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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Conditions for Bogoliubov-de Gennes Hamiltonian representation

The $H_{BdG}$ hamiltonian is described in topocondmat.org as follows: here we can see that the submatrices along the diagonal are related as negative of complex conjugate of each other, I feel that ...
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Spin Hall Effect in 2D Topological Insulator

Suppose the 2D topological insulator has the magnetic element doping in the system and the easy-axis for the magnetization is along the z axis. The the surface gap is opened due to the time-reversal ...
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Representation of symmetry operators in second quantuzation

Hamiltonian invariant under a symmetry- The action of a group $G$ on the set of Bloch momentum is given by a linear representation $T_g: k \to T_g k \equiv k_g $. Now say that a fermionic Bloch ...
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How to calculate surface states in Weyl semimetals?

I'm reading an article https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235127. Fig. 2 in this article shows band structures calculated from Eq. (9), (13), (14), (15), and (16). For example, ...
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Why are topological materials/phases "exotic"?

From what I understand, when a system has topological order, any local perturbation doesn't change the phases and thus its properties. This would suggest that it should be really easy to find ...
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Why does "chiral symmetry" in the Altland-Zirnbauer classification mean something different to other contexts?

The Altland-Zirnbauer classification of random matrices is based on three symmetries: time-reversal, charge conjugation, and a third which is sometimes referred to as "chiral" or "...
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Calculate the energy gap using Green's function

Can I calculate the energy gap of the given Hamiltonian by Green's function? Is there any basic code in MATLAB to do that?
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What is the difference between topological insulators and axion insulators

I'm currently reading a paper about an antiferromagnetic axion insulator candidate and am having trouble understanding exactly what is meant by the words axion insulator in this case. To my knowledge ...
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Quadruple index and the existence of corner state

I have been following this paper, which seems to discuss the connection between corner state and a quadruple index calculated $q(k_z)$ by equation (3) of it. At two special point $k_z=0 /\pi$, the ...
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Does gap closing and reopening guarantee a non-trivial topological phase?

I know that a Dirac point carries a Chern number of $\pm\frac{1}{2}$ and when we have a gap closure at any point in our band structure we can transfer Chern numbers depending on at how many points we ...
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Topological classification of the Kane-Mele model?

Where should the Kane-Mele Model fall in the 10-fold way topological classification? I see that it is on a honeycomb lattice which is bipartite and thus has particle-hole symmetry. Going by that ...
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The flat-band basis, Green's function projectors, and TKNN

Bernevig & Hughes' book "Topological Insulators and Topological Superconductors" have a derivation of the TKNN invariant in terms of the finite-temperature Green's function. From the ...
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Why $NA-NB=1$ when $v<w$ for SSH model

Based on A Short Course on Topological Insulators, chapter 1 for SSH model, in order to show the bulk-boundary correspondence, (the winding number equals $NA-NB$, the net zero-energy edge state number ...
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Topological hinge states at only one pair of diagonal edges?

High-order 3D topological systems can have hinge states at the edges like the figure below from this paper. But can we have only one pair of counter-propagating diagonal edges? I mean remove the red ...
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Is the derivative of Berry curvature with respect to band energy possible?

The Berry curvature is one of the most important quantities for a topological material like Dirac semimetal(DSM), Weyl semimetal(WSM) etc. Berry curvature is always momentum ($\mathbf{k}$) dependent ...
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Berry flux of magnetic monopole

I am sorry if it looks stupid question. I want to ask how sin is transferred back from spherical to Cartesian and how $F_{ij}$ tells us about magnetic monopole in magnetic field. $F_{ij}$ is berry ...
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Numerically calculating the berry curvature for graphene

i'm trying to reproduce this density plot for the Berry curvature in the Brillouin zone of graphene from this website. In order to do this I am attempting to use this equation for the berry curvature ...
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What's the relation between quantized Hall effects and topology materials?

