  O(N)-symmetric lattice scalar fields are considered, coupled to a chemical
potential and source terms. At the example of N=2, it is shown that such
systems can even in (0+1) dimensions produce infinite-range correlations and a
non-zero vacuum expectation value whenever the chemical potential assumes
certain discrete values. Different mechanisms for how the latter phenomena are
produced are discussed, depending on whether source terms are set to zero or
non-zero values. In the conclusion, the relation of these findings to the
Mermin-Wagner theorem is addressed.
