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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Flat-Band Basis (Bernevig & Hughes) for computation of Hall conductance

In Topological Insulators and Topological Superconductors (Bernevig & Hughes) a limit of an insulating Hamiltonian, the flat-band limit is used to compute the Hall conductance. For the fist we ...
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Analytically calculate the zero energy edge states of the Su-Schrieffer-Heeger model

Is it possible to analytically calculate (or verify the existence of) zero energy edge states for the SSH model in real space? This seems to be discussed in Section 1.5.2 of "A Short Course on ...
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What does QL mean in experimental condensed matter physics (thin film growth)?

In literatures of thin film growth I often see the unit QL. It often occurs at contexts like "a 64 QL film," "the growth occurs QL-by-QL," and "the growth rate was found to be ...
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Odd number of surface Fermi surfaces in strong Topological Insulators

I am now getting into topological insulators and I have read several times that a strong TI possesses always an odd number of Fermi surface bands crossing the Fermi level. However, this is not ...
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Difference between “ordinary” quantum Hall effect and quantum anomalous Hall effect

I am reading a review article on Weyl semimetal by Burkov where he writes, top of page 5: A 3D quantum anomalous Hall insulator may be obtained by making a stack of 2D quantum Hall insulators [Ref. ...
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Aharonov-Bohm effect in topological insulator in a square lattice

Does the presence of the Aharonov-Bohm (AB) effect break Time-reversal symmetry (TRS) for spinless systems in a topological square lattice? As we know that TRS protects the edge states in Topological ...
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Particle Hole Symmetry of BdG Hamiltonians

It is straight-forward to verify that any Hermitian BdG Hamiltonian of the form $$ \mathcal{H} = (c_1^\dagger, c_1, c_2^\dagger, c_2,...) \begin{pmatrix} H_{11} & H_{12} & \cdots \\ H_{21} &...
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Some questions in Asboth, Oroszlany, Palyi 's lecture notes on Topological insulators

I am reading Asboth, Oroszlany, Palyi 's lecture notes on Topological insulators. I am having some diffculty with the mathematics on page 4, Section 1.2.1. We have the SSH hamiltonian: $$ \hat H_{bulk}...
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Transverse current in a 2D topological pump?

Topological pumps (Thouless pumps) and Chern insulators are often brought up in the same context, as they both result in quantized transport, and feature a certain robustness (they are 'topologically ...
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Realization of SSH model through electrical circuits - how to measure impedance?

I am trying to reproduce the results given in this paper. The authors create a circuit whose $I-V$ equations are similar to the Hamiltonian of the SSH model. And then through impedance measurement, ...
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Majorana modes at the edge of a QSHI

I am currently looking at Fu and Kane paper: Phys. Rev. B 79, 161408(R). They write the QSHI edge states as $H_{\text{edge}}=\psi^{\dagger}(-iv\sigma_{z}\partial_{x})\psi$ where $\psi=(\psi_{\uparrow},...
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Chern number for nonintracing hamiltonian while bands crossing

Is it possible to define and calculate chern number for two bands while they're crossing each other?
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Berry curvature vanishes in TRS system

In spin 1/2 system with TR symmetry , the Berry curvature must vanish. Because Berry curvature is odd. How to prove it? \begin{equation} \langle\partial_{-k_x}u^{I}(-k)|\partial_{-k_y}u^{I}(-k)\rangle-...
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Winding number as topological invariant in Su-Schrieffer-Heeger (SSH) model

I'm studying the SSH model, here's the reference. I don't get what the definition of a topological invariant is in this case. I think the important property is that the winding number cannot be ...
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Insulator or conductor with different boundary conditions

I'm studying the 1-D SSH model. It's a toy model for a topological insulator. Here's the reference I'm using. If the hopping amplitudes $v$ and $w$ are equal, then with periodic boundary conditions we ...
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How do topological insulators violate Nielsen-Ninomiya Theorem?

I am under the impression that topological insulators have a distinguishing characteristic where they have an odd number of Dirac points that intersect band gaps at the Fermi energy. However, this ...
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How to obtain the optical constants (dielectric function or complex conductivity) of a material from its band structure?

