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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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 classes. For class D (broken time-reversal symmetry, particle-hole ...
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What is a bulk state and bulk bands?

I am a bachelor student and I started studying topology and I came across two terms I have never seen before: Bulk band structure and bulk states. Can someone explain these two terms or provide me a ...
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Why does particle-hole symmetry in 1D lead to a $Z_2$ topological invariant?

From the well-known AZ Tenfold Classification Table, a 1D system with square-positive particle-hole symmetry belong to class D and hence is characterized by a $Z_2$ topological invariant. I suppose ...
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A K-theory isomorphism

I found this identities in a paper on Floquet topological classification which the author mentioned as a "well-known K-theory isomorphism" $$K_{R}^{0,n}(S^1\times X, \{0\}\times X) = K_R^{0,...
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What spaces these metrics parameterize?

$ g_{red} :g_{\mu \nu} = \left( \begin{array}{cc} \frac{1}{8 \left(-y^2+z^2+1\right)} & 0 \\ 0 & \frac{1}{8 \left(y^2-z^2-1\right)} \\ \end{array} \right)$ $ g_{green} :g_{\mu \nu} = \left( \...
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3D topological insulator without $C_{n>2}$ rotational symmetry?

Is there any known/predicted 3D topological insulator without any $C_{n>2}$ rotational symmetry?
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What does “continuous transformation” mean with regard to the Hamiltonian of a system?

When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the ...
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What determines helicity of 3D topological insulator surface state

Consider a 3D topological insulator surface state Hamiltonian with mass possibly due to magnetic doping $$\sum_{i,j=x,y}v_{ij} k_i \sigma_j + m\sigma_z.$$ What determines the helicity $X=\mathrm{sgn}[\...
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Homotopy group for spin-1 BEC

Homotopy group can be used to classify topological defects. The procedure is Find the Lie group $G$ that leaves the free-energy functional invariant when transforming $\psi$, where $\psi$ is the ...
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Phase freedom of the edge states in topological insulator

Suppose that we consider the BHZ-like Hamiltonian of the form $$ H_{bulk}=\left(M-B k^{2}\right) \tau_{z}-A k_{x} \tau_{y}+A k_{y} \sigma_{z} \otimes \tau_{x} $$ where $\tau_i $ acts on the orbital ...
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Topological phases of matter

So according to this, scientists have discovered more than 5 states of matter we usually had that is the solid, liquid, gases, and Bose-Einstein-Condensate, and plasma. So how many topological phases ...
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Related to Topological phases of matter

I have a very fundamental doubt, I am new to this field and I learned that the quantum hall effect was the first topological state of matter discovered, so is every step in the graph of the quantum ...
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Homotopy classification in ten-fold way

I am trying to understand algebraic invariants in topological insulators and topological superconductors through homotopy. But I encounter kind of a conceptual question. Let's say we have a second ...
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Berry phase of generic two-dimensional gapless Dirac Hamiltonian

Reference: Topological Insulators and Superconductors, B. Andrei Bernevig, Taylor L. Hughes: Chapter 8, problem 1 The generic Bloch Hamiltonian $H(k)=k_i\mathcal{A}_{ij}\sigma_j$, with $i\in\{1,...
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Why is the topologically non-trivial edge state robust against off-diagonal disorder?

By numerical calculation it's easy to see that the topological non-trival edge state is robust against off-diagonal disorder, i.e., the edge state keeps exponential decay with distance in the presence ...
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Why does the Berry Phase of π cause anti localisation in Dirac fermions?

I am learning about the theory of topological insulators and one point that puzzles me is the following: The Berry Phase aqcuired by forming a closed loop on a Dirac cones is π. The argument that I do ...
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Is the Berry phase defined in terms of the periodic part of the Bloch wavefunction or the wavefunction itself?

In this paper, the berry phase is approximated to be $e^{-i\theta} = \prod_{i=1}^{N} \langle\psi_{n,k_i} | \psi_{n,k_{i+1}} \rangle$. The authors claim that "each Bloch wavefunction appears twice ...
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Is Hall conductivity time-reversal-odd at finite frequency in a topological system?

In some topological materials, e.g., the quantum (anomalous) Hall state and some related variants, the Hall conductivity $\sigma_{xy}$ is quantized and directly related to the Chern number, which ...
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Numerically calculating non-Abelian Berry curvature: Definition of multiplet in explicit $4\times 4$ system with 2-fold degeneracy?

I am trying to use eq 16 of the following paper to calculate the Chern number of a 4x4 degenerate system: https://arxiv.org/abs/cond-mat/0503172 [1]. I believe this is the standard scheme used by many....
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Time Reversal Symmetry: An Intuitive Picture

Okay so here is my confusion. When reading into TR as applied both to Classical and Quantum Physics, I have found two opinions and somehow I do not know which one is more correct than the other. The ...
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Calculating the Hall Conductance using the torus shape for the Magnetic Brillouin zone

Komoto's paper (ANNALS OF PHYSICS 160, 343-354 (1985)) on the calculation of the Hall conductance provides a clear discussion about how calculate the conductance using the torus shaped magnetic ...
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Time reversal of scalar Bloch states

There's a well-known result about time reversal symmetry in crystals that's important when studying the physics of topological insulators, but I am having trouble proving it. I'll start with notation. ...
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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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64 views

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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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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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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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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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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