Thermodynamics of Two-Dimensional Ideal Ferromagnets - Three-Loop Analysis

TL;DR

This paper calculates the low-temperature thermodynamic properties of two-dimensional ideal ferromagnets up to three-loop order using effective field theory, revealing logarithmic temperature dependencies due to spin-wave interactions.

Contribution

It provides the first detailed three-loop analysis of 2D ferromagnets within the effective Lagrangian framework, including renormalization and numerical evaluation of complex diagrams.

Findings

Logarithmic terms appear in the low-temperature series due to spin-wave interactions.

The leading correction in free energy density is proportional to T^4 ln T.

Three-loop calculations are feasible within effective field theory, surpassing microscopic methods.

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, as well as the 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 modified spin-wave theory.