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 can the localization property of the edge mode in topological insulator/quantum hall system be manifested through the effective action?

To be more specific, we can write down the Chern-Simons term from coupling the system to EM to describe the 2d quantum hall system and its derivative respect to the EM field gives the current. How can ...
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How does the effective action describing the EM response of (3+1)d topological insulator become the (2+1)d Chern-Simons term?

Mathematically, it seems to be resulting from Stokes theorom once the 3d manifold has a 2d boundary. However, the EM response described by (2+1)d CS term requires the system to break time-reversal ...
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What's the relation between quanutm hall effect and topological insulator state?

In a recent PRX paper(https://journals.aps.org/prx/abstract/10.1103/PhysRevX.10.011050), I see that in 45nm and 50nm-thick Cd3As2 films, they find quantum hall effect and say that this is because of ...
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What is the physical significant of monopole charges in Nodal Line Semimetals?

Topological nodal line semimetals is formed when the symmetry of a semimetal system enforces band touching occurring on 1-dimensional submanifolds of the Brillouin Zone. Following the topological ...
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Clarification on a statement Bernevig's textbook on Topological insulators

On page 11 of the aforementioned book Bernevig claims after doing some calculation that the integral of Berry curvature over a sphere containing a monopole is $2\pi$. Now my question is the following: ...
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Wave function of $s$-band with odd parity

How does a wave function of a $s$-band state with odd parity looks like (in real space)? To keep it simple, the restriction to a linear chain and a state at the $\vec{k}=0$-point might be useful. My ...
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Alternatives for calculating topological invariants in topological materials

My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
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64 views

How to calculate $\sigma_{xx}$ in lattice model?

It is known that one can find the Hall conductivity $\sigma_{xy}$ from a lattice model (in a magnetic field, say) using the TKNN formula (PRL 49 405-408 (1982)), i.e. by summing the Chern numbers for ...
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Green function of three band crossing topological material in energy coordinate representation do not converge

Recently I am studying the properties of Green function of three diamentional topological materials with three band crossings, whose Hamiltonian can be written as $\hat H= v_f \ \vec k \cdot \vec S $, ...
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63 views

What does the subindex $x$ in $\text{Bi}_{1-x}\text{Sb}_{x}$ mean?

I am currently reworking our condensed matter lecture. We shortly discussed topologic insulaters at the example of $$\text{Bi}_{1-x}\text{Sb}_{x}$$ I am not sure what the $x$ stands for. Normally ...
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50 views

Reality condition on Hamiltonian from Time-reversal and particle-hole inversion symmetry

Starting from a Fermionic second quantized Hamiltonian as $$ H = \sum_{A,B} \psi^{\dagger}_A \mathcal{H}_{A,B} \psi_B $$ Imposing time-reversal symmetry and charge-conjugation inversion symmetry, like ...
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Why are topological insulators interesting?

Why are topological insulators interesting? Meaning, why should an undergraduate or graduate student start working on this? What are the technological applications? I am not sure how to answer these ...
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Gravitational correction in index theorem for 3+1 time-reversal invariant TI

In Witten's review paper: Fermion path integrals and topological phases, the index theorem for 3+1 $\mathcal{T}$-conserving TI is given by $$e^{\mp i\pi \eta/2}e^{\pm i\pi(P-\hat{A}(R))}=(-1)^{\...
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What's the projection operator for topological insulator with three band crossings

For weyl semimetal whose Hamiltonian is $\hat H= v_f \ \vec k \cdot \vec\sigma $ [1,2], its eigen value is $\epsilon_ {\pm}=\pm v_f k $, thus the hamiltonian can be written as $\hat H= \frac{1}{2}\...
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How do we obtain Eq. 3.15 in Fu and Kane's PRB 74, 195312 Time Reversal polarization and a $Z_2$ adiabatic spin pump

