
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Why can we treat the Bloch hamiltonian as an effective hamiltonian?

Often when looking at topological insulators the hamiltonian is broken down into the Bloch hamiltonian and then analysed ignoring the creation/annhilation operators of the Bloch waves. Why is it okay ...
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Why is the flux quantized in 4D quantum Hall effect?

I am reading "Topological Field Theory of Time-Reversal Invariant Insulators" by Qi, Hughes, and Zhang (https://arxiv.org/abs/0802.3537). It argues that time reversal invariant (TRI) insulators in 2+1 ...
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Inversion symmetry restrictions to the Berry curvature in 2D

It is said that if a lattice has inversion symmetry, then the Berry curvature, $\vec{\Omega}(\vec{k})$ is even in $\vec{k}$, i.e. $$\vec{\Omega}(\vec{k})=\vec{\Omega}(-\vec{k})$$ I have also derived ...
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In a spinless system with time reversal symmetry, is $E_n(k)=E_n(-k)$ always true?

I am studying TR-symmetry from: "Group Theory" by Dresselhaus, Dresselhaus and Jorio and there's a point that I cannot quite understand. The point is under eq. (16.17). In general, we know that the ...
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561 views

Berry curvature and time reversal symmetry

When the time reversal operator, $\hat{\Theta}$ acts on a phase, $e^{i\phi}$ it gives $e^{-i\phi}$. Since the Berry phase factor is $e^{i\gamma}$, where $\gamma$ is the Berry phase, if the Berry ...
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How to understand the Z2 invariant in three dimensional topological insulators

I know how the $Z_2$ invariant comes into the 2 dimensional topological insulator with time reversal symmetry. But I don't understand why we need to use four $Z_2$ invariant to describe the 3 ...
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How to calculate Zak phase from numerical wavefunctions with arbitray phase?

In numerical calculations, an arbitrary gauge or phase attached to a wavefunction at a particular $\mathbf{k}$ places an obstacle in calculating the Berry connection $$\mathcal{A}(\mathbf{k})=\langle ...
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Model of Dirac semimetal without surface states?

There are continuum models of the 3D Dirac semimetal. For example, proposed in this paper, when $k_\pm=k_x\pm ik_y$ and $M=m-|\vec{k}|^2$, $$H=\begin{bmatrix} M & k_+ & 0 & 0 \\ k_- & -...
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Quantum Spin Hall Effect : Dimensional reduction from Chern Simons 3 form

I have a problem matching the two definitions of the Z2 topological invariant for a 2D system, assuming Sz symmetry. I can either define it from 1D to 2D, as a Chern number per spin: $$ C_{\rm s} = \...
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Relationship between different $Z_{16}$ classifications

I find that there exist two classifications which have a $Z_{16}$ group structure: The sixteen fold way of classifying Majorana fermions, vortex systems appearing in Kitaev's paper on his honeycomb ...
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Topological insulators with time-reversal symmetry: How to derive $\mathbb{Z}_2$-invariant from the zeros of $\langle u_i(k)|T|u_j(k)\rangle$?

Following the book ''Topological Insulators and Topological Superconductors'' by B. Bernevig (esp. chapter 10.1), I want to understand how to derive a Z2 invariant starting from the zeros of the off-...
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100 views

What are the implications that the Hamiltonian of a material lacks time reversal symmetry?

When reading about topological insulators and the quantum Hall effect, I've read that some Hamiltonians of the crystal structure representing the "materials" lack time reversal symmetry. My question ...
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Why, the Valence Bundle is non-trivial and the complete Bloch Bundle isn't?

I'm thinking about the bundles defined in Topological Insulators, and I took that sentence, present in many sources, as true: The topology of the Valence Bundle is non-trivial, i.e., the Valence ...
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1answer
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How to calculate Chern number of a band numerically?

