Topological insulator are materials formed by a insulator bulk and metallic surfaces with topological origin. These materials may be important for developments in Quantum Computing and Spintronics.

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Time-reversal transformation for two-component bosonic models

Consider a two-component bosonic model $\mathcal{H}=-t\sum_{i\sigma}{b_{i\sigma}b_{i+1\sigma}^\dagger}+h.c. +\sum_{i\sigma\sigma^\prime}U_{\sigma\sigma^\prime}n_{i\sigma}n_{i\sigma^\prime}$. Here ...
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Interpretation of negative mass in condensed matter physics

I am reading the book "Topological insulator: Dirac equation in condensed matters" by Shun-Qing Sheng. I do not know much about this topic and this is the first time I am confronted with it, so this ...
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Why bulk states in quantum hall effect do not contribute to electric conductivity

Most reviews and textbooks explain quantum hall effect as insulating bulk states and conducting edge states, as is shown in the following picture. My question is: why bulk states are insulating in ...
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Meaning of the term 'bulk'

I have recently started reading literature on 2 dimensional systems in Condensed matter. While reading, I frequently came across the word 'bulk'. Sometimes it referred to 2-D and sometimes to 3-D. I ...
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$Z_2$ invariant and Wannier states switching partner

I have been reading about $Z_2$ topological invariant recently. However, after some literature survey, I still cannot understand $Z_2$ invariant in language of time reversal polarization. Basically, ...
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$Z_2$ invariants in quantum spin hall (QSH)

In a recent literature survey, I learned that $Z_2$ topological invariant is defined as zeros of Pfaffians in half a Brillouin Zone, where Pfaffians are defined as $P(k)=Pf[<u_i(k)|T|u_j(k)>]$. ...
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why do edge states in Graphene exist between the Valence and Conduction band?

I read in a review that there are 2 Dirac points in graphene, where the conduction band and valence band touch each other. Near these points electrons obey a linear dispersion relation. Breaking of ...
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Topological insulators literature

I started learning things on topological insulators and I got lost in dozens of existing papers on this topic. Could anyone recommend me appropriate literature that explains deeply enough what ...
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Chern number of a two-level system

The bulk of my question relates to a two-level system, but I have some questions about the Chern number in general as well. The Chern number of a gapped periodic system (free fermions or mean field) ...
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Computational method for finding edge states?

I am actually interested to learn how to calculate edge states in 1D topological systems using computational methods, Q. can anyone tell me which method is best suited and easy to calculate edge ...
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What is the physics behind “Bulk-edge correspondence”?

There is a frequently mentioned concept in the field of topological insulator called "bulk-edge correspondence" or "bulk-boundary correspondence", which basically gives the relationship between the ...
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Problem with quantum Hall effect and Berry curvature

I am having trouble proving that the Hall conductivity is equal to the integral over the Berry curvature in momentum space. In the TKNN (1982) paper, using the Kubo formula $$ \sigma_{xy} = \frac{ ...
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Fermion version of Gauss-Milgram sum?

For Bosonic topological order, a very useful formula was proved to be true: $\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $ (for more detail: $d_a$ is the quantum dimension of anyon ...
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Are there topological non-trivial states in zero dimension?

The periodic table of topological insulators and superconductors suggests that there can be topological non-trivial phases in zero dimension in non-interacting system with certain symmetries. A 0D ...
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edge modes of topological insulators?

in a paper http://arxiv.org/abs/1011.2273 there is a possibility shown that gapped modes at edges can exist for non-trivial interacting topological systems and mathematically shown that gapless modes ...
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About Majorana fermion in spin-orbit coupled quantum wires

Majorana mode has attracted great theoretical and experimental interest. The experimental evidence is obtained in quantum wires. The origin theoretical proposals of quantum wires are the papers: 1、R. ...
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Polarization to define trivial and non trivial topological phases?

Polarization is well defined for particle hole symmetry systems, so can we use polarization to identify topological phases? for example polarization can have possible value $$P=0 \quad or \quad1/2$$ ...
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Electric polarization in terms of berry phase?

I was reading a text in which Electric polarization in terms of Berry phase was defined as $P=\frac{e}{2\pi}\sum_{n}\int A_n (k) dk$ under gauge transformation $P\rightarrow P+ne$ (which means ...
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Are the edge states of a topological insulator only zero energy states?

For non-trivial topological insulators, are the edge states only zero energy states? Or are boundary states of different energies also available?
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Can a symmetry-preserving unitary transformation that goes from a trivial SPT to a non-trivial SPT be local?

This question concerns the very interesting paper: ''Symmetry protected topological (SPT) orders and the group cohomology of their symmetry group'' by Chen et al., http://arxiv.org/abs/1106.4772 In ...
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How does time reversal symmetry work in topological insulator?

I am doing microelectronic devices with topological insulators. Can some one explain time reversal symmetry in a topological insulator to electrical engineering student like me?
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1answer
203 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 ...
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What is the reason for chiral anomalies in condensed matter systems?

If you consider a massless relativistic fermion theory and you perform a chiral transformation, then you realize that while the classical action remains invariant under this transformation the ...
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Chiral anomaly in Weyl semimetal

In the presence of electromagnetic fields $E$ and $B$, four current is not conserved in a Weyl semimetal i.e. $\partial_{\mu} j^{\mu}\propto E\cdot B \neq 0$. There are some proofs in the literature ...
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Topological term under electron-electron interaction

By integrating out fermions in gapped Dirac Hamiltonian, one can obtain a topological term for topological insulator. Why there is no further correction to this term when electron-electron interaction ...
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band structure of Topological insulators

The above figure is Rashba-split free electron-like surface state in a projected bulk band gap. The bellow figure is the band structure of Topological insulators. x axis is the wave vector, y axis ...
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What is the difference between a material in which the Rashba effect occurs that is not a topological insulator, and one is a TI?

