Questions tagged [quantum-hall-effect]
Questions related to the quantum Hall effect (the quantisation of resistivity observed when a 2-dimensional electron gas system is subjected to a strong perpendicular magnetic field), as well as formulations of states, topological properties, and applications.
313
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All Chern numbers as mapping degree?
The Chern number of Haldane model can be interpreted as a mapping degree from $T^2$ (1BZ) to $S^2$ (Bloch sphere). The question is whether all the Chern numbers can be interpreted in this way.
In ...
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Plateaus in Quantum Hall Effect and its Robustness
I was studying Quantum Hall Effect and there I came out with a question that why the plateaus in the plot of Hall Resistivity are robust ?
I know by solving Schrodinger equation and using Landau Gauge ...
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Does the fractional quantum Hall effect always occur when the integer quantum Hall effect occurs?
Does the fractional quantum Hall effect always occur when the integer quantum Hall effect occurs? In other words, is there an example of a material where the IQHE can be measured without the "...
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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 ...
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Why the anomalous Hall effect breaks the time-reversal symmetry?
Can someone please explain in a simple way why the anomalous Hall effect breaks the time-reversal symmetry while the Hall effect does not?
- 121
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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
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43
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Can anyons exist on a torus without any additional conditions?
While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
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Hofstadter butterfly patterns in different honeycomb lattice structures
Without loss of generality, we set vector potential $A=\left( -yB,0,0 \right)$. Here B is magnetic field. And due to Peierls substitution, $t_{ij}\rightarrow \exp \left( igaA_{r_j} \right) t_{ij}$, ...
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Why is a large magnetic field problematic (for topological physics)?
In papers studying or searching for topological order (intrinsic or symmetry-protected) in various condensed matter systems (e.g. Field-tuned and zero-field fractional Chern insulators in magic angle ...
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Correlators $\langle \psi(z_1) \cdots \psi(z_N) \sigma(w_1) \sigma(w_2) \rangle$ in Ising conformal field theory
Question: How one obtains
$$\langle \psi(z_1) \cdots \psi(z_N) \sigma(w_1) \sigma(w_2) \rangle \sim \mathrm{Pf}\left( \frac{f(z_i,z_j; w_1, w_2)}{z_i-z_j}\right) \prod_{i=1}^N (z_i-w_1)^{-1/2} (z_i-...
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The full symmetry group and infinitesimal generators of the quantum Hall effect
What is the full symmetry group and infinitesimal generators of the quantum Hall effect (in the continuum, not a lattice)? For example the kelper potential has $SO(4)$ symmetry with the angular ...
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Can anyone clarify the origin of the radial current in this discussion on the quantum hall effect?
I am trying to understand parts of this webpage on the quantum hall effect, and I am stuck on the part where they talk about Corbino geometry. So we have a conductive 2D annulus and a changing ...
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Non-unitary magnetic translation operator for Landau levels
In chapter 9 of their book [1], Altland and Simons consider the Landau Hamiltonian in the symmetric gauge,
\begin{equation}
H=\frac{1}{2m^*}\left[\left(-\mathrm{i}\partial_1-\frac{x_2}{2l_0^2}\right)^...
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Effective Theory for Matter Fields Coupled to a Chern-Simons Field
Assuming that matter fields coupled with the background Chern-Simons field (or Maxwell-Chern-Simons field), I want to obtain the effective theory in terms of matter fields only, by integrating out a ...
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The origin of spin Hall current in quantum spin Hall effect
I am trying to follow https://topocondmat.org/w5_qshe/fermion_parity_pump.html 's explanation for quantum spin Hall effect.
I am stuck on the following paragraph:
"In particular, let’s again make ...
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What is the difference between josephson junction and other weakly linked materials?
Josephson junction is made of two superconductors that are weakly coupled, but we have normal conductors which can be weakly coupled and they also show I-V characteristics similar to a Josephson ...
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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. ...
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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 ...
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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$ ...
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Why is Hall response $(\sigma_{xy}-\sigma_{yx})/2$ rather than $\sigma_{xy}$?
In this PRL paper (and other works like the review article), the Hall response is defined as the antisymmetric part $$\sigma_H=(\sigma_{xy}-\sigma_{yx})/2$$ instead of $\sigma_{xy}$ itself.
What is ...
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What barriers have prevented us from using Landau levels to make qubits?
Landau levels allow us to jam all electrons into nearly identical quantum states - these states share the same quantum numbers (e.g. orbital and spin) except for the guiding center. Furthermore, the ...
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Shubnikov-de-Haas effect and Quantum Hall effect
I am wondering if these two phenomena are two names for the same thing or whether these are distinct effects and there are situation where one appears, but the other one doesn't?
Both seem to produce ...
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What is a "statistical" gauge field?
In the Fractional Quantum Hall Effect (FQHE), one introduces a Chern-Simons (CS) gauge field and it is called statistical. Why? Another main question is below (*), but maybe I should state some things ...
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Action for boundary term in Chern-Simons theory (David Tong's note)
This question is about obtaining the boundary action from Chern-Simons theory.
While reading David Tong's chapter 6 on quantum Hall effect, I cannot derive an equation between (6.9) and (6.10) of the ...
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Monopoles in the fractual quantum Hall
What role do magnetic monopoles play in the fractional quantum Hall effect?
Also see here.
EDIT: I should add that I understand the fractual quantum Hall effect only from the perspective of minimal ...
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Landau Levels degeneracy in a finite sample
According to different sources: Tong lectures on IQHE (Tong), MIT Open courses (MIT) etc, when calculating the number of states in each Landau Level all of them impose (in the Landau gauge) periodic ...
