cond-mat0206558.tex
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{\em Paper withdrawn pending major revisions.}

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{\noindent \bf Correlation Induced Enhancement 
of Frictional Drag between Two-Dimensional Hole Layers}
 
\vspace{0.3cm} 



\vspace{0.3cm} 

In a recent Letter\cite{Pillar},
the frictional drag was measured
on double layers two-dimensional
hole systems in the regime of very small
hole concentrations $p$. At $p =1.5, 2.5$ 
and 7 $\times 10^{10}~{\rm cm}^{-2}$ 
(or $r_{s} \simeq 24.5, 19$ and 11.4)
the authors observed a 500, 200 and 20 times enhancement 
of the drag over the standard 
calculation\cite{phonon,coulomb} valid for a weakly 
interacting electron gas ($r_s < 1$). 
(Here $r_{s} = (\pi p)^{-1/2}/a_B$, where
$a_{B}$ is the Bohr radius of a 
hydrogen like acceptor.)
They suggested that
this enhancement results from 
interaction effects in the dilute
regime. In this comment,
we would like to show that for $r_s \gg 1$ indeed very
strong enhancement of drag follows from
replacing the Thomas-Fermi screening radius 
by the screening radius of the strongly correlated
hole liquid. 

The theoretical expression for the drag resistivity $\rho_{d}$ 
used in Ref.\onlinecite{Pillar} 
is derived in Refs. \onlinecite{phonon,coulomb}.
If $\tau_{d}$ is defined by $\rho_{d} = m/pe^{2}\tau_{d}$,
one has
%
\begin{equation}
\frac{1}{\tau_D}=\frac{\zeta(3)\pi}{16}
   \frac{k_{B}^2T^2}{\hbar\epsilon_{\rm F}}
\frac{1}{(k_{\rm F}d)^2}
\frac{1}{(q_{\rm TF}d)^2}.
\label{smallrs}
\end{equation}
%
Here
$m$ is the hole effective mass,
$\epsilon_{\rm F} = \hbar^{2}k_{\rm F}^{2}/2m$, 
$k_{\rm F}=(2\pi p)^{1/2}$ and 
$q_{\rm TF} = 2me^{2}/\hbar^2\kappa = 2/a_{B}$ 
are the Fermi energy, the Fermi wave vector and 
the inverse intralayer Thomas-Fermi screening radius of the  
two-dimensional gases (we assume that they have 
the same concentration $p$), 
$T$ is the temperature, 
$d$ is distance between planes of two gases,
$\kappa$ is the dielectric constant of GaAs
and $k_B$ is the Boltzmann constant.
 
We would like to emphasize that at $r_{s} \gg 1$ 
the use of $q_{\rm TF}$ in order to 
describe intralayer screening can not be justified. 
Indeed, the Thomas-Fermi screening radius equals
$a_{B}/2$.  
It is much smaller than the average 
distance between holes $(\pi p)^{-1/2} = r_{s}a_{B}$
and the hole gas definitely
can not screen at such lengths.
At $r_{s}\gg 1 $, the Coulomb interaction 
of holes dominates their kinetic energy and holes 
are strongly correlated within
each layer so that short range order of 
this liquid is close to that of the 
Wigner crystal with the same $p$.
This means that the energy of this Fermi liquid per hole
is very close to that of the
Wigner crystal, $-1.1e^{2}(\pi p)^{1/2}/\kappa$
\cite{ceperley} and the chemical potential of 
the Fermi liquid, $\mu_{FL}$, 
is close to $\mu_{WC} = -1.65 e^{2}(\pi p)^{1/2}/\kappa$.
Therefore, the inverse screening radius of the hole 
Fermi liquid, $q_{FL}=2\pi e^{2} (d\mu_{FL}/dp)^{-1}/\kappa$,
is close to the Wigner crystal one, $q_{WC} =
2\pi e^{2}(d\mu_{WC}/dp)^{-1}/\kappa = - 2.42 (\pi p)^{1/2}$.
(Using Fig. 2 of Ref. \onlinecite{ceperley} 
we found that $q_{FL}$ differs 
from $q_{WC}$ only by $2\%$ at $r_s \sim 20$).

We start from general expression of $\rho_{D}$
with help of the intralayer 
density-density response functions $\chi(\omega, q)$
(see Eqs. (18) and (21) of Ref.~\onlinecite{MacDonald}).
A strong Coulomb interaction 
leads to replacement 
of the free hole
factor $d\mu/dp$  
by $d\mu_{WC}/dp$ both in Re$\chi(\omega, q)$ 
and Im$\chi(\omega, q)$, 
because modulation of concentration of quasiparticles 
requires modulation of the hole concentration $p$.
This results in replacement of 
$q_{TF}^2$ by $q_{WC}^2$
in Eq.~(\ref{smallrs}) and 
leads to a large new 
factor $\alpha = (q_{\rm TF}/q_{WC})^{2} = 
0.69 r_{s}^{2}$. 
The interaction also renormalizes 
the mass of quasiparticles $m^{*}$ adding 
another factor $(m^{*}/m)^{2}$.
Here we neglect this weaker effect.
Then the law $\rho_D \propto p^{-3}$, which 
follows from Eq. (\ref{smallrs}),
is replaced by $\rho_D \propto p^{-4}$ for the
large $r_s$ regime.
For three samples mentioned above 
we get $\alpha = 414$, 249 and 90. The first two 
are close to the experimental enhancement 
factors (see above). 
Note that recent
experimental data\cite{Jim} 
obtained in electron double layers at smaller $r_s$
agree well with our result $\alpha = 0.69 r_{s}^{2}$.
Our theory is limited 
by the condition $k_{F}d > 1$. 
At $k_{F}d \ll 1$ the Born approximation fails,
the $d$ dependent small factors reach unity,
$1/\tau_{d} \sim k_{B}^2T^2/\hbar\epsilon_{\rm F}$ and 
$\rho_D \propto p^{-2}$. 
To be able to observe the  
$\rho_D \propto p^{-4}$ law at very small $p$ 
one has to study structures with larger $d$.
Eventually, the drag 
may allow to measure $m^{*}(r_s)$.

