Why in quantum hall effect longitudinal resistance and conductivity can be simultaneously zero? Why in quantum hall effect longitudinal resistance and conductivity can be simultaneously zero? I am puzzled about it. What's the physics meaning of it?
 A: The resistivity and conductivity are tensors
\begin{align}
\rho &= \begin{pmatrix}\rho_{xx} & \rho_{xy}\\\rho_{yx} & \rho_{yy}\end{pmatrix}\\
\sigma &= \begin{pmatrix}\sigma_{xx} & \sigma_{xy}\\\sigma_{yx} & \sigma_{yy}\end{pmatrix}
\end{align}
linearly relating the electric field and current density:
\begin{align}
E_a &= \rho_{ab}J_b,\\
J_a &= \sigma_{ab}E_b.
\end{align}
As matrices, $\rho$ and $\sigma$ are inverses,
\begin{align}
\sigma = \rho^{-1}
\end{align}
so that
\begin{align}
\sigma = \frac{1}{\det \rho}\begin{pmatrix}\rho_{yy} & -\rho_{xy}\\-\rho_{yx} & \rho_{xx}\end{pmatrix}.
\end{align}
If the longitudinal components $\rho_{xx}$, $\rho_{yy}$ of the resistivity are zero, and if the transverse $\rho_{xy}$ and $\rho_{yx}$ are non-zero so that $\det \rho \neq 0$, then the longitudinal components of $\sigma$ are also zero.
Since the longitudinal conductivity is zero, an electric field applied along the $x$ (or $y$) direction does not lead to a current in this direction. Instead, a transverse current is produced. Since the longitudinal resistivity is zero, if we pass a current along the $x$ (or $y$) direction, we don't observe a potential difference along this direction. Instead, we would measure a transverse (Hall) voltage.
Quoting Steve Girvin:

Notice that, paradoxically, the system looks insulating since $\sigma_{xx} = 0$ and yet it
  looks like a perfect conductor since $\rho_{xx} = 0$.

