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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About the $Z_2$ topological invariant

In Kitaev 2001 it is shown that the topological invariant $Z_2$ in a topological superconductor (Class D or BDI, one dimensional) can be defined as $$ (-1)^\nu={\rm sign\, Pf} [ A ]={\rm sign\, Pf}[ ...
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About Weyl superconductors and fractionalized Weyl semimetals

Recently, the experimental observations of Weyl fermion semi-metal have been made. Weyl fermion becomes very hot in condensed matter physics. I am confused about the Weyl superconductors and ...
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Quantum spin Hall effect and the edge states

In quantum spin Hall effect or Kane-Mele model, how can one get rid off the edge states without affecting the bulk?
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How to describe spin-orbital coupling in Weyl semi-metal

In three dimensional Weyl semi-metal, the Hamiltonian that describes low excitation quasi-particle is well-know Weyl Hamiltonian: +/- $k\cdot\sigma$. But if I want to add spin-orbital coupling in that ...
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Topological invariant in 1D

In 2D, with state $\psi(k_x, k_y)$, it is common to calculate measure of topology of material: 1 - Calculate Berry connection $a = -i <\psi | \partial_{\boldsymbol{k}} | \psi>$. 2 - Calculate ...
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What are the instances of usage of four color theorem in the theory of fractional statistics?

How important is four-color theorem (Hypothesis) in theory of Fractional Statistics?
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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 ...
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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 ...
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29 views

Will all linear band inversions of non-degenerate bands change chern number by one?

I have learned from literature that band touching is the source of chern number.In three dimensional material, any non-degenerate linear band crossing will form a weyl point which is a monopole of ...
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Majorana fermions in s wave superconductor

I need some help to understand the majorana fermions in $s-$ wave superconductor and to check whether following method is correct For $s-$ wave superconductor we can write the Hamiltonian as ...
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1answer
76 views

Superconductivity and time-reversal symmetry

Let us consider a system of a 1D edge of a 2D topological insulator in proximity to an s-wave superconductor. The system is described by the Hamiltonian: $$ H =\frac{1}{2} \int \mathrm{d}x \ ...
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why Hall conductance quantized

When I am studying quantum Hall effect, the quantum Hall conductance can be represented by Green function $\left(\text{up to}\ \frac{e^2}{h}\large \right)$: I cannot understand why it is an integer? ...
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114 views

1/m Laughlin state and $U(1)_M$ chiral CFT

I am a little confused that people claim that the edge theory of a 1/m Laughlin state corresponds to a $U(1)_m$ chiral CFT. This indicates there should be $m$ primary field operators in $U(1)_m$ ...
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Magnetism in Topological Insulator?

I have read many papers with the statement that on the introduction of the magnetic impurity, the gap at the Dirac point opens in the surface states. I am little confused about it. If the gap is ...
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2answers
77 views

What are the conditions to observe gapless modes at the boundary for 1D case?

We observe gapless modes at the boundary for the case of SSH model or Polyacetylene The Hamiltonian for SSH model has particle hole and time reversal symmetries, it also has a Dirac like spectrum. It ...
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2answers
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Exact Diagonalization of a BdG Hamiltonian on a Finite Lattice

I would like to numerically find the edge modes of a $p_x$ + $i p_y$ BdG Hamiltonian. The lattice version is given by H = $\sum\left[-t \left(c_{m+1,n}^{\dagger} c_{m,n} + \text{h.c} \right) - ...
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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 ...
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How to calculate Berry curvature in photonic analogue of quantum spin hall effect? [closed]

The Chern number equals zero,but the berry curvature of spin up and down is not equal to zero,how to calculate the berry curvature?
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1answer
104 views

Is there a bulk signature of topological nontriviality for a 3D free fermion band insulator?

Is there such thing as a 3D Chern invariant (or some other quantity) that I can use to test an insulating quasiparticle spectrum is a topologically trivial or non-trivial insulator? Does one exist ...
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214 views

Can a nondegenerate fermionic topological Mott insulator (TMI) state support an emergent bosonic topological order?

Based on my recent study and motivated by a recent paper, I have a naive question. Consider a 2d Hubbard model for electrons at half filling $H=\sum c_k^\dagger h_k c_k+U\sum n_{i\uparrow ...
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Conductance measurement of InAs/GaSb Quantum Spin Hall Edges

My questions are related to recent article: http://arxiv.org/ftp/arxiv/papers/1507/1507.08362.pdf I can't figure out how their sample (wafers) actually looked like. In particular I can't understand ...
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What makes a system topological?

As I understand, if the Chern number which is obtained by integrating Berry curvature over a surface with a boundary is an integer, then the Chern number is a topological invariant. So when does Chern ...
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1answer
400 views

What is a Dirac semimetal?

What is the precise definition of a Dirac semimetal? Is it sufficient for two bands to touch at a single k point with a linear crossing, or are more conditions required for a material to be called a ...
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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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1answer
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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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186 views

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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86 views

$Z_2$ 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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1answer
83 views

$Z_2$ 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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159 views

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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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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105 views

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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161 views

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
423 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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1answer
432 views

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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319 views

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 ...