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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Topological properties of Inversion symmetric Rice-Mele model
I know that the Rice-Mele model (1D chain model whose unit cell consists of two sublattices whose on-site energies are different e.g, m,-m) is a non-chiral model, so it does not have a quantized ...
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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 ...
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Meaningful topological invariants and quantities for the description of 3D topological insulators
I'm currently trying to understand the classification of 3D topological insulators (like Bi2Se3). Most reviews dealing with this topic start with the introduction of the Quantum Hall effect since this ...
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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 ...
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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 $C_1=1$, $C_2=-2$ ...
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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&...
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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 ...
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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 ...
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About the graphene's Dirac points
I am studying Hasan & Kane's colloquim about topological insulators, and a question has emerged when I reahced the 4th page of this thesis. 1
Passages below the equation (3) show that near the ...
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Zero frequency limit of Hall conductivity not quantized?
The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system, such as quantum Hall effect and Haldane model. ...
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Does Hall conductivity change sign with chemical potential?
The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system. A typical plot of $\sigma_{xy}(\mu)$ is like the ...
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Topological Insulator [closed]
What effect on the Brillouin zone (torus) after applying the magnetic field? As in real space, pressure deforms the torus and up to a certain pressure, this remains invariant topologically. Similar to ...
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Topological Invariants of Floquet Topological Insulators
I've been working through the following sources:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.155118
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.195303
where they derive new ...
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Topological superconductor definitions
Topological insulators are materials who are conductors on the boudary but not in the bulk. What is the definition of tolopological superconductor? A superconductor on the boundary but not in the ...
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Infinite stacking of integer quantum Hall systems
Let us consider a (3+1)-dimensional system $\mathcal{H}$ constructed by stacking (2+1)-dimensional integer quantum Hall systems $\mathcal{H}_\text{Hall}$, e.g., $E_8$ bosonic systems (or $\sigma_H=1$ ...
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Topological phase and Chern number
the relation between topological phase and Chern numbers is unclear to me.
For Haldane model if the Chern numbers of its two bands go from (+1,-1) to (0,0), we say that it goes from topological phase ...
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How to get the chemical potential of a Chern Insulator in PBC?
I want to calculate the Magnetization of a Chern Insulator, whose expression in k-space contains the chemical potential. If the system doesn't have the particle-hole symmetry, how can I get the ...
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Why a $4\times 4$ double degenerate Hamiltonian can be obtained with Clifford algebra matrices?
in the book - Topological Insulators and Topological Superconductors, B. Andrei Bernevig and Taylor L. Hughes, Princeton University Press (2013), pag 148 - the authors say that in order to have a $4\...
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Berry phase from Bloch wave functions in the basis of Wannier functions
The formulate to calculate berry phase for Bloch wave functions is
$$
\gamma = i \sum_{n\in occ}\int_{\mathcal{C}} dk \langle \psi_k^n|\partial_k|\psi_k^n\rangle,
$$
where $|\psi_k^n\rangle$ is a ...
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Integer central charge and entanglement entropy
I was study topological insulators using entanglement entropy, let's say 1-D SSH model, then at critical points the von Neumann entanglement entropy obeys the Calabrese-Cardy formula. For free ...
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How to define the surface current operator of a bulk system?
Consider a bulk uniform system with a boundary (edge, surface, etc.) and we want to study some transport effects mainly due to some boundary states. One possible experimental scenario might be to ...
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Must a time-reversal symmetric Hamiltonian really have $T^2 = \pm 1$?
It is apparently common knowledge in the condensed matter literature that the time reversal operator $\hat{T}$ always squares to $\pm 1$ at the first quantised level. When acting on spins the time ...
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Symmetry Class of Graphene with Spin-orbit Coupling
I know that tight-binding graphene with $\text{p}_\text{z}$-, $\text{d}_\text{xz}$- and $\text{d}_\text{yz}$-orbitals and spin-orbit coupling is a $\mathbb{Z}_2$-topological insulator. But I want to ...
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Dirac Hamiltonian and Topological Insulators
Specifically, why do we treat the Bloch Hamiltonian in a topological insulator (TI) as a two level system that possesses a linear dispersion relationship? What justifies this effective model? I can't ...
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Calculation on the squashed entanglement
The definition of squashed entanglement:link
I read a paper on this topic, and found a good result (Fig.2(a)) from it showing that the squashed entropy can indicate topological phase transition of SSH ...
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Relation betwwen the expectation value of the position operator in Bloch state with unitary position operator
Recently, I want to get some knowledge about topological insulators, so I read the A short course on topological insulators. In the chapter 3, the author try to explain why polarization connects with ...
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Proof for existence of edge (surface) states in topological materials
Many publications demonstrate the existence of edge (surface) states for certain topological model Hamiltonians and give reasonable arguments why this should hold in general. Is there a mathematically ...
