The Quantum Hall Effect In the integral quantum Hall effect, one has that in regions where $R_{xy}$ (the Hall resistance) is a constant, $R_{xx}$ surprisingly goes to zero. Why does that happen? Do impurities in the material play a role in this? 
 A: $R_{xx}$ is the longitudinal resistance, i.e. the resistance along the direction of current flow -- to be clear, this is the direction where the voltage difference from the battery or power source is. So a vanishing $R_{xx}$ means that there is dissipationless flow, and is kinda the "big thing" about the IQHE (Integer Quantum Hall Effect) and why it relates to topological insulator.
The fact that $R_{xx}=0$ is not directly caused by impurities. Impurities guarantee that there are plaeatux in $R_{xy}$ and hence large regions where $R_{xx}=0$. 
The reason why $R_{xx} = 0$ is that at that precise value for magnetic field strength $B$ and carrier number $N$, the Landau levels are completely filled. So there are literally no other states an electron could scatter into. If you had Ohmic (i.e. dissipationful) conduction, you'd have an electron being scattered around and hence moving at a constant velocity (Drude model). This would require it to change motional state at each scattering event. But if there are no states available, then it cannot scatter and it just keeps going along. Hence the carrier flow is dissipationless (and topologically protected, in that it does not matter whether you put a dent on the edge of the material, electrons will go around it and still move along the edge).
The presence of impurities just broadens the otherwise fixed-energy Landau levels, thereby guaranteeing that the dissipationless flow occurs for a range of $B$ (or $N$) and not just at the specific combination filling the Landau level.
Just to be clear, the plot we are talking about is ($R_{xy} =R_{\mathrm{H}}$ and $R_{xx} = R_x$):