This is a probably a very noob question. I attempted to derive Fu and Kane's Eq. 3.14 from the defnition of $A^I$ in (3.12) but ran into a bit of trouble with the minus signs with the $A^I(-k)$ ...
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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?
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The surface (or edge in two dimensions) of a topological insulator

The surface (or edge in two dimensions) of a topological insulator, necessarily has gapless states that are protected by time-reversal symmetry. I don't understand what is that mean?
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Hermiticity of spin-orbit coupling in real space

In the Kane-Mele model, the spin-orbit coupling is defined in real space as $$\sum_{\langle \langle i j \rangle \rangle \alpha \beta} i t_2 \nu_{ij} s^z_{\alpha \beta} c_{i \alpha}^\dagger c_{j \beta}...
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Berry curvature of a two band lattice Hamiltonian

Consider a general two band lattice Hamiltonian, which can be expressed as a $2\times2$ matrix: $$\mathcal{H}=\vec{h}(\vec{k})\cdot \vec{\sigma}$$ where $\vec{\sigma}$ is the vector of Pauli matrices. ...
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Why is the Brillouin zone a 2-torus?

I have been reading through several references on topological band theory, for example: https://arxiv.org/pdf/1510.07698.pdf http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf Topological Insulators ...
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Wiedemann-Franz law generalized to quantum Hall effects in electronic systems

Wiedemann-Franz law states a relation in a conductor between the thermal and electric conductivities by their ratio as $\kappa/\sigma=LT$ where $T$ is the temperature and $L$ is the Lorenz number ...
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Factor of 2 issue in the non-gauge invariance of Chern-Simons theory with a boundary

It is well known that the Chern-Simons (CS) theory by itself is not gauge invariant in the presence of a spacetime boundary. Concretely, suppose the flat half space $\mathcal{M}$ with $x\in \mathbb{R},...
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Strange symmetry of Haldane model edge

In the Haldane model we break both the inversion symmetry and time reversal symmetry, as a consequence I didn't not have any expectations when it comes to symmetry of the energy bands. However, to my ...
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Does flat band imply localization?

Consider the Kitaev chain, whose Hamiltonian is as follows: $$ H = -\mu \sum_n c_n^\dagger c_n -t\sum_n (c_n^\dagger c_{n+1} + \mathrm{h.c.}) +\Delta \sum_n (c_n c_{n+1} + \mathrm{h.c.}) $$ I have ...
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About symmetry constraints in momentum space

When people study symmetry protected topological phases, certain symmetry constraints are enforced on the Hamiltonian. Specifically, for non-interacting fermionic systems, we could focus on the ...
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Meaning of complex pairing terms in Kitaev chain

I am studying some properties of the one dimensional Kitaev chain, which has the following form: $ H = -\mu \sum_n c_n^\dagger c_n - t \sum_n (c_{n+1}^\dagger c_n + h.c.) + \Delta \sum_n (c_n c_{n+1} ...
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399 views

Why Kubo formula can be applied to calculate conductivity?

It seems that Kubo formula is widely adopted to calculate conductivity, or at least Hall conductivity [for example, in the famous paper by TKNN: PRL 49 405-408 (1982)]. However, the derivation of ...
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Low energy continuous model to lattice model(tight binding) in solid physics state

It is widely employed to link the low-energy continuous model to lattice tight-binding model using this strategy as shown in Section IIA Page.1063 of RMP 83, 1057 : Just replace $k_i$ by using $\frac{...
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Conditions for zero mode edge state to appear

Consider a non-interacting translationally invariant system described by H(k), k is the crystal momentum. The dimensionality of the system is denoted as d. I was thinking about what are the general ...
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Particle Hole symmetry in Interacting superconductor

For noninteracting superconductors (fermion hopping plus fermion pairing), there is a particle hole (PH) symmetry which is a redundancy. The redundancy says that the BdG Hamiltonian satisfies the ...
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Algebra of Time Reversal and Particle Hole Symmetry in 10-fold Classification of Topological Insulator/superconductor

