# 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?

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