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
0
votes
0answers
28 views
+50

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 ...
4
votes
1answer
135 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. ...
2
votes
0answers
38 views

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 ...
2
votes
2answers
26 views

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: &...
0
votes
0answers
16 views

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 ...
1
vote
1answer
157 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, ...,...
0
votes
1answer
12 views

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 ...
1
vote
0answers
23 views

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 ...
0
votes
1answer
22 views

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 ...
0
votes
0answers
16 views

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 ...
2
votes
3answers
73 views

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} &...
3
votes
0answers
37 views

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 ...
1
vote
1answer
51 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, ...
1
vote
1answer
36 views

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}...
4
votes
2answers
188 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
20 views

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 ...
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....
2
votes
1answer
150 views

Sum of the Chern number of all bands

My understanding is that, if we sum the Chern numbers of all the bands in systems, they add up to zero. I believe this is a rigorous mathematical result. Is it possible to understand this physically?
0
votes
1answer
51 views

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?
0
votes
0answers
47 views

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},...
0
votes
1answer
142 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 ...
2
votes
1answer
241 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 ...
0
votes
0answers
19 views

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-...
3
votes
0answers
48 views

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 ...
1
vote
1answer
17 views

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 ...
1
vote
0answers
37 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
1answer
167 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 $$-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$$ ...
3
votes
0answers
24 views

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} \\\...
1
vote
1answer
149 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 ...
3
votes
1answer
54 views

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 ...
4
votes
3answers
2k views

What's the reason for particle-hole symmetry operator to be anti-unitary?

I have been looking at some literature on Topological Superconductor, where the BdG Hamiltonian is frequently used, the $H_{BdG}$ has the so-called particle-hole symmetry, which is commonly defined ...
2
votes
2answers
706 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. ...
4
votes
1answer
58 views

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 $$\mathcal{T}^2= (-1)^B$$ ...
1
vote
0answers
45 views

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 ...
0
votes
0answers
15 views

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 ...
2
votes
2answers
2k views

Edge states for SSH model?

We can write the Hamiltonian for SSH model as $H=\sum_i(t+\delta t)c_i^{\dagger} c_{i+1}+(t-\delta t)c_{i+1}^\dagger c_i+h.c$ We know that there are two topological phases $N_1=0$ for $\delta ...
3
votes
1answer
186 views

Why one can observe Quantum Hall Effect in 3D Topological Insulators in an external magnetic field when TRS is broken?

In magnetotransport experiments scientists have observed the Quantum Hall effect in 3D topolgical insulators. Using a standard hall-bar geaometry in an external magnetic field they see plateaus in the ...
9
votes
3answers
2k 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 ...
2
votes
4answers
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 ...
3
votes
2answers
361 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-...
0
votes
0answers
19 views

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 ...
7
votes
2answers
155 views

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 ...
0
votes
1answer
45 views

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 ...
7
votes
1answer
228 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
56 views

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 ...
0
votes
1answer
68 views

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.
0
votes
0answers
29 views

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=\...
2
votes
0answers
60 views

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 ...
1
vote
1answer
41 views

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 ...
0
votes
0answers
20 views

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)....

1
2 3 4 5
8