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Gauge fixing in derivation of fractional QHE action
I'll copy the text from a relevant question:
This follows the discussion in Altland and Simons Condensed Matter
Field Theory -- section 9.5 on deriving the Chern-Simons action for
FQHE.
Starting with ...
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Prerequisites for Chern-Simons approach to the Fractional Quantum Hall Effect
I am interested in learning the Chern-Simons approach to the fractional quantum Hall effect right from the basics. I have learnt about Lie groups and Weyl quantisation and am currently learning ...
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Quantum Hall Effect: Two-terminal resistance and thermodynamic equilibrium
In Goerbig's lecture notes on the quantum hall effect, Fig. 3.3 describes the so-called two-terminal measurement. The figure and accompanying text show us that in a two-terminal measurement, the ...
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Fractional quantum Hall effect: is new physics required to understand it?
Is it true that some of the experimentally observed states in fractional quantum Hall effect are unexplainable by current physics? If so, does this point towards a revision of quantum mechanics, and / ...
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Landau levels in graphene - unbounded from below spectrum
Considering the case of monolayer graphene in a perpendicular magnetic field arises LL in it. The final spectrum is given by
$$
\epsilon_n=\mathrm{sign}(n)\hbar\omega\sqrt{|n|}
$$
where $n\in\mathbb{Z}...
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Gaps in FQHE bulk spectrum
In David Tong's note on Quantum Hall Effect, Tong states on page 85 that the incompressible Laughlin wavefunction is "responsible for the gap in the bulk spectrum" of FQHE (Fractional ...
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Spectral flow in IQHE
I am encoutering some problems in understading the argument in David Tong's QHE lecture notes that the spectral flow of extended states give rise to currents in IQHE. First, Tong argues in page 36 ...
- 215
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What's the difference of flux tube and vortex in FQHE (especially in Jain wavefuntion)
In the book Composite Fermion by Jainendra K.Jain, he mentioned the motivation of Jain wavefunction: attach flux tube of 2p flux quantum to fermions to form composite fermions. Naively, this is done ...
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Edge States in Integer Quantum Hall Effect (IQHE)
As part of an exercise we have been asked to consider a free particle in a uniform magnetic field $\vec B = (0,0,B)$, bounded by a strong confining potential:
$$H = \frac {(-i\hbar\nabla^ -e \vec A)^2}...
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Filling Fermions into the Lowest Landau Level and the Laughlin Wave Function
For a particle moving in the $xy$-plane in the presence of a uniform magnetic field $B$ along the $z$-direction, if one chooses the symmetric gauge of the vector potential, then one can use the ...
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Term in Hamiltonian when electric field is turned on
In David Tong's notes on Quantum Hall Effect, in his derivation of the Kubo formula, he says that turning on the electric field has the effect of adding a $- \mathbf{J} \cdot \mathbf{A}$ to the ...
- 1,891
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Edge mode currents in quantum hall effect
In David Tong's notes on the quantum hall effect, when calculating the current due to edge modes, he writes it's expression as
$$I_y = -e \int \frac{dk}{2 \pi} v_y(k)$$
where $v_y$ is the drift ...
- 1,891
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Why are the plateaus in the Quantum Hall horizontal rather than diagonal
I was wondering why the plateaus of $\rho_{xy}$ in the integer quantum Hall effect are horizontal and do not scale linearly with the magnetic field $B$ since the Lorentz force should still be acting ...
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Spectral Flow vs. Edge currents in the integer quantum Hall effect
In the integer quantum Hall effect, there are at least two perspectives given in the scientific literature, which are related. One picture is from the Laughlin-Halperin construction where we imagine ...
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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 ...
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Magnetic field with periodic boundary conditions, torus and magnetic monopoles
Suppose I have a 2D square lattice in the xy-plane, and I apply a uniform magnetic field in the z-direction. To simplify calculations, I would like to assume periodic boundary conditions in both the x ...
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Laughlin gauge argument, integer quantum hall and periodic boundary conditions
In every treatment I have seen of the Laughlin gauge argument, it is suggested that as a flux quantum, $\Phi_0 = h/e$, threads through the cylinder or ribbon, that one unit of charge is pumped from ...
- 2,777
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103
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$U(1)$ Chern-Simons from simple integer quantum Hall arguments, basics
I am trying to understand how the Chern-Simons term appear in $U(1)$ effective theory of integer quantum Hall effect (IQHE). I usually read in (90% of) lecture notes "$A dA $ is the Chern-Simons ...
- 115
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Laughlin wavefunction
I have just started reading about the Fractional Quantum Hall Effect (FQHE) and had these doubts:
In the review article [1, 2] by Prof. Wen, he writes that electron always dances anti-clockwise so ...
- 61
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Would there be plateaus in IQHE in haldane graphene model
It's my understanding that the plateaus observed in the integer quantum hall effect are due to scattering due to impurities in the material. The haldane model for graphene does not include a model for ...
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Holography vs edge modes in Chern-Simons theories
There are two facts I have heard about Chern-Simons theories
They are dual to a WZW theory on the boundary (e.g. Relation between CS/WZW and AdS/CFT).
They require a chiral theory on the boundary due ...
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2
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Landau levels degeneracy in symmetric gauge
I'm reading David Tong's lecture notes on the Quantum Hall Effect.
When symmetric gauge taken, a basis of the lowest landau level wave functions is
$$\psi_{LLL,m}\sim\left(\frac{z}{l_B}\right)^m e^{-|...
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282
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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 ...
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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 ...
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