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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Trivial examples for the Chern number from the potential for quantized transport

I'm trying to understand the phenomena of quantized electron transport better. The difficult step is that for any Hamiltonian (where $V(x,t)$ is periodic in both arguments and is a slow function of $t$...
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Argument for number of edge states as topological invariant for SSH model

I am currently reading the book "A short introduction to Topological insulators" by Asboth etal. In the first chapter on SSH model, they argue (see sec 1.5.3) that number of edge states is a ...
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Time-reversal and inversion operators commuting, topological insulators

See Eq. (1.26-1.27) in https://phy.ntnu.edu.tw/~changmc/Teach/Topo/latex/06.pdf, where it seems the parity and time-reversal operators commute. I was wondering how does this exactly come? Spins are ...
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Why does the conduction band conduct in the SSH model?

I am reading "A short course on topological insulators" https://arxiv.org/abs/1509.02295. There is a basic point that I don't understand about conduction and the band structure. ...
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About the degeneracy and the time reversal invariant point of Brillouin Zone

I'm confusing about some concept of the Band degeneracy in the band theory. As I know the Kramer Theorem guarantee that as long as your system is time reversal invariant and even the Spin orbial ...
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How can we judge the topological property of a material by looking at it's band structure?

I am a beginner of studying topological insulator. I want to ask some general question in this area to clarify my understanding. May be I am asking wrong, hope you can point me out. If certain ...
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About Chern Insulator

I know when we talk about Insulator, U(1)charge symmetry naturally exists. But in this article:Quantum phase transitions of topological insulators without gap closing, the author claims that: "...
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Some questions about Axion Electrodynamics

We know that the action of Maxwell's equation can add another term, that is $\theta$ term: S$\theta$=$\theta\int\vec{E}\cdot\vec{B}$. My questions are: 1.Why in TI(topological insulator), the $\theta$ ...
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Proof of necessity of Band Inversion in Topological Insulators

Is there any rigorous proof/argument for the necessity of Band Inversion in topological insulators? From a mathematical point of view, I am trying to find an answer in terms of the topology of the ...
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Topological Insulator vs Topological Band Insulator

Is there any conceptual difference between topological insulator (TI), and topological band insulator (TBI)? And if there is, what would it be?
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Chiral symmetry in Su-Schrieffer-Heeger (SSH) model

We know that the hamiltonian SSH model in the presence of on-site potential(V) can be written on the basis of the Pauli matrix. $$h(k)=V\sigma_0+h_x\sigma_x+h_y\sigma_y,$$ and the term V breaks the ...
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How to calculate Berry connection of a Hamiltonian numerically?

I hope there is a method to calculate berry connection of a Hamiltonian numerically, in order to calculate $\theta$-term like: $$\theta=\frac{1}{4 \pi} \int d^{3} k \epsilon^{i j k} \operatorname{Tr}\...
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Landau level's Chern number

I got some confusions reading the materials on integer quantum Hall effect and Berry curvature in Modern Condensed Matter Physics by Steven Girvin and Kun Yang. The following is in Chapter 13, ...
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Must helical edge states be protected by time-reversal symmetry?

In a lattice system that exhibits quantum spin Hall effect (QSHE), like topological insulators in 2D or 3D, we call a pair of counter-propagating gapless edge states with opposite spin helical edge ...
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Numerical solution to Harper's equation

I want to learn how to solve Harper's equation and plot the Hofstader butterfly spectrum numerically. Can anyone suggest references/links where this is done?
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Topology of Helium 3A and 3B

The question concerns the topology and dimensions of Helium 3A and 3B A. The Helium 3A phase shows the same low energy excitations as those of a 2 spatial dimensional chiral p-wave superconductor --- ...
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Single crystals for observing toplogical phase

The topological devices are mostly fabricated by single crystal growth technique. Is it necessary to have monocrystal for observing topological phase? If yes, why so? Else, can we observe the ...
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Calculating Winding Number Numerically in MATLAB

How can I calculate the Winding number numerically in Matlab? I mean, I want to calculate the $$W=\frac{1}{2\pi i}\int_{-\pi}^\pi dk \frac{d}{dk}Ln h(k)$$ by using a loop that changes k from $-\pi$ to ...
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About the $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect)

We know that $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect) can be calculated by the Berry curvature, but we also know that $\sigma_{xy}$ is calculated by the ...
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55 views

Why topologically non-trivial materials are robust againist any external perturbations or defects?

Topologically non-trivial materials are insensitive to perturbations or defects. How can I prove it mathematically? I thought of making the first-order perturbation term zero. $$\left< \psi \right|...
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Inversion Symmetry on Internal Hilbert Space of Bulk Lattice

I am currently studying Short Course On Topological Insulator by J. K. Asb´oth, L. Oroszl´any, A. P´alyi. In Chapter 3.2, I have a few questions to ask regarding it: Is the phase $e^{i\phi}$ in (3.31)...
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Time reversal symmetry and quantum spin Hall effect

As it knows the edge states in QSHE are protected by TR symmetry, so any perturbation that are symmetric under time reversal cannot destroy these states. A key point is that for $T^{2}=-1$ we have a ...
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Staggered Zeeman field in topological magnetic insulators

I was reading the following paper. However, I do not understand a crucial part of their argumentation. They add a parity (P) and time (T) symmetry breaking term to the Hamiltonian in eq (2). Then they ...
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How do I see if the material is a Topological Insulator from the band structure?

In this paper1 the following bandstructure of Bi$_2$Se$_3$ is shown: In "a" they show the bands without Spin orbit coupling (SOC) and in "b" they include SOC. It is said that: &...
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Current in a 2D p-wave superconductor

If leads are connected to the edge of a D>1 topological superconductor, will the current flow differently from a conventional s-wave superconductor ? Will the current flow through the gapless edge ...
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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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63 views

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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