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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 does the Berry curvature relate to the hopping strengths in the Haldane model?

Take Haldane's Hamiltonian, as quoted from Fruchart et al.'s An Introduction to Topological Insulators: 3.5.3. Haldane's Hamiltonian The first quantized Hamiltonian of Haldane's model can be ...
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Magnetic Flux in Unit Cell of Haldane Model

I am trying to understand how magnetic fluxes arise due to NN and NNN hopping in Haldane's model. In An Introduction to Topological Insulators by Fruchart et al., we see the following figure: ...
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Gapless modes at the boundary between topological insulator and normal insulator

I am currently learning about topology in condensed matter physics. I think I understand most of how topological indeces come about and differences between Z and Z2 indeces and the symmetries that ...
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2D BHZ tight binding model for Quantum spin Hall insulator

I am currently reading this article : https://arxiv.org/abs/cond-mat/0611341 and want to derive the k-space tight binding model of 2D BHZ. The tight binding model is written as \begin{equation} H = \...
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Choice of Unit Cell on Band Diagram (Brillouin Zone Folding)

I am looking at photonic band diagrams specifically, but my question relates to band diagrams in general. For a honeycomb lattice, I can pick a (primitive) rhombic unit cell or a hexagonal unit cell. ...
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Arriving at the symmetry classes for topological insulators

I'm new to the subject of topological insulators. I've been trying to understand how they come about the Cartan classification for topological insulators. I've been following these slides kitp.ucsb....
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Understanding the Su-Schrieffer-Heeger (SSH) model and Topological insulators regarding invariants

I'm studying the topic of Topological insulators, I'm having a very hard time understanding what is the relationship between the fact that topological invariants are different from $0$ and the ...
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Is the quantum Hall state a topological insulating state?

I am confused about the quantum Hall state and topological insulating states. Following are the points (according to my naive understanding of this field) which confuse me: Topological insulator has ...
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Topological Thouless pumping

I can't understand why the number of electrons pumped per cycle of a quantum pump is protected topologically. As for me this number is an integer in any case, because the number of electrons in ...
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Massless Dirac fermions vs helical Dirac fermions

Some papers when, dealing with graphene, write about charge carriers called helical Dirac fermions that have a conical energy–dispersion relation and a conserved quantity $\sigma\cdot k$ (pseudospin–...
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Source Berry Curvature Chern Insulator

Why is there non-zero hall conductance for a Chern insulator? From section 2.3 of Bernevigs book 'Topological insulators and topological superconductors' I learned one can view degeneracies are ...
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Weyl semimetal lattice model with real hoppings only?

Is there any lattice model of Weyl semimetal with real hopping amplitudes only? I was trying to find such a simplest model with only two Weyl points. As far as I've tried or read in papers, no ...
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Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form $$\mathcal{H}=\hat{\Psi}^\dagger H \hat{\Psi}$$ with $$\hat{\Psi}=\left(a_1, a_2, ...,...
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Topological indices for systems that lack translational invariance

I have a 1D discrete, finite system that lacks translational invariance. It appears to have edge states, in much the same way as an SSH model has edge states. In the SSH model we can study the ...
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Experimental confirmation of Majorana modes in Kitaev chain

I'm confused about majorana modes at the edge of Kitaev chain, what do we seek in experiment? When we first define this one we write the creation and annihilation operators as: $$a^{+}=\frac{1}{2}(\...
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Attempt at proving $-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$ from Kane and Fu's paper

I am trying to prove result (3.4) of the following paper: http://li.mit.edu/S/2d/Paper/Fu07Kane.pdf namely, that $$-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$$ ...
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Is the classification of (Symetry Protected) Topological Order for 3 band models different than for two band models?

I was reading this article: https://arxiv.org/abs/1512.08882 on the 10 fold way which gives a nice explanation of the possible topological phases for each of the symmetry classes. The example ...
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APS $\eta$-invariant and spin-Ising TQFT

I am interested in the relation between the Atiyah-Patodi-Singer-$\eta$ invariant and spin topological quantum field theory. In the paper Gapped Boundary Phases of Topological Insulators via Weak ...
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Second Chern class in 2D Haldane model from Atiyah-Singer Index Theorem?