I have a tight-binding Hamiltonian with an aperiodic potential which shows non-trivial topological properties. I wanted to calculate the Chern number of it's bands. How can I do it numerically?
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The relation between U(1) charge conservation and electronic insulator

I am now learning the topological classification and have some problems on the paper (https://arxiv.org/abs/1303.1843) I am reading now. In this paper, there is a statement in the second part ( II. ...
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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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348 views

How do spectrum gaps relate to topological protected states?

In particular, I want to understand what fundamental (mathematical) structure gives rise to topological mechanical metamaterials, or topological protected states in general. According to a recent PNAS ...
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Physical meaning of topological invariant

What does it mean in terms of band structure when we say that any topological invariant of some system is non-zero? For example what does it mean when we say that Chern number=1 in case of IQHE? Does ...
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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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Is the non-trivial topology on the torus reflected on the Bloch sphere?

Almost every text on topological insulators have the Bloch sphere example of a two level system showing the non triviality of the bundle of an eigenvector over the sphere: we can't define an ...
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Why is Berry connection a connection?

The Berry connection, following the derivation of the Berry phase for a non degenerate system, is $\mathcal{A}_{k}(\lambda) = i \langle n|\frac{\partial}{\partial \lambda^{k}}|n\rangle$ This result ...
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Non-integer topological invariants

I am reading the lecture notes of Asboth on topological insulators. There he defines a topological invariant as an integer that does not change under adiabatic deformation of the parameters in the ...
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How to calculate Edge states of Topological insulators

Topological insulators are novel state of matter in which bulk is insulator and edges are gapless. How do we calculate these gapless states? I am reading the following PRL Feng Liu and Katsunori ...
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Haldane model, why the nearest hopping is not changed?

I am leaning the Haldane model : https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2015 Haldane imaged threading magnetic flux though a graphene sheet, and the net flux of a unit cell is ...
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What is topological quantum matter? (Answer to which was given by TED-Ed)

What is topological quantum matter? Today I saw the answer to this question in a video by TED-Ed on YouTube. But I couldn't make any sense of it (for I am still in high-school). Could someone provide ...
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Landau-Peierls Substitution in Haldane Model

In his 1988 paper "Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the parity anomaly" [1], Haldane performs a Landau-Peierls Substitution $\hbar\delta\mathbf{k}\...
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Lifting degeneracy in Topological Insulators

I was reading the review by M.Z. Hasan and C.L. Kane on topological insulators and had a question regarding degeneracy lifting. In the review, they mention that Haldane imagined breaking time reversal ...
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Laughlin's topological argument

I have a confusion about understanding the Laughlin's topological argument on Hall conductivity quantization. This argument states that the Hall conductivity is $$ \sigma_{xy}=\frac{e}{h}Q, $$ where $...
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How to derive energy dependent Gamma and Delta of topological nanowire?

I read two articles, G. Tkachov, and E. M. Hankiewicz. “Spin-helical transport in normal and superconducting topological insulators.” physica status solidi (b) 250, no. 2 (2013): 215. (arXiv) G. ...
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What is the physical interpretation of Z and Z2 in topological periodic table?

Different classes of topological materials are marked with 0, Z, Z2 in columns representing dimensions of these materials. What is the physical meaning of 0, Z, Z2 and what they represent? Thanks
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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 ...
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Chiral symmetry vs quantized Zak phase

I've been doing some condensed matter research about the topological phases in one dimension system and have some questions. I've heard that the chiral symmetry leads to the $\pi$-quantization of Zak ...
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1answer
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Units related to chemical potential and orbital magnetization

I am studying this paper: Physical Review B 74, 024408 (2006) (arxiv) ABSTRACT We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator (i.e., one in which ...
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866 views

Why is disorder essential for the Integer Quantum Hall effect IQHE?

The title already gives away the question. I see that disorder effects that the Landau levels are broadened out. They allow states to be either extended through the whole solid or localized to a ...
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Integral Approximation in Peierls Substitution

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (...

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