# Zero Resistance in Quantum Hall Effect and Superconductivity

What is the difference between the zero resistance of $R_{xx}$ in integer quantum Hall effect and the zero resistance in superconductivity?

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In a naive picture, the zero resistance in superconductivity arises from the macroscopic superposition of the wavefunction of the $N$ Cooper pairs, which are bosons. Below the critical temperature $T_c$, the ensemble of bosons condensate into a Bose-Einstein condensate and the response to an electric field is dissipationless because of macroscopic coherence.
In the case of the integer quantum Hall effect, the zero resistance is due to the momentum-locking of the eigenstates in the direction parallel to the electric field. This can be understood as a single particle phenomenon which appears thanks to the external applied magnetic field $\mathbf{B}$: states in each edge of the $2$D sample move in opposite directions and backscattering between edges is forbidden except at the percolation transition inbetween the Hall plateaux.
As it is said below in the comment by wsc, there is also an important distinction between both systems from the experimental point of view. In a superconductor below $T_c$, the resistance is strictly zero while in the Hall plateaux there is always a residual resistance at $T>0$. In the latter, one cannot overemphasize the crucial role of disorder for the drop of longitudinal resistance and quantization of the Hall component.
Moreover, the dimensionality plays an important role in the tensorial nature of the resistance tensor (in $2$D, one can loosely identify resistance and resistivity due to dimensionality reasons). In this case, inverting the relation $\mathbf{E}=\bar{\rho}\mathbf{j}$ shows that $\rho_{xx} \propto \sigma_{xx}$ and a small resistivity in the longitudinal direction also involves a small non vanishing conductivity.
This is a fine answer, but incomplete. It's extremely important that in superconductors, $R_{xx} = 0$ at $T>0$, which is not true in QH plateaus (where at any nonzero $T$, $R_{xx}$ is just very small). Furthermore, a source of much confusion for people is that the conductivity of a superconductor is infinite, the conductivity of a QH state is not $\rho_{xx}^{-1}$, but the inverse of the whole resistivity tensor, and is also a small, finite number. –  wsc Mar 9 '13 at 17:41