I know that the complex conductivity ($\sigma = \sigma_1+i\sigma_2$) is related to the dielectric function ($\epsilon = \epsilon_1+i\epsilon_2$) by: $$ \epsilon_1 = 1 - \frac{4\pi\sigma_2}{\omega} \\\...
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Why does an energy band crossing the Fermi energy mean the gap closes?

This online course on topology in condensed matter states the following: We say that two gapped quantum systems are topologically equivalent if their Hamiltonians can be continuously deformed into ...
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What is a topological Dirac semimetal?

I have just started learning about a topological Dirac semimetal. Then I'm wondering that the Dirac point always crosses the Fermi energy in a topological Dirac semimetal. If the Dirac point does not ...
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Computing Two-point correlation functions and the 'Quantum Loop Topography' ML method

In this paper by Y. Zhang and E-A. Kim, the authors have designed a novel pre-processing step for machine learning topological classification. The gist of the paper is that the authors compute chained ...
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Bosonic SPT phases with time reversal and a $Z_2$ symmetry

Consider a bosonic system with time reversal symmetry $\mathcal{T}$ and a unitary on-site $\mathbb{Z}_2$ symmetry. Suppose the symmetry is realized in a special way such that $$\mathcal{T}^2= (-1)^B$$ ...
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Can I manually break the spin-momentum locking in topological insulator by large magentic and electric field

Spin-momentum locking forces the spin and momentum to be orthogonal. If I apply a strong magnetic field and a strong electric field in the same direction, will the locking be broken and spin&k are ...
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Winding number in Su-Schrieffer-Heeger (SSH) model

I'm studying the SSH model from this review and on page 14, equation (1.38), they give a formula from evaluating the winding number saying it's easy to check it. Now, I've done the math and came up ...
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What are the differences between Haldane phase, non-interacting topological insulator/superconductor, and SPT order?

Haldane phase, and non-interacting topological insulator/superconductor are often regarded as examples of symmetry protected topological (SPT) orders.
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What does “parity eigenvalue” mean in Fu-Kane formula?

I'm studying the online course "Topology in Condensed Matter", in the QSHE section (<https://topocondmat.org/w5_qshe/fermion_parity_pump.html>), I've studied the Fu-Kane formula $$ Q=\...
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How are Majorana zero-modes protected against fermionic operators?

I am learning about Majorana fermions in topological quantum computation, and more particularly about the Kitaev chain, described by $$ H = -\mu \sum_{i=1}^N c_i^\dagger c_i - \sum_{i=1}^{N-1} \left(t ...
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The “basic hamiltonian” of topological systems

I am currently studying topological insulators and repeatedly found the claim (e.g. here), that the "basic hamiltonian" of a topological system in $d$ spatial dimensions can be written using ...
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Topological Insulator - why does a band have to be isolated to have a Chern number of 1?

I'm trying to understand the principle of topological insulator. Why a band has to be isolated to have a Chern number of 1? More precisely, why, in the case of Haldane Model, all the bands in the ...
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Interacting helical edge state and scattering process

Imagine if you have a 2D topological insulator system where you can bring together the helical edge states from the opposite boundaries to interact (say, by a voltage gate or by a narrow constriction)....
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What are the most general symmetries that a Hamiltonian of the form $H=\vec{k}\cdot\vec{\sigma}$ can have?

Hamiltonians of the form $H=\vec{k}\cdot\vec{\sigma}$ with $\vec{k}$ being the crystal momentum and $\sigma_i$ being the $i$-th Pauli matrix (an $su(2)$ generator), are pretty common in the study of ...
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Why are energy levels half filled in SSH model?

The SSH model describes states of electrons in a polyacetylene chain, which is modeled as a lattice with two orbitals per site. Now, in many articles it is claimed that in the ground state, half of ...
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Is $(x,y)\rightarrow (-x,-y)$ an inversion transformation?

Does anyone know whether $(x,y)\rightarrow (-x,-y)$ is an inversion transformation or not? I know that the standard inversion (parity) transformation in two dimensions should be something like $(x,y)\...
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Can Berry phase been carried by bulk electrons in TIs?

