Why is there longitudinal response in a partially-filled Landau level?

Question

Suppose I consider an infinite, non-interacting (so no FQHE should happen) 2DEG in the magnetic field $\vec B=B\hat z$ with a non-integer filling factor, say 0.13 or whatever. Suppose I apply an electric field $\vec E=E\hat y$, apparently I should expect response current in both $x$ (Hall) and $y$ (longitudinal) directions.

However, I do not understand the very existence of the longitudinal response along $y$. Here I provide a simple argument against it. I can perform a Galilean transformation to a reference frame with $\vec v$ along $\hat x$ such that $\vec E=-\vec v\times \vec B$. In this reference frame electric field should disappear. In the non-relativistic limit, in this frame there is only magnetic field $\vec B$. So there should not be any net current in this frame. However we know there is longitudinal $y$ current! I cannot seem to find where this contradiction comes from. Any help? Am I missing something simple?

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PS: I think I understand the "standard" explanation about gapless excitations etc. But I would appreciate it if someone can point out to me the inconsistency in my logic above, or mistakes in assumption.

Answer 1

Your argument is correct, in fact you can use it to show that the Hall conductivity is set by the density. However, there is an underlying assumption when you apply Galilean transformation, that is translation invariance. In reality there are impurities that can backscatter electrons and cause the current to dissipate. So your argument immediately fails for any real systems with disorder.

EDIT: In fact, one can use a Drude-type classical model to calculate the conductivity tensor. Assume the charge and mass of the carrier is $e$ and $m$, with carrier density $n$, the result is

$ \begin{pmatrix} J_x\\ J_y \end{pmatrix}=\frac{\sigma}{1+(\omega_c\tau)^2} \begin{pmatrix} 1 & \omega_c\tau\\ -\omega_c\tau & 1 \end{pmatrix} \begin{pmatrix} E_x\\ E_y \end{pmatrix}, \sigma=\frac{ne^2\tau}{m} $

Here $\tau$ is the relaxation time and $\omega_c$ is the cyclotron frequency. For (unrealistic) clean system, $\tau\rightarrow \infty$, the longitudinal conductivity vanishes.