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.

Filter by
Sorted by
Tagged with
1
vote
1answer
39 views

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 $-...
1
vote
1answer
65 views

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,...
0
votes
1answer
43 views

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 ...
2
votes
1answer
193 views

Attempt at proving $-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$ from Kane and Fu's paper

I am trying to prove result (3.4) of the following paper: http://li.mit.edu/S/2d/Paper/Fu07Kane.pdf namely, that $$-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$$ ...
1
vote
1answer
143 views

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, ...
1
vote
1answer
962 views

Does time reversal symmetry hold for (Kitaev model) 1D spinless $p$-wave superconductor?

The hamiltonian 1D spinless $p$-wave superconductor can be written as $$ H=\sum_k \phi_k^\dagger \begin{pmatrix} \xi(k) & 2i\Delta \sin(k)\\ -2i\Delta \sin(k ) & -\xi(k)\end{pmatrix}\phi_k$$ ...
1
vote
1answer
584 views

Choice of Unit Cell on Band Diagram (Brillouin Zone Folding)

I am looking at photonic band diagrams specifically, but my question relates to band diagrams in general. For a honeycomb lattice, I can pick a (primitive) rhombic unit cell or a hexagonal unit cell. ...
3
votes
1answer
244 views

Is spin-orbit coupling really necessary for topological insulators

I have heard that for an insulator to be non-trivial, large spin-orbit coupling is necessary. However, I have read the definition of $Z_2$ topological invariant and chern number. In no way can I ...
1
vote
1answer
50 views

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: $$ |\...
1
vote
1answer
48 views

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 ...
2
votes
1answer
51 views

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....
1
vote
1answer
38 views

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 ...
8
votes
1answer
261 views

How to perform stroboscopic measurements for Floquet topological insulators?

Floquet topological insulators (arXiv:1008.1792, arXiv:1211.5623) have attracted much research interests in condensed matter physics. The goal is to realize topological insulators from trivial ...
1
vote
0answers
50 views

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 ...
11
votes
4answers
3k views

Bulk boundary correspondence = difference in Chern numbers?

In topological insulators the bulk boundary correspondence is frequently stated as the principle that the number of edge modes equals the difference in Chern numbers at that edge. I found this e.g. in ...
4
votes
1answer
219 views

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. ...
3
votes
1answer
96 views

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: "...
1
vote
1answer
212 views

Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form $$\mathcal{H}=\hat{\Psi}^\dagger H \hat{\Psi}$$ with $$\hat{\Psi}=\left(a_1, a_2, ...,...
1
vote
0answers
30 views

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 ...
2
votes
1answer
31 views

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,...
0
votes
0answers
46 views

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( \...
0
votes
0answers
19 views

3D topological insulator without $C_{n>2}$ rotational symmetry?

Is there any known/predicted 3D topological insulator without any $C_{n>2}$ rotational symmetry?
0
votes
2answers
67 views

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 ...
0
votes
0answers
38 views

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}[\...
0
votes
1answer
43 views

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 ...
1
vote
1answer
79 views

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 ...
1
vote
0answers
319 views

2D BHZ tight binding model for Quantum spin Hall insulator

I am currently reading this article : https://arxiv.org/abs/cond-mat/0611341 and want to derive the k-space tight binding model of 2D BHZ. The tight binding model is written as \begin{equation} H = \...
1
vote
0answers
14 views

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 ...
7
votes
1answer
2k views

How to determine the parity eigenvalues of time-reversal invariant momenta point from first principle calculation when we judge topological insulator?

This is a question of topological insulator. Liang Fu and C. L. Kane proposed a method to judge whether an inversion symmetric insulator is a topological insulator or not in their article(L. Fu and C....
15
votes
4answers
12k views

Chern insulator vs topological insulator

What is the basic distinction between a Chern Insulator and a Topological Insulator? Right now I know that a Chern Insulator has "topologically non-trivial band structure" and that a Topological ...
1
vote
2answers
86 views

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 ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
205 views

Source Berry Curvature Chern Insulator

Why is there non-zero hall conductance for a Chern insulator? From section 2.3 of Bernevig's book 'Topological insulators and topological superconductors' I learned one can view degeneracies are ...
0
votes
0answers
25 views

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 ...
3
votes
1answer
515 views

Regimes of Josephson junction

There are several formulas to describe critical current in Josepshon S-N-S junction mainly based on Eilenberger and Usadel equations for quasi-classical Green's functions. The starting point is the ...
5
votes
2answers
294 views

Why, the Valence Bundle is non-trivial and the complete Bloch Bundle isn't?

I'm thinking about the bundles defined in Topological Insulators, and I took that sentence, present in many sources, as true: The topology of the Valence Bundle is non-trivial, i.e., the Valence ...
0
votes
0answers
51 views

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,...
2
votes
2answers
1k views

Time reversal of Bloch Hamiltonian (at TRIM points)

I know that time-reversal symmetry requires the Bloch Hamiltonian $H(\textbf{k})$ to transform as: $$ \Theta H(\textbf{k}) \Theta^{-1}=H(-\textbf{k}) $$ where $\Theta$ is the time-reversal operator. ...
3
votes
1answer
353 views

Why hopping amplitude with no negative sign?

I'm learning SSH model now. I notice people use tight-binding model of this form, $$H=t\sum_{<i,j>} c_i^†c_j+\mathrm{H.c}$$ where $t>0$ in Lecture 1 : 1-d SSH model, or A Short Course on ...
2
votes
5answers
2k views

Understanding the Su-Schrieffer-Heeger (SSH) model and Topological insulators regarding invariants

I'm studying the topic of Topological insulators, I'm having a very hard time understanding what is the relationship between the fact that topological invariants are different from $0$ and the ...
5
votes
1answer
130 views

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 ...
2
votes
1answer
318 views

Parity of Bloch states at TRIM points

There is an argument presented in Fu and Kane's paper on inversion symmetric topological insulator which I have not yet convinced myself. Just below Eq.(3.6), the authors said that because of ...
1
vote
0answers
16 views

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 ...
4
votes
2answers
364 views

Why does proximity to a superconductor open a gap in the surface states of topological insulators

I have read in many places that the gapless surface states of 3D topological insulators are robust to perturbations which do not break time-reversal symmetry. I have recently also seen many papers (...
2
votes
0answers
84 views

How to interpret overlap in Hamiltonian if it is not a degeneracy?

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ h_x +...
9
votes
3answers
804 views

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 ...
0
votes
0answers
34 views

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 ...
4
votes
2answers
428 views

Topological insulators with time-reversal symmetry: How to derive $\mathbb{Z}_2$-invariant from the zeros of $\langle u_i(k)|T|u_j(k)\rangle$?

Following the book ''Topological Insulators and Topological Superconductors'' by B. Bernevig (esp. chapter 10.1), I want to understand how to derive a Z2 invariant starting from the zeros of the off-...
1
vote
0answers
29 views

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 ...
1
vote
1answer
76 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 ...

1
2 3 4 5
9