  In this Letter, we show that at low temperatures, zero-point fluctuations of
the plasmon modes of two mutually coupled 2-D planar Wigner crystals give rise
to a novel long-range attractive force. For the case where the distance $d$
between two planar surfaces is large, this attractive force has an unusual
power-law decay, which scales as $d^{-7/2},$ unlike other fluctuation-induced
forces. Specifically, we note that its range is longer than the ``standard''
zero-temperature van der Waals interaction. This result may in principle be
observed in bilayer electronic systems and provides insight into the nature of
correlation effects for highly charged surfaces.