Instead of theoretical calculation, we could use
experimental values of
$d\mu_{FL}/dp$ obtained in measurement of
the compressibility\cite{Sivan,Yacoby} 
for the same range of $p$ in order 
to calculate
the enhancement factor $\alpha$.
Values of the low temperature $d\mu_{FL}/dp$
obtained in Ref.\onlinecite{Sivan} agree very well with 
the used above Wigner crystal one, so that using these
data will make no difference.
On the other hand, the
absolute values of the 
$d\mu_{FL}/dp$ measured in Ref.\onlinecite{Yacoby}
are almost twice smaller.
Surprisingly, they agree
with the Hartree-Fock theory 
which at large $r_s$ gives $d\mu_{FL}/dp \simeq 0.55~
d\mu_{WC}/dp$. There is no reason to expect that the
Hartree-Fock theory is valid at $r_s \sim 20$.

We are grateful to E. Abrahams, J. Eisenstein, H. Noh, M. Pustilnik,
U. Sivan and B. Spivak for useful discussions.
This work was supported by NSF DMR-9985785.

\vspace{0.3cm}
\noindent   T. T. Nguyen, B. I. Shklovskii \\
\indent {\small Theoretical Physics
Institute, University of Minnesota}\\
\indent {\small 116 Church St. Southeast, Minneapolis,
Minnesota 55455}\\

\vspace{-0.18in}
\begin{references}
\vspace{-0.68in}
%
\bibitem{Pillar} R. Pillarisetty, Hwayong Noh, 
D. C. Tsui, E. P. De Poortere, E. Tutuc,
and M. Shayegan, Phys. Rev. Lett. {\bf 89},
016805 (2002).
%
\bibitem{phonon}  T. J. Gramila, J. P. Eisenstein, A. H. MacDonald,
L. N. Pfeiffer and K. W. West, Phys. Rev. B {\bf 47}, 12957
(1993).
%
\bibitem{coulomb} A. Jauho and H. Smith, Phys.
 Rev. B {\bf 47}, 4420 (1993).
%
\bibitem{ceperley} B. Tanatar, and D. M. Ceperley, Phys. 
Rev. B {\bf 39 }, 5005 (1989).
%
\bibitem{MacDonald} Lian Zheng, A. H. MacDonald Phys. 
Rev. B {\bf 48}, 8203 (1993).
%
\bibitem{Jim} M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, K. W. West
cond-mat/0206547.
%
\bibitem{Sivan} S. Shapira, U. Sivan, P. M. Solomon, E. Buchstab, 
M. Tischler, and G. Ben Yoseph, Phys. Rev. Lett. {\bf 77}, 3181 (1996).
%  
\bibitem{Yacoby} S. Ilani, A. Yacoby, D. Mahalu, and Hadas Shtrikman,
Phys. Rev. Lett. {\bf 84}, 3133 (2000).


\end{references}
\end{multicols}
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cond-mat0206558.extracted_bib
\bibitem{Pillar} R. Pillarisetty, Hwayong Noh, 
D. C. Tsui, E. P. De Poortere, E. Tutuc,
and M. Shayegan, Phys. Rev. Lett. {\bf 89},
016805 (2002).
%

\bibitem{phonon}  T. J. Gramila, J. P. Eisenstein, A. H. MacDonald,
L. N. Pfeiffer and K. W. West, Phys. Rev. B {\bf 47}, 12957
(1993).
%

\bibitem{coulomb} A. Jauho and H. Smith, Phys.
 Rev. B {\bf 47}, 4420 (1993).
%

\bibitem{ceperley} B. Tanatar, and D. M. Ceperley, Phys. 
Rev. B {\bf 39 }, 5005 (1989).
%

\bibitem{MacDonald} Lian Zheng, A. H. MacDonald Phys. 
Rev. B {\bf 48}, 8203 (1993).
%

\bibitem{Jim} M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, K. W. West
cond-mat/0206547.
%

\bibitem{Sivan} S. Shapira, U. Sivan, P. M. Solomon, E. Buchstab, 
M. Tischler, and G. Ben Yoseph, Phys. Rev. Lett. {\bf 77}, 3181 (1996).
%  

\bibitem{Yacoby} S. Ilani, A. Yacoby, D. Mahalu, and Hadas Shtrikman,
Phys. Rev. Lett. {\bf 84}, 3133 (2000).


