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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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How to numerically calculate Zak phase for SSH3 model?

The k-space hamiltonian of SSH3 model with nearest neighbour hopping is given by H(k)= \begin{bmatrix} 0 & u & w e^{-ika} \\ u & 0 & v \\ w e^{ika} & v & 0 \end{bmatrix} ...
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Why does the solution to 2D Dirac equation have four but not two components?

I can understand that in QFT, we have four components and we use $4\times 4$ Dirac matrix. When I am reading the book Topological Insulators - Dirac Equation in Condensed Matter by Shen Shun-Qing, I ...
Susstring W's user avatar
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Is there a topological invariance/winding number for non-translation invariance system?

My question is: Is there a topological invariance/winding number for non-translation invariance system? For example, if we modify the interacting parameter in SSH model, such that it depends on the ...
feng lin's user avatar
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Weyl crystal with broken time reversal symmetry (TRS) and spin-momentum locking

We know that the stability of Weyl points crucially requires that the bands involved are non-degenerate. Otherwise, there can be terms that cause band hybridization within the degenerate subspaces and ...
Gabriel Elyas's user avatar
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Meaning of wavevector in SSH model

If we solve the hamiltonian of the following form in the SSH model, You will get an eigen vector of the following form. I am having difficulty interpreting this eigen vector. According to this eigen ...
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Can topologically non-trivial edge states exist without an energy gap?

I am relatively new to the field of topological materials, and I came across a paper that claims that they found topologically non-trivial states in a material. Basically, the authors found a type-I ...
Mikhail Petrov's user avatar
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Can the hybridization of edge states in the 1D SSH model be observed numerically?

So I was reading the lecture notes by Asboth on topological insulators . In the first chapter the SSH model is discussed : $H_{SSH} = \sum_{i = 1}^N v|i,A\rangle \langle i,B | + h.c. + \sum_{i = 1}^{N-...
Sayan Mondal's user avatar
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Plotting edge modes for a tight binding Hamiltonian with superconducting pairing

I would like to obtain a plot such as the one in the first picture for the eigenstates of the Hamiltonian reported in the second picture. The problem here is that we have superconducting pairing and I ...
Alberto Zorzato's user avatar
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Is there a Majorana representation for toric code

Kitaev's toric code is known to be the Z2 gauge field theory, which suggests that there might exists a Majorana representation for the toric code, e.g., Majorana + Z2 gauge field. Hence, I wonder if ...
Richard's user avatar
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How do I find the matrix for the Su-Schrieffer-Heeger model's Hamiltonian?

I will not bother to write down the tensor product in the joint basis of A and B here in this post, where A and B are atoms/electrons and m denotes a unit cell. The Hamiltonian for this model is given ...
Despaxir's user avatar
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How can a Soliton be neutral in the Su-Schrieffer-Heeger (SSH) model when it carries an electron?

I apologize if this is a trivial question. I am currently studying Girvin and Yang's book where they try to explain Spin-charge separation for the Solitons in the SSH model. To my understanding, a ...
FrustratedSpin's user avatar
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Dealing with discontinuous phase issue in computing winding number numerically

Consider a 1D SSH model with winding number given by $$\nu = \frac{1}{2\pi i}\int_{-\pi}^\pi d\phi,$$ where $d\phi$ is the change in phase of the eigenvectors between nearby $k$ points. The phase is ...
Sean's user avatar
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Why does mirror/inversion symmetry exclude nearest-neighbor spin-orbit coupling in the Kane-Mele model?

I've been reading about the Kane-Mele model and noticed that it does not include nearest-neighbor spin-orbit coupling, with explanations often pointing to mirror symmetry as the reason. Could someone ...
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Finite temperature Greens function simplification

I am currently studying topological insulators using Topological insulators and superconductors by Bernevig. In chapter 3, section 3.2.2, he has derived the finite temperature Green's function using ...
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Is the flat band of zigzag ribbon topological or just a state of topological origin?

In my understanding, an edge state has topological origin if its existence is bound to a nontrivial topological invariant. An edge state is topological if it has topological origin. The flat bands of ...
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Justification for Wick rotation for topological insulator

In Appendix B of the paper (1), the authors compute the second Chern number $C_2$ of a band structure by manipulating the ground- and excited-state projection operators $P_{\text{G}}(\mathbf{k})$ and $...
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Find smooth gauge numerically in the presence of symmetries

Let's assume a gapped two-band model in $2$D characterized by a Bloch-Hamiltonian $h(k_x,k_y)$ in the presence of reflection symmetry along the $x$ direction represented on the orbital degrees of ...
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Gauge constraint in the definition of the $\mathbb{Z}_2$ invariant

Cross-posted at MMSE. In Fu and Kane's paper from 2006, the authors define the $\mathbb{Z}_2$ invariant for time-reversal invariant topological insulators as an obstruction to Stoke's theorem, $$ \...
Sounak Sinha's user avatar
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Where to find model parameters of a simple Weyl semimetal?

