All Questions
Tagged with topological-insulators quantum-hall-effect
52
questions
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Deriving an alternating expression for Hall Conductivity
I was reading the paper (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.3372) in which Thouless and Wu show that the hall conductivity is a topological invariant. My question is about the ...
- 1,778
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1
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187
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Deriving the non-abelian Berry connection
I'm slightly confused about a manipulation in Section 1.5.4 of Tong's notes on the Quantum Hall Effect.
This concerns the derivation of the non-abelian Berry phase.
Setup:
We have an $N$-dimensional ...
- 155
3
votes
0
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78
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Zero frequency limit of Hall conductivity not quantized?
The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system, such as quantum Hall effect and Haldane model. ...
- 3,701
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110
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Does Hall conductivity change sign with chemical potential?
The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system. A typical plot of $\sigma_{xy}(\mu)$ is like the ...
- 3,701
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0
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75
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Infinite stacking of integer quantum Hall systems
Let us consider a (3+1)-dimensional system $\mathcal{H}$ constructed by stacking (2+1)-dimensional integer quantum Hall systems $\mathcal{H}_\text{Hall}$, e.g., $E_8$ bosonic systems (or $\sigma_H=1$ ...
- 813
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The flat-band basis, Green's function projectors, and TKNN
Bernevig & Hughes' book "Topological Insulators and Topological Superconductors" have a derivation of the TKNN invariant in terms of the finite-temperature Green's function. From the ...
3
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1
answer
439
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What's the relation between quantized Hall effects and topology materials?
The quantized Hall effects (ignoring fractional Hall effect) include:
Quantum Hall effect;
Quantum anomalous Hall effect;
Quantum spin Hall effect.
All these effects are just describing the ...
- 1,757
3
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1
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228
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Is the Integer Quantum Hall Effect a distinct phase of matter?
In the Landau classification scheme, phases of matter differ in terms of symmetry. However, we know of many instances where this classification scheme does not apply. I have often heard topological ...
- 2,877
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43
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Noncontractable loops in the 2D Brilluoin zone and the Chern number
I'm getting quite twisted around trying to figure out what all is quantized exactly in IQH looking at it from the Chern number perspective.
Let's suppose quantum hall on a torus -- I can apply a large ...
- 123
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votes
2
answers
332
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What does "continuous transformation" mean with regard to the Hamiltonian of a system?
When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the ...
- 423
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2
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377
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Topological phases of matter
So according to this, scientists have discovered more than 5 states of matter we usually had that is the solid, liquid, gases, and Bose-Einstein-Condensate, and plasma. So how many topological phases ...
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70
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Is Hall conductivity time-reversal-odd at finite frequency in a topological system?
In some topological materials, e.g., the quantum (anomalous) Hall state and some related variants, the Hall conductivity $\sigma_{xy}$ is quantized and directly related to the Chern number, which ...
- 3,701
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0
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141
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Calculating the Hall Conductance using the torus shape for the Magnetic Brillouin zone
Komoto's paper (ANNALS OF PHYSICS 160, 343-354 (1985)) on the calculation of the Hall conductance provides a clear discussion about how calculate the conductance using the torus shaped magnetic ...
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133
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About the $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect)
We know that $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect) can be calculated by the Berry curvature, but we also know that $\sigma_{xy}$ is calculated by the ...
- 61
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484
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Difference between "ordinary" quantum Hall effect and quantum anomalous Hall effect
I am reading a review article on Weyl semimetal by Burkov where he writes, top of page 5:
A 3D quantum anomalous Hall insulator may be obtained by making a stack of 2D quantum Hall insulators [Ref. ...
- 508
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1
answer
198
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The quantum hall effect and Hofstadter's butterfly spectrum
What is the connection between the quantum Hall effect and the Hofstadter's butterfly spectrum? I mean, can I understand something about the quantum Hall effect in the Hofstadter's butterfly spectrum?
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268
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Is there any heat loss in chiral edge channels of topological insulators?
If we are working with nontrivial topological insulator with broken time reversal symmetry then we can expect that we have some chiral edge states. Chiral states have the property that the current can ...
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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 ...
- 11
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66
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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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1
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244
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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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46
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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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1
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391
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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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227
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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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426
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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 ...
- 1,335
1
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1
answer
360
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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 Bernevig's book 'Topological insulators and topological superconductors' I learned one can view degeneracies are ...
- 13
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0
answers
65
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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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107
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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 ...
- 1,369
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1
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150
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What are the implications that the Hamiltonian of a material lacks time reversal symmetry?
When reading about topological insulators and the quantum Hall effect, I've read that some Hamiltonians of the crystal structure representing the "materials" lack time reversal symmetry. My question ...