The quantized Hall effects (ignoring fractional Hall effect) include: Quantum Hall effect; Quantum anomalous Hall effect; Quantum spin Hall effect. All these effects are just describing the ...
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What does the toy model Hamiltonian for 3D weyl seminmetal mean?

I have been going through this Wannier tools tutorial for observation of weyl nodes in basic toy model Hamiltionan. It represents the toy model Hamiltonian as: $$H = A(k_x\sigma_x+k_y\sigma_y)+[M_0-...
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Detection of topological phases

In the book A Short Course on Topological Insulators (https://arxiv.org/abs/1509.02295) the authors start with a simple toy model, the SSH-Chain, which is a bipartite one-dimensional lattice with ...
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What does Dirac cones at the $K$ and $K'$ point mean?

Graphene is a topological insulator with Dirac cones located at the points $K$ and $K'$ in the Brillouin zone. The bands at these points correspond to conducting edge modes. However, if we consider a ...
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Why number of left-moving and right-moving edge states on a finite lattice system is equal?

I read an arguments about number of left-movers and right-mover in finite system in paper titled as Antichiral Edge States in a Modified Haldane Nanoribbon. In second paragraph of introduction, it ...
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The $\rm SO(8)$ invariant interaction piece in Fidkowski and Kitaev's model

In this paper (arXiv link), the authors demonstrate the existence of a quartic interaction $W$ involving the 8 majorana operators $c_1 \ldots c_8$ (eq. 8) which is invariant under an $\rm SO(7)$ ...
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Kitaev chain and fermionic parity

I am learning the free open course on topological insulator on edX. And I am reading the section: Bulk-edge correspondence in Kitaev Chain. I am stuck at the subsection: Connecting the bulk invariant ...
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Is the Integer Quantum Hall Effect a distinct phase of matter?

In the Landau classification scheme, phases of matter differ in terms of symmetry. However, we know of many instances where this classification scheme does not apply. I have often heard topological ...
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What is the physical meaning of adiabatically varying the wavevector $k$ as a parameter to calculate the Chern number for topological effects?

Could it mean something like applying a weak electric field and perturbing the band structure? Or some other weak perturbation? Or is that the wrong idea?
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How to describe SSH chain with odd number of sites?

Usually when we discuss SSH(Su-Schrieffer–Heeger) chain, we discuss a chain with 2N atoms, with v the intra-cell coupling and w the inter-cell coupling. When N is infinite, the system becomes bulk, ...
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How to see that the trivial insulator is trivial?

I have been trying to better understand gapped phases of matter — which may be "topological" or "trivial" — and I have run into the problem that I don't really understand the ...
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From momemtum Hamiltonian to real space Hamiltonian

I know how to calculate a bulk momentum space Hamiltonian from a real space one. For example, given a SSH model $$H=v\sum_ic^\dagger_{iB}c_{iA}+w\sum_ic^\dagger_{i+1,A}c_{iB}+h.c.,$$ its bulk momentum-...
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What's the origin of the chirality selection rule?

I'm reading this paper on Circular Photogalvanic Effect (https://www.nature.com/articles/nphys4146) and they mention a "chiral selection rule": Chirality selection rule: Right-handed ...
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Calculation of the Fermi Arc

Can I ask two questions, regarding the Fermi Arc? Suppose there are only three atoms in the bulk Weyl Semimetal system and the Fermi Arc of the system is within the x-y plane. If there is only one ...
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Group velocity (open vs periodic boundary conditions) [duplicate]

I'm trying to understand the meaning of the group velocity for Bloch electrons given by $$ \mathbf{v}=\frac{1}{\hbar}\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}} $$ where $E(\mathbf{k})$ is the ...
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kp Model - Topological Quantum Chemistry

I have the following problem. I want to be able to construct Hamiltonians using the irreducible band representations provided by the TQC website. Say, for example, the model Weyl semimetal TaAs https:/...
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Equivalence between Hamiltonians of different dimensions

Consider a 2D lattice Hamiltonian $H_2$ of symmetry class A and a 1D ladder Hamiltonian $H_1$ of class AIII having the same number of bands and the same TKNN number for each band. Can $H_2$ and $H_1$ ...
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Inversion symmetry in Kane-Mele model?

