  Within the framework of effective Lagrangians we calculate the free energy
density for an O($N$) antiferromagnet in 2+1 dimensions up to three-loop order
in the perturbative expansion and derive the low-temperature series for various
thermodynamic quantities. In particular, we show that the magnon-magnon
interaction in the O(3) antiferromagnet in $d$=2+1 -- the O(3)-invariant
quantum Heisenberg antiferromagnet on a square or a honeycomb lattice -- is
very weak and repulsive and manifests itself through a term proportional to
five powers of the temperature in the free energy density. Remarkably, the
corresponding coefficient is fully determined by the leading-order effective
Lagrangian ${\cal L}^2_{eff}$ and does not involve any higher order effective
constants from ${\cal L}^4_{eff}$ related to the anisotropies of the lattice --
the symmetries are thus very restrictive in $d$=2+1. We also compare our
results that apply to O($N$) antiferromagnets in 2+1 dimensions with the those
for O($N$) antiferromagnets in 3+1 dimensions. The present work demonstrates
the efficiency of the fully systematic effective Lagrangian method in the
condensed matter domain, which clearly proves to be superior to spin-wave
theory. We would like to emphasize that the structure of the low-temperature
series derived in the present work is model-independent and universal as it
only relies on symmetry considerations.