I'm studying 3D topological insulators and more in particular, weak antilocalization (WAL) effects, so I know that they are characterized by a $\pi$ Berry phase that gives rise to destructive ...
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Why the energy bands for edge states are not periodic?

Consider one edge state for a Chern insulator. It could have the band structure as shown below. We can see that the band for the edge state does not obey $E_{-\pi/a}=E_{\pi/a}$. I remember the ...
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Floquet bandstructure calculation

In this paper "Photonic Floquet Topological Insulators" the authors calculate the bandstructure of a time-periodic Hamiltonian. They create a time-dependent tight-binding Hamiltonian via the ...
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Question about the proof of different contractions for the effective Brillouin zone differing by an even Chern number

I am reading Moore and Balents'paper (DOI: 10.1103/PhysRevB.75.121306) which proves the Z2 invariant is the parity of the Chern number for the effective Brillouin zone. I am confused about some ...
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The quantum hall effect and Hofstadter's butterfly spectrum

What is the connection between the quantum Hall effect and the Hofstadter's butterfly spectrum? I mean, can I understand something about the quantum Hall effect in the Hofstadter's butterfly spectrum?
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Fourier Transform of a Superlattice Hamiltonian

In a paper by Gábor B. Halász and Leon Balents they derive the energy band structure for a Hamiltonian that models a time reversal invariant realization of the Weyl semimetal phase. The model is a ...
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Is there any heat loss in chiral edge channels of topological insulators?

If we are working with nontrivial topological insulator with broken time reversal symmetry then we can expect that we have some chiral edge states. Chiral states have the property that the current can ...
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Topological invariants, what's that?

What's the difference between the Berry phase, the Euler number,the winding number and the Chern number? As far as I know they can all be computed by the same integral, but there seems to be some ...
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Mutual statistics between dyons (charge-monopole composite)

I am asking for some intuitive understanding between two dyons with $(e,m)$ in 3-dimensional space. Here the magnetic charge $m$ is normalized as \begin{eqnarray} m=\int_{S^2}\frac{B}{2\pi}\in\mathbb{...
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How can the localization property of the edge mode in topological insulator/quantum hall system be manifested through the effective action?

To be more specific, we can write down the Chern-Simons term from coupling the system to EM to describe the 2d quantum hall system and its derivative respect to the EM field gives the current. How can ...
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How does the effective action describing the EM response of (3+1)d topological insulator become the (2+1)d Chern-Simons term?

Mathematically, it seems to be resulting from Stokes theorom once the 3d manifold has a 2d boundary. However, the EM response described by (2+1)d CS term requires the system to break time-reversal ...
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What's the relation between quanutm hall effect and topological insulator state?

In a recent PRX paper(https://journals.aps.org/prx/abstract/10.1103/PhysRevX.10.011050), I see that in 45nm and 50nm-thick Cd3As2 films, they find quantum hall effect and say that this is because of ...
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What is the physical significant of monopole charges in Nodal Line Semimetals?

Topological nodal line semimetals is formed when the symmetry of a semimetal system enforces band touching occurring on 1-dimensional submanifolds of the Brillouin Zone. Following the topological ...
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Clarification on a statement Bernevig's textbook on Topological insulators

On page 11 of the aforementioned book Bernevig claims after doing some calculation that the integral of Berry curvature over a sphere containing a monopole is $2\pi$. Now my question is the following: ...
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Wave function of $s$-band with odd parity

How does a wave function of a $s$-band state with odd parity looks like (in real space)? To keep it simple, the restriction to a linear chain and a state at the $\vec{k}=0$-point might be useful. My ...
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Alternatives for calculating topological invariants in topological materials

My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
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How to calculate $\sigma_{xx}$ in lattice model?

It is known that one can find the Hall conductivity $\sigma_{xy}$ from a lattice model (in a magnetic field, say) using the TKNN formula (PRL 49 405-408 (1982)), i.e. by summing the Chern numbers for ...
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Green function of three band crossing topological material in energy coordinate representation do not converge

Recently I am studying the properties of Green function of three diamentional topological materials with three band crossings, whose Hamiltonian can be written as $\hat H= v_f \ \vec k \cdot \vec S $, ...

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