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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Integral Approximation in Peierls Substitution

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (...
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Hall Conductance and Chern Number Sign Convention

I have a simple question regarding sign conventions pertaining to the Chern number and Hall conductance (and what seems to be inconsistencies in the literature). In a 2D band insulator, the Chern ...
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Calculation of Hall Conductance from Feynman Diagram

I'm trying to understand the calculation of the Hall conductance for a Chern insulator from a field theory standpoint. Specifically, I want to understand how, when integrating out the fermions from ...
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Bulk boundary correspondence = difference in Chern numbers?

In topological insulators the bulk boundary correspondence is frequently stated as the principle that the number of edge modes equals the difference in Chern numbers at that edge. I found this e.g. in ...
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Why is the Berry curvature odd under time reversal?

The question is: Why is the Berry curvature, defined as $$\mathcal{F}=-\mathrm i\, \epsilon_{ij}\, \left\langle\partial_{ki}u_{n}(k)\mid \partial_{kj}u_{n}(k) \right\rangle ,$$ odd if I apply time ...
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Topological insulators: What does the scattering matrix at an topological edge tell about the Chern number?

In class we were briefly discussing, that one way to see if the edge of a TI carries a state is to consider the scattering of a lead that is attached to this edge. In fact the argument was more ...
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Is this a topological $\mathbb Z_2$ (Majorana-)invariant in *any* dimension?

Consider a non-interacting superconducting Hamiltonian in an arbitrary dimension. This is most conveniently expressed in terms of Majorana modes, which are defined as $$\gamma_{2n-1} = c_n + c_n^\...
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How to show that Chern number gives the amount of edge states?

When talking about topological insulator and talking about bulk-edge correspondence, it seems to be widely accepted conclusion that the band Chern number (winding number) is equal to, when the ...
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683 views

Time reversal, particle-hole and chiral symmetry

i looked for time reversal operator,and i read in some quantum books that $T^2=+1$ for even number of fermions and $T^2=-1$ for odd number of fermions, so does $T^2$ depend on the hamiltonian or not? ...
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Gauge invariance of the Fu-Kane-Mele invariant for 2D topological insulators

Context: In the simplest case, one can consider the Fu-Kane-Mele (FKM) invariant for a four-band model in two dimensions where two bands are filled. Let $|\psi_0(\boldsymbol k)\rangle$ and $|\psi_1(\...
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Derivation of Kubo Formula for Hall Conductance

I am trying to derive the result of the TKNN formula but am experiencing difficulty in deriving the Kubo formula. The Kubo formula used in the TKNN paper is, $$ \sigma_{xy}= \frac{ie^2}{\hbar} \sum_{E^...
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624 views

Protection of Majorana zero-modes in Kitaev chain

Is there a deeper reason, that there exist Majorana zero-modes in the whole topological phase of a Kitaev chain, which then disappear in the trivial phase. The Hamiltonian of the Kitaev chain with ...
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Chern number for the systems with open boundary conditions

For two-dimensional materials with periodic boundary conditions, we can solve the Bloch states and substitute them into the definition of Chern number, as shown in the picture: In the case of open ...
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Chern number in one-dimensional system

As the title, could we define Chern number for condensed matter systems with one spatial dimension? E.g. the 1D Su-Schrieffer–Heeger (SSH) model.
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Resources on Topological Insulators, Dirac and Weyl semimetals

I want to start studying about topological insulators and go all the way up to Dirac and Weyl semi-metals. What are some good resources(preferable textbooks if there are any) that cover these(don't ...
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Resources on non-Abelian gauge transformations and etc

I am currently studying topological insulators and occasionally the concept of non-Abelian phases and non-Abelian gauge transformations comes up in related literature. However, my understanding of ...
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How the classifying spaces of Cartan class AI and D is calculated?