I am working on topological insulator (TI) materials and I always have trouble understand the time reversal symmetry, spin orbit coupling in TI. As to my understanding, the TI material property ...
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How the Vortex containing majorana bound state is non-abelian statistics

Recently,I read some papers about non-abelian statistics of majorana fermion, such as: Majorana Returns F. Wilczek http://www.nature.com/nphys/journal/v5/n9/full/nphys1380.html and Non-Abelian ...
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Winding number for SSH model

The Hamiltonian for SSH model can be written as $h(k)=\begin {pmatrix}0&t_1+t_2exp^{-ika}\\t_1+t_2 exp^{ika}&0 \end{pmatrix}$ for finding the topological invariant Why we only calculate the ...
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236 views

Topology of Fermi surface

In The universe in a Helium droplet, Grigory Volovik relates the stability of a fermi surface to topology of a Green function. There he gives the example of a Fermi gas and says that the Green ...
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About recent experimental evidence of Majorana edge states in topological superconductors

I have a couple of question about the recent experimental evidence of Majorana edge states in topological superconductors. Which are the main differences between the experimental signatures of ...
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1answer
77 views

Topological invariant for interacting systems using single particle green functions?

Why Single particle green's function is (preferred) used to find topological for interacting systems? $N_1 =\frac{\epsilon_{ijk}}{24 \Pi ^2} \int dw d^3k G \partial_i ...
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Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
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What makes a superconductor topological?

I have read a fair bit about topological insulators and proximity induced Majorana bound states when placing a superconductor in proximity to a topological insulator. I've also read a bit about ...
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does Hall plateus require the existence of impurity in the sample?

While studying Hall conductivity with The Qantum Hall effect written by S.M.Girvin, I read a sentence "We have shown that the random impurity potential(and by implication Anderson localization) ...
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Equivalence classes of mappings from $T^{2}$ to an arbitrary space $X$

I was reading the paper "Homotopy and quantization in condensed matter physics", by J.E Avron et al. ( http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51). There they have classified the ...
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Bloch Hamiltonian of low energy edge mode of a 2D topological insulator

First time to pose a question here. It's a Hamiltonian appears in this paper https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.87.137 The equation (71) $$ h_0(q) = v q\sigma^y+m\sigma^x-\mu ...
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$\mathbb{Z}_2$ topological insulators which obey inversion symmetry as well

According to Fu & Kane (2006), systems with simultaneous time-reversal invariance and inversion symmetry have their $\mathbb{Z}_2$ topological invariant given by the product of the parity ...
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Compute $Z_2$ Invariant of 2D Topological Insulators without Computing the Eigenstates

For 2D Time-Reversal Invariant systems ($T H(\vec{k}) T^{-1} = H(-\vec{k}) $), there is a formula by Fu-Kane-Mele in order to determine whether the system belongs to either one of distinct topological ...
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How are topological invariants constructed?

I've seen several different definitions for what are called topological invariants, for instance in the context of Majorana unpaired modes, by Kitaev: http://arxiv.org/abs/cond-mat/0010440 ...
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What is the physical mechanism of the topological phase transition driven by temperature?

The topological property of topological insulators (TIs) is characterized by the non-trivial topological invariants of their band structures, such as $Z_{2}$ topological invariants. While it's clearly ...
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Questions on gapless edge excitations in symmetry-protected topological state

I am studying a one-dimensional bosonic system with $ U(1) \rtimes Z_2^T$ symmetry numerically, which might has a symmetry-proteced toplogical(SPT) phase. I have several questions about the ...
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How to find dispersion relation for 1 d topological insulator?

Is it correct to write the dispersion relation for following Hamiltonian where $\sigma_{x}$ act in spin space and $\tau_{x}$ acts in pseudo spin particle hole spin $H_{BdG} (k)=(\xi_{k}+B\sigma_{x}+u ...
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What is the different between topological order and Landau's order in a system

I have thought about topological order for a long time, but I am still confused it. Roughly speaking in my understanding, the topological state is the eigen-state of some special symmetry such time ...
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Which Symmetry class and what kind of topological invariant for $2D -p+ip$?

What kind of topological invariants are there for $2D-p+ip$ topological superconductor and to which symmetry class it belongs to?
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Interacting Chiral topological invariants using Green function

We can calculate the topological invariants for 1D interacting topological insulators as $n=\frac{\text{Tr}}{2\pi i}\oint_cG\partial_kG^{-1} $ where as for interacting chiral topological ...
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Does time reversal symmetry hold for (kitaev model) 1D spinless $p-$ wave superconductor?

The hamiltonian 1D spinlesss p wave superconductor can be written as $$ H=\sum_k \phi_k^\dagger \begin{pmatrix} \xi(k) & 2i\Delta \sin(k)\\ -2i\Delta \sin(k ) & -\xi(k)\end{pmatrix}\phi_k $$ ...
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What symmetry class does 1D spinless $p$-wave superconductor belongs to?

$Z_{2}$ topological invariant exist for Kitaev model. What symmetries does it conserve? And to what symmetry class it belongs to? The hamiltonian for kitaev model can be written as $$ H=\sum_k ...
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What is meant by time reversal invariant momenta (TRIM) point of the Brillouin zone (BZ) and how to determine them in 3D reciprocal space?

In literatures related to topological insulators, I have frequently encountered the term "time reversal invariant momenta" or TRIM points in the Brillouin zone of the crystal. In many literatures, I ...
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What can we learn from a band structure diagram?

Other than the band gap and its magnitude, what are the things that we can immediately learn about the properties of the material just by glancing at its band structure? Can we say something about ...