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Coupling between unpaired Majorana modes in Kitaev chain
In this course,
https://topocondmat.org/w1_topointro/1D.html, to argue when the edge will disappear, the authors say:
The only possibility to move the energy levels from zero is to couple the two ...
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Are there any experimental realizations of semi-Dirac materials?
My question is there any experimental proof of semi-Dirac dispersion beside optical lattice?
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Current operator, why is this form valid up to second order in $q$?
Context
In Bernevig´s textbook Topological insulators and topological superconductors, an approximate form of the current operator in momentum space is derived. It is said to be valid to second order ...
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Why Aharonov-Bohm interference loop in topological insulator nanowire/nanoribbon could be formed by two paths with opposite spin?
When applying a magnetic field along the axis of a topological insulator nanowire/nanoribbon, AB oscillations are observed in previous experiments, e.g., https://doi.org/10.1038/nmat2609. I am a ...
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Concise formula for Skyrmion number of a four-band Hamiltonian
The Haldane model on the honeycomb lattice is often written as
$H=\sum_{i=0}^3 \vec{d}_i\cdot \vec{\sigma}_i$
where $\vec{\sigma}$ is the vector of Pauli matrices and $\vec{d}$ is a vector of ...
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Is graphene a topological material?
I understand that graphene is a great material with many exotic properties. Hoever, does it have a non-trivial band topology? It's mentioned that graphene has a weak spin-orbit coupling, which is ...
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What is a p-wave superconductor in 1-dimension often referred in the Kitaev model? What term in the hamiltonian makes it P wave?
I am currently reading papers related to Majorana zero modes observation in 1D nanowire systems. I am very new to the field and I read everywhere that the presence of spin-orbit interaction, magnetic ...
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Edge state protection in Chern insulator
I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
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$R$ projection operator in entanglement entropy
I was reading a paper about entanglement entropy. The author introduced a real space cutoff operator $R$ which he claimed to project onto a real space subregion. Then he used $\bar P:=RPR$ as an ...
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Topological properties of twisted TMD homobilayers
I'm reading this article about twisted TMD homobilayers (https://arxiv.org/abs/1807.03311) and there are certain topological properties that I don't understand:
On page 3, in the paragraph next to Fig ...
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Absence of topology in semi-dirac materials
Good morning everybody, I am facing a problem when calculating the topological invariant in a semi-dirac system, whose Hamiltonian is:
$$
H=k_x^2\sigma_x+k_y\sigma_y
$$
My question is that this ...
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Conditions for Bogoliubov-de Gennes Hamiltonian representation
The $H_{BdG}$ hamiltonian is described in topocondmat.org as follows:
here we can see that the submatrices along the diagonal are related as negative of complex conjugate of each other, I feel that ...
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Spin Hall Effect in 2D Topological Insulator
Suppose the 2D topological insulator has the magnetic element doping in the system and the easy-axis for the magnetization is along the z axis. The the surface gap is opened due to the time-reversal ...
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Representation of symmetry operators in second quantuzation
Hamiltonian invariant under a symmetry- The action of a group $G$ on the set of Bloch momentum is given by a linear representation $T_g: k \to T_g k \equiv k_g $. Now say that a fermionic Bloch ...
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How to calculate surface states in Weyl semimetals?
I'm reading an article https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235127. Fig. 2 in this article shows band structures calculated from Eq. (9), (13), (14), (15), and (16). For example, ...
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Why are topological materials/phases "exotic"?
From what I understand, when a system has topological order, any local perturbation doesn't change the phases and thus its properties. This would suggest that it should be really easy to find ...
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Why does "chiral symmetry" in the Altland-Zirnbauer classification mean something different to other contexts?
The Altland-Zirnbauer classification of random matrices is based on three symmetries: time-reversal, charge conjugation, and a third which is sometimes referred to as "chiral" or "...
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Calculate the energy gap using Green's function
Can I calculate the energy gap of the given Hamiltonian by Green's function?
Is there any basic code in MATLAB to do that?
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What is the difference between topological insulators and axion insulators
I'm currently reading a paper about an antiferromagnetic axion insulator candidate and am having trouble understanding exactly what is meant by the words axion insulator in this case. To my knowledge ...
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Quadruple index and the existence of corner state
I have been following this paper, which seems to discuss the connection between corner state and a quadruple index calculated $q(k_z)$ by equation (3) of it. At two special point $k_z=0 /\pi$, the ...
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Does gap closing and reopening guarantee a non-trivial topological phase?
I know that a Dirac point carries a Chern number of $\pm\frac{1}{2}$ and when we have a gap closure at any point in our band structure we can transfer Chern numbers depending on at how many points we ...
