  The thermodynamic properties of ferromagnetic spin chains have been analyzed
with a variety of microscopic methods over the years: Bethe ansatz, spin-wave
theory, Schwinger-boson mean-field theory, Green functions and renormalization
group methods. Surprisingly, in all these different studies, to the best of our
knowledge, the manifestation of the spin-wave interaction in the
low-temperature series for the thermodynamic quantities has been ignored. In
the present work, we address this problem by following a different path, based
on the systematic effective Lagrangian method. We evaluate the partition
function up to two-loop order and derive the low-temperature expansion of the
energy density, entropy density, heat capacity, magnetization and
susceptibility in the presence of a weak external magnetic field. Remarkably,
the spin-wave interaction only manifests itself beyond two-loop order. In
particular, there is no term of order $T^2$ in the low-temperature series of
the free energy density. This is the analog of Dyson's statement that, in the
case of three-dimensional ideal ferromagnets, there is no term of order $T^4$
in the low-temperature series of the free energy density. The range of validity
of our series is critically examined in view of the Mermin-Wagner theorem. We
also compare our results with the condensed matter literature and point out
that there are some misleading statements.
