Thermodynamics of ferromagnetic spin chains in a magnetic field: Impact of the spin–wave interaction

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 the 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 T5/2. In the pressure, the coefficient of the T5/2-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.