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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Calculation on the squashed entanglement

The definition of squashed entanglement:link I read a paper on this topic, and found a good result (Fig.2(a)) from it showing that the squashed entropy can indicate topological phase transition of SSH ...
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Relation betwwen the expectation value of the position operator in Bloch state with unitary position operator

Recently, I want to get some knowledge about topological insulators, so I read the A short course on topological insulators. In the chapter 3, the author try to explain why polarization connects with ...
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Proof for existence of edge (surface) states in topological materials

Many publications demonstrate the existence of edge (surface) states for certain topological model Hamiltonians and give reasonable arguments why this should hold in general. Is there a mathematically ...
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Coupling between unpaired Majorana modes in Kitaev chain

In this course, https://topocondmat.org/w1_topointro/1D.html, to argue when the edge will disappear, the authors say: The only possibility to move the energy levels from zero is to couple the two ...
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Are there any experimental realizations of semi-Dirac materials?

My question is there any experimental proof of semi-Dirac dispersion beside optical lattice?
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Current operator, why is this form valid up to second order in $q$?

Context In Bernevig´s textbook Topological insulators and topological superconductors, an approximate form of the current operator in momentum space is derived. It is said to be valid to second order ...
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Why Aharonov-Bohm interference loop in topological insulator nanowire/nanoribbon could be formed by two paths with opposite spin?

When applying a magnetic field along the axis of a topological insulator nanowire/nanoribbon, AB oscillations are observed in previous experiments, e.g., https://doi.org/10.1038/nmat2609. I am a ...
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Concise formula for Skyrmion number of a four-band Hamiltonian

The Haldane model on the honeycomb lattice is often written as $H=\sum_{i=0}^3 \vec{d}_i\cdot \vec{\sigma}_i$ where $\vec{\sigma}$ is the vector of Pauli matrices and $\vec{d}$ is a vector of ...
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Is graphene a topological material?

I understand that graphene is a great material with many exotic properties. Hoever, does it have a non-trivial band topology? It's mentioned that graphene has a weak spin-orbit coupling, which is ...
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What is a p-wave superconductor in 1-dimension often referred in the Kitaev model? What term in the hamiltonian makes it P wave?

I am currently reading papers related to Majorana zero modes observation in 1D nanowire systems. I am very new to the field and I read everywhere that the presence of spin-orbit interaction, magnetic ...
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Edge state protection in Chern insulator

I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
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$R$ projection operator in entanglement entropy

I was reading a paper about entanglement entropy. The author introduced a real space cutoff operator $R$ which he claimed to project onto a real space subregion. Then he used $\bar P:=RPR$ as an ...
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Topological properties of twisted TMD homobilayers

I'm reading this article about twisted TMD homobilayers (https://arxiv.org/abs/1807.03311) and there are certain topological properties that I don't understand: On page 3, in the paragraph next to Fig ...
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Absence of topology in semi-dirac materials

Good morning everybody, I am facing a problem when calculating the topological invariant in a semi-dirac system, whose Hamiltonian is: $$ H=k_x^2\sigma_x+k_y\sigma_y $$ My question is that this ...
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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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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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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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