  We study the relation between the spectral gap above the ground state and the
decay of the correlations in the ground state in quantum spin and fermion
systems with short-range interactions on a wide class of lattices. We prove
that, if two observables anticommute with each other at large distance, then
the nonvanishing spectral gap implies exponential decay of the corresponding
correlation. When two observables commute with each other at large distance,
the connected correlation function decays exponentially under the gap
assumption. If the observables behave as a vector under the U(1) rotation of a
global symmetry of the system, we use previous results on the large distance
decay of the correlation function to show the stronger statement that the
correlation function itself, rather than just the connected correlation
function, decays exponentially under the gap assumption on a lattice with a
certain self-similarity in (fractal) dimensions D<2. In particular, if the
system is translationally invariant in one of the spatial directions, then this
self-similarity condition is automatically satisfied. We also treat systems
with long-range, power-law decaying interactions.