The $AI$ class means that the systems with $T^{2}=1$ symmetry. Which means that we can write our Hamiltonian in a real gauge, such that its eigenvectors are real. So we can write our flat band ...
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Transfer matrix approach to the topological phases

The transfer matrix contains all the information. i.e., information about the edges and bulk. What new insight does the transfer matrix approach provide in the study of the topological phases of ...
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Does any Hamiltonian that breaks time-reversal symmetry is isomorphic to a IQHE

The bulk Hamiltonian of the 2D Chern insulator in is given by \begin{equation} H=\sin k_{x}\sigma^{x}+\sin k_{y}\sigma^{y}+(2-m-\cos k_{x}-\cos k_{y})\sigma^{z} \end{equation} This Hamiltonian breaks ...
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Under what conditions are topological edge/interface states immune to random disorder?

Recent studies classified "intrinsic", "symmetry protected", and other topological properties of matter. The paper (Topological states in photonic systems) claims that "The transport of many ...
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How to know chern number of Massive Dirac Hamiltonian

I need your help. I consider this Hamiltonian $H=v(\xi k_y \sigma_x+k_x\sigma_y)+m\sigma_z $ $\xi$ is valley index. It is +1 (at K' valley) or -1 (at K valley ). I want to know Chern number of ...
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Calculating Chern number of Massive Dirac Hamiltonian

Given a massive Dirac Hamiltonian of $H = k_x\sigma_x + k_y\sigma_y + m\sigma_z$ The Chern number can be found as $Q = \frac{1}{2\pi}\int d^2k (\nabla_k\times A(k))$ Where $A(k)$ is the Berry's ...
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(Topological) Superconductor is an Insulator

Superconductor may be viewed as an Insulator, if the Superconductor is formed by the BCS pairing where the excitations around the Fermi surface are gaped everywhere. However, superconductor also ...
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How are topological insulators connected to quantum computing?

I see lots of references in news articles on how topological insulators can be used to make a working quantum computer but I'm not sure how a protected edge state is going to lead to a quantum ...
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What is “quantum” in topological insulators?

When I'm looking at descriptions of topological insulators, (non interacting just in case anybody ascribes interactions), I'm essentially looking at single particle quantum mechanics on a lattice. ...
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Why is Brillouin zone a torus?

I am told that Brillouin Zone is a torus, which plays an important role in defining chern number and other topological invariants. The argument is that Bloch wave function $\psi_k$ is periodical in ...
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Is the 'Chern number' of a topological Kondo insulator an integer?

If you calculate the anomalous Hall conductance $\sigma_{xy}/\sigma_0$ for a simple complex hopping model at a whole band filling, this will equal an integer Chern number (given e=h=1). I would like ...
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AKLT state and Nobel physics prize 2016

The AKLT Hamiltonian and the chain is described in Wikipedia, and also the page 17 of this year Nobel Prize advanced information I have questions concerning the info released by nobelprize.org, and ...
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Chern-Simons term: Integrating out Gapped fermions in 2+1 dimensions coupled to external gauge field

Suppose we have the bulk of a topological insulator, in $2+1$ dimensions, described by a quadratic Hamiltonian in the fermion field operators, namely $$H=\sum_{i,j}\psi_i^{*} h_{ij}\psi_j$$ (the ...
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3D Weyl semimetals: lattice models

Due to the graphene, honeycomb lattice is famous for host of 2D Dirac particles, with Hamiltonian: $H = \sigma_x p_x + \sigma_y p_y$ where $\sigma_i$ are Pauli matrix. Last few years, great ...
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Carrier density in topological insulators

I'm currently doing my master thesis in physics, studying Topological Insulators. For the study of Shubnikov-de Haas oscillations visible in the magnetoresistance, it is usual to perform a Landau ...
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Topological invariant for graphene

I try to understand the concept of topological invariants in condensed matter specially in the case of the $\mathbb{Z}_{2}$ invariant and graphene. From Fradkins book I know that the $\mathbb{Z}_{2}$ ...
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What's the reason for particle-hole symmetry operator to be anti-unitary?