3
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2
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800
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Questions about Berry Phase
I'm learning about the Berry Phase from the original paper, and from the TIFR Infosys Lectures The Quantum Hall Effect by David Tong (2016).
I have some questions regarding the original derivation of ...
- 903
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463
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Physical meaning of topological invariant
What does it mean in terms of band structure when we say that any topological invariant of some system is non-zero? For example what does it mean when we say that Chern number=1 in case of IQHE?
Does ...
- 1,335
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How to calculate Edge states of Topological insulators
Topological insulators are novel state of matter in which bulk is insulator and edges are gapless. How do we calculate these gapless states? I am reading the following PRL
Feng Liu and Katsunori ...
- 831
4
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0
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878
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Laughlin's topological argument
I have a confusion about understanding the Laughlin's topological argument on Hall conductivity quantization.
This argument states that the Hall conductivity is
$$
\sigma_{xy}=\frac{e}{h}Q,
$$
where $...
- 2,333
2
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1
answer
2k
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Why is disorder essential for the Integer Quantum Hall effect IQHE?
The title already gives away the question. I see that disorder effects that the Landau levels are broadened out. They allow states to be either extended through the whole solid or localized to a ...
- 808
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0
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541
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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 ...
7
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1
answer
2k
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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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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,
$$ \sigma_{xy}= \frac{ie^2}{\hbar} \sum_{E^...
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Does any Hamiltonian that breaks time-reversal symmetry is isomorphic to a IQHE
The bulk Hamiltonian of the 2D Chern insulator in is given by
\begin{equation}
H=\sin k_{x}\sigma^{x}+\sin k_{y}\sigma^{y}+(2-m-\cos k_{x}-\cos k_{y})\sigma^{z}
\end{equation}
This Hamiltonian breaks ...
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0
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253
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why Hall conductance quantized
When I am studying quantum Hall effect, the quantum Hall conductance can be represented by Green function $\left(\text{up to}\ \frac{e^2}{h}\large \right)$:
I cannot understand why it is an integer?
...
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7
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2
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5k
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Why bulk states in quantum hall effect do not contribute to electric conductivity
Most reviews and textbooks explain quantum hall effect as insulating bulk states and conducting edge states, as is shown in the following picture.
My question is: why bulk states are insulating in ...
- 1,431
1
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1
answer
482
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Problem with quantum Hall effect and Berry curvature
I am having trouble proving that the Hall conductivity is equal to the integral over the Berry curvature in momentum space. In the TKNN (1982) paper, using the Kubo formula
$$
\sigma_{xy} = \frac{ ie^...
- 81
3
votes
1
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446
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Does Hall plateus require the existence of impurity in the sample?
While studying Hall conductivity with The Quantum 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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5
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1
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151
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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 ...
3
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1
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596
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How to determine the orientation of the massive Dirac Hamiltonian?
In the calculation of the Chern number within a 2D lattice model, let's take the Haldane model for example, the Chern number$=\pm1$ has 2 contributions coming from 2 Dirac points described by
$$h_1(\...
- 3,744
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1k
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TKNN invariant changes due to continuous deformation of parameter space
Naively, I would assume that a topological invariant remains invariant under continuous deformations of whatever space the invariant belongs to. In the case of topological insulators, this space is ...
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1
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2k
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Determining spectra of edge states numerically
Normally we write a Bloch Hamiltonian $H(\mathbf{k})$ for the bulk and determine the spectrum which gives us various bands i.e we basically obtain $E=E(\mathbf{k})$ for the bulk only.
Also in the ...
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2
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1k
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A simple conjecture on the Chern number of a 2-level Hamiltonian $H(\mathbf{k})$?
For example, let's consider a quadratic fermionic Hamiltonian on a 2D lattice with translation symmetry, and assume that the Fourier transformed Hamiltonian is described by a $2\times2$ Hermitian ...
- 3,744
18
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1
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1k
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Hall conductivity from Kubo: Bulk or edge?
Using the Kubo formula, Thouless, Kohmoto, Nightingale, and den Nijs (TKNN, PRL 49 405-408 (1982)), proved that upon summing all the contributions of the filled states of an insulator, the Hall ...
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Whis is the difference between charge fractionalization in 1D and 2D?
Both 1D Polyacetelene and 2D fractional quantum Hall state can support fractional excitations.
But as I can see, there are some differences: the ground state of Polyacetelene breaks translational ...
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3
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8k
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Why are there chiral edge states in the quantum hall effect?
The most popular explanation for the existence of chiral edge states is probably the following: in a magnetic field, electrons move in cyclotron orbits, and such such cyclotron orbits ensure electrons ...
- 1,055
3
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0
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911
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Simple model of edge states for a two-dimensional topological insulator
Quantum spin Hall states or, topological insulators are novel states of matter that have insulating bulk and gapless edge states. Are there any simple models that show these features?
See e.g. the ...
