  Within the effective Lagrangian framework, we explicitly evaluate the
partition function of two-dimensional ideal ferromagnets up to three loops at
low temperatures and in the presence of a weak external magnetic field. The
low-temperature series for the free energy density, energy density, heat
capacity, entropy density, as well as the magnetization are given and their
range of validity is critically examined in view of the Mermin-Wagner theorem.
The calculation involves the renormalization and numerical evaluation of a
particular three-loop graph which is discussed in detail. Interestingly, in the
low-temperature series for the two-dimensional ideal ferromagnet, the spin-wave
interaction manifests itself in the form of logarithmic terms. In the free
energy density the leading such term is of order $T^4 \ln T$ -- remarkably, in
the case of the three-dimensional ideal ferromagnet no logarithmic terms arise
in the low-temperature series. While the present study demonstrates that it is
straightforward to consider effects up to three-loop order in the effective
field theory framework, this precision seems to be far beyond the reach of
microscopic methods such as modified spin-wave theory.
