Low-Temperature Properties of Ferromagnetic Spin Chains in a Magnetic Field

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.