Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson’s analysis

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.

Figure 1

Feynman graphs related to the low-temperature expansion of the partition function for a ferromagnet up to order $p^{10}$ in dimension $d$ $=$ 3 $+$ 1. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Vertices associated with the leading term ${\mathcal{L}}_{\mathit{eff}}^2$ are denoted by a dot. Note that ferromagnetic loops are suppressed by three-momentum powers in $d=3+1$.

Figure 2

Feynman graphs related to the low-temperature expansion of the partition function for a ferromagnet at order $p^{11}$ in dimension $d$ $=$ 3 $+$ 1. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Vertices associated with the leading term ${\mathcal{L}}_{\mathit{eff}}^2$ are denoted by a dot.

Figure 3

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

Figure 4

Two Feynman graphs related to the low-temperature expansion of the partition function for a ferromagnet at order $p^{12}$ and $p^{13}$ in dimension $d$ $=$ 3 $+$ 1. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Note that there are further Feynman graphs of order $p^{12}$ and $p^{13}$, which we have not displayed.