I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is ...
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Topological materials and fractionalized excitations

I've been told several times that topological materials (such topological insulators) must have "fractionalized" excitations. Equivalently, if a material does not have fractionalized excitations, it ...
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How does on-site energy M influence Berry curvature and topological transitions in Haldane's model?

SOLUTION: The following papers almost fully-answer my question: https://arxiv.org/pdf/0904.2117.pdf https://arxiv.org/pdf/1111.5020.pdf Essentially, the Dirac points move and merge as M changes. I ...
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What does the sign of Berry curvature physically mean?

For Haldane’s model, I plotted the Berry curvature as follows: [UPDATE II: It appears as if the four peaks on the sides are due to Dirac points being in the process of moving as the on-site energy ...
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What is the closed-loop line integral of Berry curvature in a two-level model?

I am aware that integrating the Berry curvature over the entire Brillouin zone gives us the Chern number. However, I wonder what a closed loop line integral of the Berry curvature means. I think ...
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Are Kondo insulators topological insulators? What are their relationship to superconductivity?

Reading about the Kondo effect and Kondo insulators, I found they are strikingly similar to topological insulators. Are Kondo insulators topological insulators? Related question, due to their ...
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Book recommendations - Topological Insulators for dummies

Is there a pedagogical explanation of what is a topological insulator for those that do not even know what the Berry phase is but have a basic understanding of quantum mechanics and solid state ...
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Visualizing k-space tori in 3D

In many introductions to topological insulators (in the exposition of Haldane’s model, for example), we represent the parameter space, a torus, on a plane with axes running from $0$ to $2\pi$. In an ...
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How does Laughlin argument for hierarchical fractional quantum Hall effect work?

For 1 level and 1 layer $1/q$ FQHE let's say $q=5$ we have the following argument for Laughlin gauge principle. It says that if we adiabatically increase the flux from $0$ to $q\phi_0$ of a corbino ...
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Relation between topological insulators and breaking of time reversal symmetry

Whenever one talks about topological insulators, the breaking of time reversal symmetry is always mentioned. Is there an intuitive reason as to why one need time reversal symmetry to be broken in ...
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Real space chern number for two orbital system

The real space Chern number is given by $C = 12\pi i \sum_{j \in A}\sum_{k \in B}\sum_{l \in C} (P_{jk}P_{kl}P_{lj}-P_{jl}P_{lk}P_{kj})$, where $A, B,C$ are three regions arranged anticlockwise, ...
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Bulk-boundary correspondence in topological insulators

I'm not an expert in the area; just recently I checked this paper because of my research. What puzzles me a lot is the so called bulk-boundary correspondence. Can anyone explain in succint terms what'...
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Bands above and below the fermi surface carry the same or opposite winding number?

Non-interacting topological insulator (TI) in odd dimensions is characterized by winding number. It is said in many papers that bands above and below the fermi surface has the opposite winding number ...
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2D Chern Simons action by integrating out fermions

In Qi, Hughes, and Zhang's paper (https://arxiv.org/abs/0802.3537), they show how the Chern number appears as a coefficient of response function. Given the Hamiltonian (49) of a (2+1) or (4+1)D ...
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“Weak” and “Strong” topological insulators

For translationally invariant systems, we can define some topological invariant based on the translational symmetry, which is referred to "weak" topological invariant. For example, according to Kitaev'...
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The origin of $Z_2$ in topological insulators

In condensed matter physics, a study of topological insulators use the term $Z_2$ (defined as topology index) very frequently. What is the origin of $Z_2$? Why not $Z_3$ or $Z$? My understanding is ...
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Weyl Semimetal and Fermi Velocity

In a Weyl semimetal, the Nielson-Ninomiya theorem enforces the fact that number of positive and negative chirality Weyl points must be equal. Is there any restriction on the form of the Weyl points? ...
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Is topological surface state always tangential to bulk bands?

