12
votes
Accepted
Haldane pseudopotential
Let's do a change of variables, from $\vec{r}_1,\vec{r}_2$ to $\vec{\Delta}=\vec{r}_1-\vec{r}_2$ and $\vec{\Sigma}=\vec{r}_1+\vec{r}_2$. Note that the Jacobian associated with this transformation is $...
- 55.5k
8
votes
Accepted
Why is disorder essential for the Integer Quantum Hall effect IQHE?
The usual plot of the IQHE has conductivity on the vertical axis and electron density on the horizontal axis. Thus, the plateaus correspond precisely to increasing the electron density yet having the ...
- 2,104
8
votes
Quantum Hall Effect: Why the spikes of the longitudinal resistance appear every time when Hall conductance jumps?
At the plateaus, the Landau levels are completely filled and the system is gapped in terms of charge transport. Therefore at low temperatures $\sigma_{xx}=0$. For longitudinal resisitivity, note that
$...
- 1,515
7
votes
How to understand the Chern-Simons effective theory in Fractional Quantum Hall Liquid?
It is an old question but very important so I will answer.
The latter one, is not an effective low energy theory it is an exact model.
However the wen's theory low energy effective theory.
you can ...
- 1,429
7
votes
Has the existence of anyons been experimentally verified?
The only direct experimental evidence for anyons which the scientific community broadly accepts is the evidence described here for $\nu = 1/3$ Laughlin states in the fractional quantum Hall effect.
...
- 49.5k
7
votes
Accepted
Rigorous Laughlin pumping argument
I solved the issue. I will explain in detail how the existence of an operator conserved by the adiabatic evolution allows us to make sense of the spectral flow argument.
Let's start with the problem ...
- 781
7
votes
Accepted
Quantum Hall Effect: Why the spikes of the longitudinal resistance appear every time when Hall conductance jumps?
Intuitive answer.
At the plateau, the Landau levels are filled. There are no available states for particles to scatter into. So they can only do the skipping orbit at the edge, which is hence ...
- 25.3k
7
votes
Accepted
Why is Hall response $(\sigma_{xy}-\sigma_{yx})/2$ rather than $\sigma_{xy}$?
The conductivity tensor $\sigma$ can be decomposed as the sum of a scalar part proportional to the identity matrix, an antisymmetric part, and a traceless symmetric part. As an explicit example in 2D,
...
- 71.5k
6
votes
How to understand topological order at finite temperature?
Here is a field theory point of view:
A gapped system has topological order if in the IR it flows to a nontrivial TQFT. We can model temperature in Euclidean field theory by using a circular (...
- 8,144
6
votes
Accepted
Derivation of Kubo Formula for Hall Conductance
The Kubo formula applies to the position dependent case and can be used to compute a momentum-dependent Hall conductivity
$$\sigma_{xy}(\boldsymbol{q})= \frac{\mathrm{i}e^2}{\hbar} \sum_{E_a < E_F &...
- 12k
6
votes
Noninteracting electrons in 2D in presence of a magnetic field and a periodic potential?
This the famous Hofstadter problem. (Douglas Hofstadter is the author of Godel-Escher-Bach).
- 56.6k
5
votes
2 Particles in a Landau level interacting via a Central Potential
An argument to justify this solution is given in the review by David Tong. However, his argument is very brief, and misses a part related to the Peierls substitution. Here I bring a full explanation:
...
- 31.2k
5
votes
Accepted
Do Chern Insulators (QAHE) have topological order (long-range quantum entanglement)?
Chern Insulator = QAHE = IQHE. Chern Insulator has "invertible" topological order and long-range entanglement as defined in https://arxiv.org/abs/1004.3835 .
Chern Insulator does not need any ...
- 13.5k
5
votes
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Why do we go beyond two-body interaction?
In condensed matter, n-body interactions are the rule more than the exception. How do they appear, taking into account that at a fundamental level the relevant interaction is the pair-wise Coulomb ...
5
votes
Accepted
Why gapped systems are called incompressible?
The electronic compressibility $\kappa$ is defined as
$$\kappa =\frac{\partial \rho}{\partial \mu} $$
where $\rho$ is the electron density, and $\mu$ the chemical potential. The region where the $\...
- 771
5
votes
What's the relation between quantized Hall effects and topology materials?
The label "topological material" has been fairly liberally applied to a wide class of materials, and there is not always a clear agreement what should qualify as topological. However, there ...
- 1,041
4
votes
The Corbino effect
The Corbino effect is the same phenomenon as the homopolar generator. In both cases in a rotating conductive material and under the influence of an external magnetic field electrons get deflected.
It'...
- 10.8k
4
votes
First Chern number, monopoles and quantum Hall states
The original reference is here (1945!). Note that before Chern classes came the Stiefel-Whitney classes, which give $\mathbb{Z}_2$ invariants of real manifolds. Chern wanted invariants of complex ...
- 8,144
4
votes
How to show that Chern number gives the amount of edge states?
Yes, there is a proof of that. The first one appeared by a beautiful paper of Hatsugai in 1993 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.3697 for a particular special classes of ...
- 2,104
4
votes
Why is the composite fermion not included in the anyon contents of FQH topological orders?
Anyons are physical excitations in FQH states (by definition, something that can be trapped locally by a potential well), while composite fermions are just symbols in a theory of FQH states. Such ...
- 13.5k
4
votes
Accepted
Intuition on why quantum hall effect?
The simplest explanation requires basic knowledge about quantum point contacts. QPCs are narrow openings between two electron gases, where transverse motion of electrons is quantized. The conductance ...
- 65.1k
4
votes
Accepted
Why harmonic oscillator levels for noninteracting electrons in 2D in an applied magnetic field?
Why should harmonic oscillator levels arise out of the blue?
Edit after reading new comments: if you write down the Hamiltonian of the system using the standard minimal coupling of the momentum $\vec{...
- 3,089
4
votes
Where do interactions enter the composite fermion theory in the fractional quantum Hall effect?
The interaction is needed for the validity of the mean field approximation, which assumes that the gap remains intact. It is clear that without interaction the argument fails for reasons you have ...
4
votes
Hofstadter butterfly patterns in different honeycomb lattice structures
Your Hofstadter butterfly for nearest-neighbor hopping on the honeycomb lattice (first figure) is almost correct, however some aperiodicity remains in the spectrum. A common cause is using artificial ...
- 141
4
votes
Are the number of states available equal to the number of particles?
Let's say we have a 2d sea of electrons.
Better yet, let's consider a 1d system. Better yet, let's consider the simplest 1d system, the "particle in a box."
If the "box" extends ...
- 23.3k
3
votes
Electron positions in the lowest Landau level
The Laughlin wavefunction is a solution to some replusive-interaction many-body Hamiltonian in the entire Hilbert space, in which $x$ and $y$ still commute. It is constructed out of single particle-...
- 56.6k
3
votes
Accepted
Physical meaning of topological invariant
Dynamics of the physical systems are governed by differential equations. As such, usually it means that evolution is a continuous and unique function of initial or boundary parameters, and time. ...
- 2,083
3
votes
How to understand the Chern-Simons effective theory in Fractional Quantum Hall Liquid?
Wen's theory is 'dropped from sky'.
Upon integrating out gapped fermions (no matter relativistic or non-relativistic), the left theory is Wen's Chern-Simons theory.
Chern-Simons has the correct ...
- 1,055
3
votes
Why does spin-orbit coupling lead to a nonzero Berry curvature?
The reason behind this has to with symmetries.
If you have time reversal (TR) and space inversion symmetry, the Berry curvature vanishes identically at all k-points. The Rashba term arises from ...
- 1,080
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