  We consider many-body systems with a global U(1) symmetry on a class of
lattices with the (fractal) dimensions D<2 and their zero temperature
correlations whose observables behave as a vector under the U(1) rotation. For
a wide class of the models, we prove that if there exists a spectral gap above
the ground state, then the correlation functions have a stretched exponentially
decaying upper bound. This is an extension of the McBryan-Spencer method at
finite temperatures to zero temperature. The class includes quantum spin and
electron models on the lattices, and our method also allows finite or infinite
(quasi)degeneracy of the ground state. The resulting bounds rule out the
possibility of the corresponding magnetic and electric long-range order.
