  The low-temperature properties of systems characterized by a spontaneously
broken internal rotation symmetry, O($N$) $\to$ O($N$-1), are governed by
Goldstone bosons and can be derived systematically within effective Lagrangian
field theory. In the present study we consider systems living in two spatial
dimensions, and evaluate their partition function at low temperatures up to
three-loop order. Although our results are valid for any such system, here we
use magnetic terminology, i.e., we refer to quantum spin systems. We discuss
the sign of the Goldstone boson interaction in the pressure, staggered
magnetization, and susceptibility as a function of an external staggered field
for general $N$. As it turns out, the $d$=2+1 quantum XY model ($N$=2) and the
$d$=2+1 Heisenberg antiferromagnet ($N$=3), are rather special, as they
represent the only cases where the spin-wave interaction in the pressure is
repulsive in the whole parameter regime where the effective expansion applies.
Remarkably, the $d$=2+1 XY model is the only system where the interaction
contribution in the staggered magnetization (susceptibility) tends to positive
(negative) values at low temperatures and weak external field.
