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?
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.