Infinite-range correlations in 1D systems with continuous symmetry

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.