In the ten fold classification of TI/TSC, when time reversal symmetry $\mathcal{T}$ and particle hole symmetry $\mathcal{P}$ are both present, i.e., in the symmetry classes BDI, DIII, CII, CI, for all ...
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Number of edge states as Topological Invariant

Can anyone please direct me to some sources which provide some definite rigorous proofs for the fact that the number of edge states ($N_A-N_B$ in case of SSH Model) is a topological invariant
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Time-reversal (explicitly) broken surface of $(3+1)$-dimensional topological insulator

Let us consider the surface of $(3+1)$-dimensional topological insulator, which is protected by the charge conservation $U(1)_Q$ and a time-reversal symmetry $\mathbb{Z}_2^T$. Such a surface, if not ...
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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 ...
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70 views

How to know which topological invariant is in play?

I'm currently working on the Haldane model where I've worked through the math to find that when the condition $$ \frac{M}{t_2} = 3 \sqrt{3} sin (\phi) $$ is satisfied the gap closes, meaning there ...
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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 ...
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Why are degenerate ground states interesting?

Studying the Su-Schrieffer-Heeger chain I have learned that the model has two different phases, one which is called topological and the other one trivial. In the notes it says that these phases are ...
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What “manifold in band parameters ” means?

I was reading an article https://arxiv.org/abs/0907.0500 in which they write about manifold in band parameter ,like in first line in my picture , and then they call it band parameter . can some ...
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What makes a topological insulator topological?

I understand that a topological insulator is one with an insulating bulk and conducting surface but I don't understand why or how the topological part comes into it. All of the resources I've found ...
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Topological superconductors

I have been reading a lot about quantum computation by using topological materials and I could see that the standard approach is to engineer a p-wave superconductor by using a 1-D semiconductor wire ...
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Reference for topology for topological insulators

In the field of topological insulators What topological space do they talk of? Looking for some resources that sheds light on the topology part of topological insulator
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Pulling apart the atoms of a topological insulator

Consider a topological insulator. In order to destroy a topological phase, the band gap of the bulk system should close at some point (passing thru a conducting state), but if the atoms that make up ...
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What happens to topological insulators at finite temperature?

There is a similar question here, but I had a few things I wanted to ask. So basically pretty much all analysis/ theory of topological insulators is for pure wave-functions and conservative ...
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What are the applications of edge states in 1D topological systems?

In 2D, we get robust conducting edges. In 3D, we get robust conducting surfaces. These are interesting because we can possibly utilise this robustness for protected electron transport (or light ...
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Examples of Simple Hamiltonians Giving Different Phases?

I am trying to study about simple condensed matter models that I can simulate numerically and use to calculate some topological invariant that defines a (topological) phase. My interest is in ...
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Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
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What's the difference between cylindrical geometry and disk with a hole for topological chiral $p_x\!+\!i p_y$ superconductor?

We know that for a topological chiral p-wave superconductor with a cylindrical geometry, i.e. one conserved momentum $k$ and one open boundary direction, there exists edge modes with opposite ...
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The surface states and Fermi arcs in Weyl semimetals

I'm confused about surface states in Weyl semimetals. Assume that we have a single pair of Weyl points and the Fermi level turned to this points. In this https://arxiv.org/abs/1301.0330 paper the ...
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Why there exists four Z2 invariants for a three-dimensional topological insulator (TI) and one Z2 invariant for a two-dimensional TI?

As Kane Mele suggested that there exist four Z2 invariants for a three-dimensional topological insulator (TI) while only one for two dimensional TI. But what is the reason for the same? Thanks in ...
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Is it absolutely insulating in its interior of topological insulator?

If putting a 3D topological insulator (TI) into a sandwich testing structure (electrode-TI-electrode), can we detect any leakage current in its interior like the ordinary insulator? Is it absolutely ...

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