  We present an effective field theory formulation for a class of condensed
matter systems with crystalline structures for which some of the discrete
symmetries of the underlying crystal survive the long distance limit, up to
mesoscopic scales, and argue that this class includes interesting materials,
such as $Si$-doped $GaAs$. The surviving symmetries determine a limited set of
possible effective interactions, that we analyze in detail for the case of
$Si$-doped $GaAs$ materials. These coincide with the ones proposed in the
literature to describe the spin relaxation times for the $Si$-doped $Ga As$
materials, obtained here as a consequence of the choice of effective fields and
their symmetries. The resulting low-energy effective theory is described in
terms of three (six chiral) one-dimensional Luttinger liquid systems and their
corresponding intervalley transitions. We also discuss the Mott transition
within the context of the effective theory.
