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

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 $T^4\ln T$: 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.

Figure 1

Feynman graphs related to the low-temperature expansion of the partition function for a two-dimensional ferromagnet up to order $p^8$. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Vertices associated with the leading term ${\mathcal{L}}_{\mathrm{eff}}^2$ are denoted by a filled circle. Note that ferromagnetic loops are suppressed by two powers of momentum in two spatial dimensions, $d_s=2$.

Figure 2

The function $k\left(\sigma \right)$, where $\sigma$ is the dimensionless parameter $\sigma =\mu H/T$.