  Mean field approach, although a generally reliable tool that captures major
short range correlations, often fails in symmetric low dimensional strongly
correlated electronic systems like those described by the Hubbard model. In
these situations a symmetry is \almost broken". The problem is linked to the
restoration of the symmetry due to strong uctuations (both quantum and thermal)
on all scales. The restoration of symmetry in statistical models of scalar
\order parameter" fields was treated recently successfully on the gaussian
approximation level by symmetrization of the correlators. Here the idea is
extended to fermionic systems in which the order parameter is composite.
Furthermore the precision of the correlators can be improved perturbatively.
Such a scheme (based on covariant gaussian approximation) is demonstrated on
the 1D and 2D one band Hubbard models by comparison of the correlator with
exact diagonalization and MC simulations respectively.
