  The thermodynamic properties of ferromagnetic spin chains have been the
subject of many publications. Still, the problem of how the spin-wave
interaction manifest itself in these low-temperature series has been neglected.
Using the method of effective Lagrangians, we explicitly evaluate the partition
function of ferromagnetic spin chains at low temperatures and in the presence
of a magnetic field up to three loops in the perturbative expansion where the
spin-wave interaction sets in. We discuss in detail the renormalization and
numerical evaluation of a particular three-loop graph and derive the
low-temperature series for the free energy density, energy density, heat
capacity, entropy density, as well as the magnetization and the susceptibility.
In the low-temperature expansion for the free energy density, the spin-wave
interaction starts manifesting itself at order $T^{5/2}$. In the pressure, the
coefficient of the $T^{5/2}$-term is positive, indicating that the spin-wave
interaction is repulsive. While it is straightforward to go up to three-loop
order in the effective loop expansion, the analogous calculation on the basis
of conventional condensed matter methods, such as spin-wave theory, appears to
be beyond reach.
