Spontaneous Magnetization of an Ideal Ferromagnet: Beyond Dyson's Analysis

TL;DR

This paper uses effective field theory to systematically evaluate the low-temperature expansion of spontaneous magnetization in an ideal ferromagnet, revealing that spin-wave interactions contribute at order T^{9/2}, beyond Dyson's original T^4 result.

Contribution

It provides the first fully systematic and rigorous calculation of higher-order spin-wave interaction effects on spontaneous magnetization beyond Dyson's analysis using effective Lagrangian techniques.

Findings

Next-to-leading correction appears at order T^{9/2}.

Coefficient of correction is determined by leading-order Lagrangian and independent of lattice anisotropies.

Higher-order corrections are also computed, clarifying the manifestation of spin-wave interactions.

Abstract

Using the low-energy effective field theory for magnons, we systematically evaluate the partition function of the O(3) ferromagnet up to three loops. Dyson, in his pioneering microscopic analysis of the Heisenberg model, showed that the spin-wave interaction starts manifesting itself in the low-temperature expansion of the spontaneous magnetization of an ideal ferromagnet only at order $T^4$. Although several authors tried to go beyond Dyson's result, to the best of our knowledge, a fully systematic and rigorous investigation of higher order terms induced by the spin-wave interaction, has never been achieved. As we demonstrate in the present paper, it is straightforward to evaluate the partition function of an ideal ferromagnet beyond Dyson's analysis, using effective Lagrangian techniques. In particular, we show that the next-to-leading contribution to the spontaneous magnetization resulting from the spin-wave interaction already sets in at order $T^{9/2}$ -- in contrast to all claims that have appeared before in the literature. Remarkably, the corresponding coefficient is completely determined by the leading-order effective Lagrangian and is thus independent of the anisotropies of the cubic lattice. We also consider even higher-order corrections and thereby solve -- once and for all -- the question of how the spin-wave interaction in an ideal ferromagnet manifests itself in the spontaneous magnetization beyond the Dyson term.