  The manifestation of the spin-wave interaction in the low-temperature series
of the partition function has been investigated extensively over more than
seven decades in the case of the three-dimensional ferromagnet. Surprisingly,
the same problem regarding ferromagnets in two spatial dimensions, to the best
of our knowledge, has never been addressed in a systematic way so far. In the
present paper the low-temperature properties of two-dimensional ideal
ferromagnets are analyzed within the model-independent method of effective
Lagrangians. The low-temperature expansion of the partition function is
evaluated up to two-loop order and the general structure of this series is
discussed, including the effect of a weak external magnetic field. Our results
apply to two-dimensional ideal ferromagnets which exhibit a spontaneously
broken spin rotation symmetry O(3) $\to $ O(2) and are defined on a square,
honeycomb, triangular or Kagom\'e lattice. Remarkably, the spin-wave
interaction only sets in at three-loop order. In particular, there is no
interaction term of order $T^3$ in the low-temperature series for the free
energy density. This is the analog of the statement that, in the case of
three-dimensional ferromagnets, there is no interaction term of order $T^4$ in
the free energy density. We also provide a careful discussion of the
implications of the Mermin-Wagner theorem in the present context and thereby
put our low-temperature expansions on safe grounds.
