Thermodynamics of two-dimensional ideal ferromagnets: Three-loop analysis

Abstract

Within the effective Lagrangian framework, we explicitly evaluate the partition function of two-dimensional ideal ferromagnets up to three loops at low temperatures and in the presence of a weak external magnetic field. The low-temperature series for the free energy density, energy density, heat capacity, entropy density, and magnetization are given and their range of validity is critically examined in view of the Mermin-Wagner theorem. The calculation involves the renormalization and numerical evaluation of a particular three-loop graph, which is discussed in detail. Interestingly, in the low-temperature series for the two-dimensional ideal ferromagnet, the spin-wave interaction manifests itself in the form of logarithmic terms. In the free energy density the leading such term is of order ${T}^{4}\mathrm{ln}T$: remarkably, in the case of the three-dimensional ideal ferromagnet no logarithmic terms arise in the low-temperature series. While the present study demonstrates that it is straightforward to consider effects up to three-loop order in the effective field theory framework, this precision seems to be far beyond the reach of microscopic methods such as spin-wave theory.