I am trying to understand how the famous Kane-Mele spin-dependent hopping term in Quantum Spin Hall state respects the parity symmetry. As far as I understand, the spin does not change sign under ...
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What is the topological space in “topological materials/phases of matter”?

I’m embarrassed to admit that after sitting in on several “topological physics” seminars, I still don’t understand the basic ideas of the area. In particular, when physicists talk about the “topology” ...
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Position of the Wannier center in tight-binding model

With the Fourier transform of the annihilation and creation operators, $$c_{m,n} = \frac{1}{\sqrt{N}} \sum_{k_x} \sum_{k_y} c_{kx,ky} e^{-i\mathbf{k}\cdot \mathbf{r}_{m,n}} \quad\text{and}\quad c^{\...
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Noncontractable loops in the 2D Brilluoin zone and the Chern number

I'm getting quite twisted around trying to figure out what all is quantized exactly in IQH looking at it from the Chern number perspective. Let's suppose quantum hall on a torus -- I can apply a large ...
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Quantization of the Berry Phase

Let's consider the Aharanov-Bohm effect. Following Girvin & Yang, an infinitely long, very thin flux tube running along the $\hat z$ axis is surrounded by a strong potential barrier preventing ...
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Inversion Symmetry in Periodic Lattices

I am studying Short Course On Topological Insulator by J. K. Asboth, et.al. In the context of inversion symmetry in section 3.2, the effect of inversion symmetry, $\Pi$, on the external degree of ...
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What is helical Dirac nature?

A concept in Spintronics which can not be found on Wikipedia. The picture is from a review of Spintronics of 2016 by Fert.
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Confusion in change of basis for anyon fusion

I'm doing some introductory reading on anyons, and I'm a bit confused by the way the basis are changed. Suppose we have the Ising anyons, which obey $1\times1=1$, $1\times\Psi = \Psi$, $1\times\sigma =...
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Chern number for electronic energy bands with orthogonal states

In the "lecture notes on Topological insulators by Asboth et all" the Chern number is defined on the basis of phase change of non-orthogonal states on a closed torus. Nevertheless, in ...
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Is there a difference between zero resistance and dissipationless transport?

Zero resistance in superconductors is what I have learned in university and in textbooks. But in the field of topological insulators, I have seen the phrase "dissipationless transport" get ...
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Topological insulators, Chern number

Chern number calculation by discretized brillouin zone method as discussed in Fukui paper, anybody can give example where this detail analysis of this method has been used? The paper is Takahiro Fukui,...
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Chiral symmetry in the SSH model

According to "A short course on topological insulators", chapter 1, in the SSH model, the consequence of chiral symmetry for the states with $E\ne 0$ is the presence of another state with $-...
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Topological insulators

https://arxiv.org/abs/1504.05280 in this paper author derived numerically orbital magnetization of 2d thin topological insulators say graphene like system numerically. I have tried to reproduce this ...
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Adiabatic approximation

The celebrated adiabatic theorem states that for a system initially in the eigenstate $|\psi(0)\rangle = |n(0)\rangle$ for $t=0$, it will stay in that state afterward under adiabatic evolution: $$ |\...
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Calculation of Bulk and edge states in SSH model

I am reading “A Short Course on Topological Insulators” by János K. Asbóth. et.all., and want to calculate the Bulk and edge state of the SSH model (Chapter 1) to drive the energy spectrum in Fig. 1....
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Physical examples of topology class D

In the course of writing a paper, I am compiling a list of various physical examples of the ten different topological symmetry classes1. For class D (broken time-reversal symmetry, particle-hole ...
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