Coulomb drag in compressible quantum Hall states

We consider the Coulomb drag between two layers of two-dimensional electronic gases subject to a strong magnetic field. We first focus on the case in which the electronic density is such that the Landau-level filling fraction $\nu$ in each layer is at, or close to, $\nu =$$\frac{1}{2}$. Discussing the coupling between the layers in purely electronic terms, we show that the unique dependence of the longitudinal conductivity on wave vector, observed in surface acoustic waves experiments, leads to a very slow decay of density fluctuations. Consequently, it has a crucial effect on the Coulomb drag, as manifested in the transresistivity ${\rho }_D.$ We find that the transresistivity is very large compared to its typical values at zero magnetic field, and that its temperature dependence is unique — ${\rho }_D\propto T^{4/3}$. For filling factors at or close to $\frac{1}{4}$ and $\frac{3}{4}$, the transresistivity has the same $T$ dependence, and is larger than at $\nu =$$\frac{1}{2}$. We calculate ${\rho }_D$ for the $\nu =$$\frac{3}{2}$ case, and propose that it might shed light on the spin polarization of electrons at $\nu =$$\frac{3}{2}$. We compare our results to recent calculations of ${\rho }_D$ at $\nu =$$\frac{1}{2}$, where a composite fermion approach was used and a $T^{4/3}$ dependence was obtained. We conclude that what appears in the composite fermion language to be drag induced by Chern-Simons interaction is, physically, electronic Coulomb drag.