  We introduce transition operators that in a given basis of the single-site
states of a many-body system have a single non-vanishing matrix element and
introduce their correlation functions. We show that they fall into groups that
decay with the same rate. The mutual information defined in terms of the von
Neumann entropy between two sites is given in terms of these so-called
generalized correlation functions. We confirm numerically that the
long-distance decay of the mutual information follows the square of that of the
most slowly decaying generalized correlation function. The main advantage of
our procedure is that, in order to identify the most relevant physical
processes, there is no need to know a priori the nature of the ordering in the
system, i.e., no need to explicitly construct particular physical correlation
functions. We explore the behavior of the mutual information and the
generalized correlation functions for conformally invariant models and for the
SU($n$) Hubbard model with $n=2,3,4$, and 5, which are, in general, not
conformally invariant. In this latter case, we show that for filling $f=1/q$
and $q<n$, the ground state consists of highly entangled $q$-site units that
are further entangled by single bonds. In addition, we extend the picture of
the two-site mutual information and the corresponding generalized correlation
functions to the $n$-site case.