I have been looking at some literature on Topological Superconductor, where the BdG Hamiltonian is frequently used, the $H_{BdG}$ has the so-called particle-hole symmetry, which is commonly defined ...
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Topological Invarient Z2

For a condensed matter person, it is easy to understand the term called 'topological invariant z2' for a topological insulator. However, is there anyway so that material scientist can easily ...
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Derivation of expression for Berry curvature

Many texts quote the expression for the Berry curvature for a two-level system, with Hamiltonian $\mathbf{h}(\mathbf{k})=(h_x,h_y,h_z)$ in terms of $\mathbf{k}=(k_x,k_y)$, as something like \begin{...
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Homotopy Theory for Topological Insulators

I'm trying to understand topological insulators in terms of homotopy invariants. I understand that in 2 spatial dimensions, we have Chern insulators since $$\pi_2(S^2) = \mathbb{Z}$$ One subtlety that ...
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How to convert a Hamiltonian for the tight-binding model in silicene into $k$-space?

I am trying to convert the Hamiltonian from the paper "A topological insulator and helical zero mode in silicene under an inhomogeneous electric field" (also on arXiv) into $k$-space. $$ \begin{array}{...
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Definition of winding numbers for Hamiltonian

In Witten's lectures at the PSSCMP/PiTP summer school, there is a use of winding number for "bad points" at sec.1.3. The formula is $$ w = \int_S\frac{d^2p}{4\pi}\left[\epsilon^{\mu\nu}\epsilon^{abc}...
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Is edge states of topological insulators superconducting?

I am told edge states of topological insulators are free from back scattering. Does this mean topological insulators have no resistance if only edge states are taken into account?
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Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The Berry phase accumulated on a path can be described by a matrix when we look at adiabatic time evolution with a Hamiltonian with degenerate energy levels. The Berry phase matrix is given by $$ \...
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difference between weak and strong topologiccal insulators

Does someone know what the difference is between weak and strong topological insulators? (And do both exist in any dimension?).
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Derivation of TKNN's main result from Kubo formula

I have a question about a small but meaningful (to me at least) step in the original TKNN paper (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405). I understand the construction of the ...
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Largest values of spin-orbit coupling

Which are some of the materials with the largest ratio spin-orbit coupling constant/hopping constant? I'm trying to compute energy bands for different values of $t$ (hopping constant) and $\lambda_{SO}...
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What is the topological number of 1D Haldane phase?

According to the topological classification work [e.g. Chen et al. Science 338, 1604 (2012)], the 1D Haldane phase should have a topological number $Z_2$, which has close relationship with the edge ...
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why pseudo wave functions can be used to calculate berry connection

Berry connection plays a very important role in topological insulators. Berry connection $A(k)$ is defined to be $i\langle u(k)|\nabla_k|u(k)\rangle$, where $|u(k)\rangle$ is the periodic part of ...
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Current operator in continuum model of graphene

For the graphene hamiltonian with NNN hopping, the wavefunctions are of the form: $(\psi_A ,\psi_B)^T$. The current from A(i) to B(j) site in the lattice model is given by: \begin{equation} J_{ij}=\...
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Imaginary argument in bessel function for a wavefunction

I am solving for the continuum model of haldane model with one of the site being a potential well. The Dirac equation for a topologically non trivial case gives a solution for the states in the band ...
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964 views

eigenvectors of tight binding Hamiltonian

I am trying to calculate berry connection using tight binding method. The most important part is to calculate $\partial_k u_k(x)$, where $u_k(x)$ is the periodic part of bloch waves, i.e. $\psi_{nk}(x)...
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chern number as an obstruction to choose a smooth gauge

In condensed matter physics, I heard that if chern number of a band $n$ is non zero, it is impossible to choose a gauge such that $\psi_{nk}$ is smooth in the whole brillouin zone. However, it is ...
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Berry phase in 1D materials

The Berry phase $\phi_B$ is the phase that an eigenstate acquires after its momentum vector goes around a circle at constant energy around the Dirac point. It is defined as $\phi_B = -i \int \langle\...

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