Think of a topologically nontrivial $D$-dimensional system. Its bulk bands form a $D+1$-dimensional manifold ($+1$ from energy). Its surface/edge bands form a $D$-dimensional one. Is the latter always ...
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Time reversal of Bloch Hamiltonian (at TRIM points)

I know that time-reversal symmetry requires the Bloch Hamiltonian $H(\textbf{k})$ to transform as: $$ \Theta H(\textbf{k}) \Theta^{-1}=H(-\textbf{k}) $$ where $\Theta$ is the time-reversal operator. ...
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Parity of Bloch states at TRIM points

There is an argument presented in Fu and Kane's paper on inversion symmetric topological insulator which I have not yet convinced myself. Just below Eq.(3.6), the authors said that because of ...
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Topological Classification of Band Insulators (in terms of Green's functions)

I am currently reading Topological Classification and Stability of Fermi Surfaces by Y. X. Zhao and Z. D. Wang (PRL 110, 240404 (2013)). They remark that the Green's function (along the complex ...
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Formula for the topological invariant for each of the symmetry classes

Is there a reference that systematically derives the topological invariant/winding number for all the ten symmetry classes in Altland and Zirnbauer's periodic table? For example, in this answer, there ...
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$\mathbb Z_2$ or $\mathbb Z$ invariant for the Su-Schrieffer-Heeger (SSH) model

I am trying to understand topological insulators and topological invariant. The Su-Schrieffer-Heeger (SHH) model is often invoked as a protoypical topological insulator in 1D that carries localized ...
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Linking phase of flux lines and excitation energy of monopole

I am reading this paper and on the left-hand side of pp.10 it states the following relation between linking phase and excitation energy of monopole: Now the $\theta = \pi$ term in the bulk implies ...
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Winding number of SSH model 3

SSH model can be written as $$H=-\sum_n\big[Jc_n^\dagger d_n + J'd_n^\dagger c_{n+1}\big]+h.c.$$ in Fourier space $$H(k)= \begin{bmatrix} c_k^\dagger && d_{k}^\dagger \end{bmatrix} \begin{...
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What is the physical significance of the edge states in the SSH model?

The SSH (Su Schrieffer Heeger) model, with a definite edge and an infinite bulk, is given by (in first quantisation) $$H=(t-\delta t)\sum^\infty_{n=1}(\lvert n,A\rangle\langle n,B\rvert +\lvert n,B\...
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What does the Chern number physically represent?

In 2D the Chern number can be written as $$C_m=\frac 1{2\pi}\int_{BZ}\Omega_m(\mathbf k)\cdot d^2 \mathbf k$$ where we are integrating over the Brillouin zone. In 2D this is equivalent to finding ...
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Why can we treat the Bloch hamiltonian as an effective hamiltonian?

Often when looking at topological insulators the hamiltonian is broken down into the Bloch hamiltonian and then analysed ignoring the creation/annhilation operators of the Bloch waves. Why is it okay ...
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Intuitive reason for existance of edge states in SSH model?

The SSH (Su Schrieffer Heeger) model is given by (in first quantisation) $$H=(t-\delta t)\sum^\infty_{n=1}(\lvert n,A\rangle\langle n,B\rvert +\lvert n,B\rangle\langle n,A\rvert )+(t+\delta t)\sum^\...
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Why band inversion happens only at surface of a topological insulator?

According to my naive understanding of topological insulators, an inverted band structure implies the existence of a gapless surface state at the surface. Band inversion happens due to strong spin-...
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Why is the flux quantized in 4D quantum Hall effect?

I am reading "Topological Field Theory of Time-Reversal Invariant Insulators" by Qi, Hughes, and Zhang (https://arxiv.org/abs/0802.3537). It argues that time reversal invariant (TRI) insulators in 2+1 ...
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Inversion symmetry restrictions to the Berry curvature in 2D

It is said that if a lattice has inversion symmetry, then the Berry curvature, $\vec{\Omega}(\vec{k})$ is even in $\vec{k}$, i.e. $$\vec{\Omega}(\vec{k})=\vec{\Omega}(-\vec{k})$$ I have also derived ...