A famous continuum model of Dirac semimetal is essentially two copies of the following Hamiltonian (up to some chirality reversal) $$h=\begin{pmatrix} M(k) & Ak_+ \\ Ak_- & -M(k) \end{pmatrix}$...
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How to compute the Chern number of a quantum dot (zero-dimensional topological insulator) in AI class?

Looking at the periodic table of topological insulators, the AI class (only time reversal symmetry is preserved) has a $\mathbb{Z}$ invariant for zero-dimensional topological insulators. In the review ...
Mohit Kumar's user avatar
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Landau levels versus Bloch states

People talk a lot about quantum Hall effects without Landau levels. It seems that there are two kinds of quantum Hall effects, with Landau levels or without Landau levels. In the second case, it is ...
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Why Zeeman coupling can be used in Bogoliubov-De Gennes superconducting Hamiltonian and Josephson junctions model for Majorana?

It is well-known that type-1 superconductivity holds perfect diamagnetism, which means that magnetic field is expelled by the superconductor. However, there are cases, especially in the topological ...
M. Plot's user avatar
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Time reversal in a two-band system

Suppose I have a 3D system of spinless fermions described by the following two-band model Hamiltonian: $$ H(\vec{k})=\vec{d}(\vec{k}) \cdot \vec{\sigma} $$ where $\vec{d}=\left(-\sin k_{x},-\sin k_{y},...
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Deriving an alternating expression for Hall Conductivity

I was reading the paper (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.3372) in which Thouless and Wu show that the hall conductivity is a topological invariant. My question is about the ...
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Topological band crossing and central charge

Conventionally for a 1D topological insulator, SSH model for example, the low energy assumes a linear dispersion i.e. Dirac cone at a topological phase transition point, thus obeys the massless Dirac ...
renn's user avatar
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Hexagonal lattice chern number calculation

Can someone please explain how to find chern number of hexagonal lattice system say graphene Hamiltonian using fukui method?
Physik's user avatar
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Deriving the non-abelian Berry connection

I'm slightly confused about a manipulation in Section 1.5.4 of Tong's notes on the Quantum Hall Effect. This concerns the derivation of the non-abelian Berry phase. Setup: We have an $N$-dimensional ...
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Non-abelian Berry connection : clashing time-ordering conventions, and component-wise form

Let $\mathcal{M}$ be a $k$-dimensional parameter space associated to a quantum system with an $N$-dimensional ground state. As usual, we assume the system is subject to some adiabatic tuning of ...
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1D Hermitian Hamiltonians are topologically trivial?

I am currently studying the following paper: https://arxiv.org/abs/1706.07435 The authors state: "The Z/2 classification we found for separable nonHermitian Hamiltonians in one dimension is in ...
TheQuantumMan's user avatar
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Link variables in numerical Chern number calculation

A well known method to calculate the Chern number numerically is the one proposed by Fukui et al. (http://arxiv.org/abs/cond-mat/0503172). Here they make use of 'U(1) Link variables' to obtain a gauge ...
Lumen Eek's user avatar
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1 answer
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Gauge freedom issues when numerically calculating overlap of states

Calculations of quantities of the form: $$\langle\psi(k)|\partial_k|\psi(k) \rangle \ ,$$ require a smooth choice of the wave function $|\psi(k)\rangle$ over the whole BZ. This is not a problem. ...
Lumen Eek's user avatar
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Relation between Displacement Operator and Winding number

I am trying to implement a paper [https://arxiv.org/abs/2003.06086] using quantum computing techniques. In the supplementary material[SM] with the main paper, they introduce a displacement operator ...
CuriousMind's user avatar
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Chern Insulator in Magnetic Field

Recently I was reading the book "Topological Insulators and Topological Superconductors" by Andrei. Bernevig & Taylor Hughes and in section 8.7, Chern Insulator in Magnetic Field, a ...
Mohammad. Reza. Moghtader's user avatar
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Numerical calculation of Chern number for low energy continuum models

