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.

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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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Quantum Hall state at $\nu=0$ in graphene

I don't understand the meaning of the observed quantum Hall (QH) state at filling fraction $\nu=0$ in graphene at a high magnetic field. A high magnetic field lifts the four-fold degeneracy of the ...
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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 ...
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Spectral unfolding for Aubry-Andre model

While I was studying on random matrix theory recently, I have tried to unfold the spectrum of Aubry-Andre model. If the amplitude $\lambda$ of quasiperiodic onsite potential is less than 2, then it is ...
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Bulk-edge correspondence in 4D synthetic system

I have been following this paper, which discuss the bulk-edge correspondence in the Harper Model. The Harper model is a model realized on a 2D square lattice, with the following Hamiltonian (see ...
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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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Position expectation values of charged particle in magnetic field and harmonic potential

I am dealing with a charged particle in a magnetic field and harmonic potential which can described with: $$H = \frac {(-i\hbar\nabla -e \vec A)^2}{2m} + \frac {m\omega_0^2}{2}(x^2+y^2)$$ Using the ...
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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 ...
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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 ...
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Room-temperature Quantum Hall Effect (QHE) in graphene

Quantum Hall plateaus can be observed in graphene with magnetic fields smaller than $20\,\mathrm{T}$ even at $300\,\mathrm{K}$. In the experiment, at room temperature, $h\omega_c$ exceeded the thermal ...
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Effective Lagrangian for integer quantum Hall state

Non-relativistic Lagrangian of electrons in a magnetic field is written as \begin{equation} L=i\psi^{\dagger}\partial_{t}\psi + \frac{1}{2m}\psi^{\dagger}\big(\partial_{i}-ieA_{i}\big)\psi \end{...
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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 ...
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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 ...
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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 ...
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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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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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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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Can we construct a Chern-Simons theory provided that the ground state is degenerate and gapped, like the Abelian fractional quantum Hall effect?

I am studying the Chern-Simons approach to fractional quantum Hall effect, which a special focus on the topological order in the context of Abelian fractional quantum Hall effect. To me, the logic to ...
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Confusion regarding band insulator and integer quantum Hall states

As we know if an energy band is completely filled and there is a finite energy gap to next band then the material behaves an insulator. Let's consider 2d electron gas subject to perpendicular magnetic ...
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What are the differences between spin-momentum locking and the spin hall effect?

Or, are they equal? I understand that the spin hall effect implies spin-momentum locking, but does spin-momentum locking also imply spin hall? Thanks.
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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 ...
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Why spin Hall effect (SHE) is mathematicaly represented by the antisymmetric part of conductivity tensor?

The spin current $\mathbf{J^\gamma}$ due to the electric field $\mathbf{E}$ is mathematically given by $$ \mathbf{J^\gamma = \sigma^\gamma E} $$ here $\gamma$ shows spin-component, and $\mathbf{\sigma^...
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Rigorous Hall conductance

I have been trying to understand the rigorous argument for calculating the hall conductance by averaging over two fluxes by reading this paper {1}. I think I understand the entire derivation, except ...
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More Rigorous Laughlin pumping argument

I'm having a hard time understanding how the concept of spectral flow helps us compute the hall conductivity, and in particular, Laughlin's pumping argument. I think that this question summarizes the ...
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QHE from Kubo's formula

I'm following David Tong's lectures on the Quantum Hall Effect, in which he rederives the TKNN formula using the Kubo formula. The notes are a understandably hand-wavy with notation, so let me provide ...
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Chern-Simon level quantization and quantum Hall effect

It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is ...
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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 ...
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Vortices in the effective field theory of Fractional Quantum Hall Effect

I am a grad student tasked with explaining what vortices are in the Zhang Hansson and Kivelson effective field theory of fractional quantum hall effect. I am explaining them to a final year bachelors ...
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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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Chern-Simons Lagrangian and gauge-fixing

Main question: Consider (2+1)D Chern-Simons action $$S = \int dt d^2\mathbf r \frac{k}{4\pi} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda.$$ Assuming the Coulomb gauge $\nabla\cdot \mathbf a ...
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Confustion between different types of Spin Hall Effects (SHE)

Spin Hall Effect (SHE) is the flow of transverse spin current due to applied electric field. There are several other effects that are somehow connected with SHE, for example, intrinsic SHE extrinsic ...
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Why do we call fracton by its name?

I am reading on fractons. In the literature, it is said that factons are fractionalized excitations. My understanding about fractons is that it is energetically costly to move fractons, and in this ...
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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 ...
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Landau level edge states as simple harmonic oscillator with boundary

Halperin 1982 investigated the Landau level edge state spectrum by solving the harmonic oscillator on a half-infinite line $x \in (-\infty,s]$, where varying $s$ is equivalent to varying the momentum ...
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How does the generalized Haldane statistics give rise to the admissible configurations of fractional quantum Hall ground states?

It's often stated in fractional quantum Hall and fractional Chern insulator literature (see for example, Phys. Rev. X 1, 021014 (2011) and arXiv:1308.0343) that ground states of fractional quantum ...
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Can the word "doping" refer to controlling the Fermi energy via a gate field?

In the Integer Quantum Hall Effect with a fixed value of the perpendicular magnetic field, I believe that the filling factor $\nu$ which determines the Hall conductivity is experimentally controlled ...
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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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Which one is the correct representation of the Landau levels?

Sometimes the Landau levels in a finite 2D sample drawn as in the figure below: where the energy $E$ is graphed against the width $x$ of the sample in real space where $x=0$ and $x=W$ are the two ...
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What are Landau bands as opposed to Landau levels?

I think, I understand what Landau levels are. They are quantized harmonic oscillator levels that arise when a bunch on noninteracting electrons are subjected to an external magnetic field. Can ...
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Laughlin's Gauge Argument and Quantum Spin Hall Effect

I'm reading C.L. Kane and E.J. Mele's original paper of QSHE(DOI:https://doi.org/10.1103/PhysRevLett.95.226801). When proving the existence of edge state, they use the Laughlin's gauge argument which ...
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