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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 field. Here we will get Landu levels. Let's fill completely the lowest Landau level ($\nu=1$ integer quantum Hall state). There is a finite energy gap to the next higher Landau levels.So effectively it is an insulator. Now if we apply electric field (let's say along x direction) , we will get electric current along y-direction. But how can an insulator conduct current?

Here let's don't consider any edge states. With out considering any edge states, it can be shown that there is a finite current under the application of electric field . It has been derived in David Tong's lecture note on "Quantum Hall effect" at page number '43'. Here i have attached the link: http://qst.theory.tifr.res.in/tifr_lectures/2016-01-28_David_Tong.pdf

My question is how a completely filled lowest Landau level having finite energy gap (insulator) conducts electric current under the application of electric field?

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If we solve the Schrödinger equation for an electron in a magnetic field in Landau gauge, the solutions are discrete bands, infinitely degenerate in transverse momentum. (In addition, there is free movement along the direction of the field, which in QHE is frozen by confining the electrin gas to two dimensions.) Since there are plenty of available momentum states, the system is conducting: it is not gapped, but rather the band is flat. This is conventional Hall effect.

Now, if there is an additional confining potential constraining the momentum states, or, if the magnetic field is so strong that we have to consider the sample edges as such confinement potential, we are left with one-dimensional edge channels. These exhibit conductance quantization, similar to that of the quantum point contacts, hence the QHE.

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  • $\begingroup$ Thanks for the answer. Lets consider a finite size , so it will restrict the number of transverse momentum value (Let's say $k_{y}$). But don't consider any edge potential. Then also if we calculate expectation value of velocity in an eigenstate of Hamiltonian lets say lowest Landau level eigenstates in the presence of electric field we get non zero answer. It is same for all lowest Landau level states (in fact for all states.). So there is a finite current. $\endgroup$
    – Rakesh
    Commented Oct 8, 2021 at 5:41
  • $\begingroup$ Now my confusion is that if we completely fill the some Landau level, lets consider lowest Landau level, then there is a finite current in the system. In this scenario it seems to me that bulk of the material also conducts current , since all eigenstates contribute to current. But since we have completely filled level and there is a finite gap to next level , so it is an band insulator. It shouldn't carry current. In this sense it seems contradictory to me . would you please clarify on this? $\endgroup$
    – Rakesh
    Commented Oct 8, 2021 at 5:41
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    $\begingroup$ @PRakeshKumarDora In case when we have the edge states, the electrons are pushed towards the edges - all the momentum states are at the edges, there is nothing in the bulk. $\endgroup$
    – Roger V.
    Commented Oct 8, 2021 at 7:34

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