Can Fukui method be employed for the Chern number calculation in a low-energy continuum model? Let's consider the low-energy continuum version of the Kane-Mele model. If not what are some other ways ...
Arjuna's user avatar
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Sublattice symmetry and the Fermi level

I am a math student who is learning topological phases from this website. Let's assume the fermi level is zero. For the graphene, the sublattice symmetry $\sigma_z H \sigma_z = -H$ makes the ...
Justin Lien's user avatar
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Lorentz Violation in Condensed Matter Systems

I have heard from a few different sources that certain condensed matter systems like topological superconductors and insulators violate Lorentz invariance. Can someone explain how that happens?
vyali's user avatar
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Physical consequence of classification based on anti-unitary symmetries in physics

I am trying to read some materials about topological insulators. It seems the ten-fold ways of classifying topological insulator relies on anti-unitary symmetry as introduced in the following ...
Dango's user avatar
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Weyl semimetal where dispersion crosses Weyl point energy not at Weyl point

Consider a two-band tight-binding Weyl semimetal model $$H=\sum_\alpha a_\alpha(\vec k)\,\sigma_\alpha$$ for $\alpha=0,1,2,3$. For instance, one with one pair of Weyl points (WPs) could be $$H=\sin{...
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How can I calculate this equation if we know that there is non zero Berry-phase between the valence and conduction band

The geometric phase can be interpreted as a Berry curvature in the momentum space. My guess is $(q^2+\text{Berry-phase}/\text{lattice constant}^2)/\text{direct gap}$. $$\langle\psi_{n',\mathbf{k}+\...
Negyedes Diána's user avatar
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What exactly does it mean that the states are protected by the symmetry?

I am trying to understand symmetries in the case of topological insulators. And I have a huge problem with the interpretation of symmetries. Let's say we consider quantum spin Hall state, where we ...
blahblah's user avatar
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Can the Zak phase approximately protect edge states in an SSH model when its value is close to pi but not quantized?

The SSH model is a well-known topological model in one dimension, whose topological invariant is the Zak phase. The Zak phase is defined as: $$ \gamma = \int_{-\pi}^{\pi} dk\ A(k) $$ where $A(k)$ is ...
lsdragon's user avatar
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What exactly is the Surface Brillouin zone?

Everyone knows the Brillouin zone. But the term 'Surface Brillouin zone' seems ambiguous; it is used in many papers, including Wan et. al. (2011) in the classic paper first describing Weyl semimetals, ...
Eero's user avatar
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Problem with Asboth lecture with bulk momentum space hamiltonian

I am trying to understand the SSH model. I am studying the book A Short Course on Topological Insulators by J. Asbóth. I have a problem because he claims that eigenstates may be decomposed into ...
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What does band inversion mean for occupation?

I am thinking about Topological Insulators at the moment and am not clear about how to understand the occupation of the inversed band. I understand that due to energetic and symmetry reasons the ...
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How to understand the use of minimal Dirac Hamiltonian in the ten-fold way topological classification?

Chiu et al. published a paper in 2016 [1] to provide a classification scheme for topological insulators and superconductors based on non-spatial symmetries. The main idea is to use the minimal Dirac ...
Spores's user avatar
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Understanding of Band Inversion in $\rm Bi_2Se_3$ Topological Insulator [closed]

I'm currently trying to understand Fig. 2 in this Paper (http://dx.doi.org/10.1103/PhysRevB.82.045122) which aims to explain the origin of band inversion in Bi2Se3. I really want to get an ...
Mika R.'s user avatar
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Topological Insulators with Multiple Band Gaps

Assume we have a $\mathbb{Z}$-topological insulator, i.e. a Chern insulator. Let its band structure have three bands with two gaps between them. Is it possible to have Chern numbers $C_1=1$, $C_2=-2$ ...
Fred's user avatar
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Gauging away phases of SSH Hamiltonian

I'm reading 'A short course on Topological Insulators' in which it talks about adding phases to the SSH model that can easily be gauged away $$ H= \begin{bmatrix} 0 & v & 0 & 0\\ v &0&...
userman5000's user avatar
1 vote
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Topological Insulators with different spin band

To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
sined's user avatar
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1 answer
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How are topological materials considered "protected by the gap" if the electrons couple to gapless phonons?

My understanding is that Hamiltonians are usually classified topologically by whether they can be continuously transformed into each other without closing the band gap. I then usually hear claims ...
Aaron Dunbrack's user avatar

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