The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

TL;DR

This paper rigorously proves that in the large spin limit, the free energy of the 3D quantum Heisenberg ferromagnet aligns with the predictions of spin wave theory, confirming the effective bosonic behavior of magnons.

Contribution

It provides a rigorous proof connecting the free energy of the quantum Heisenberg ferromagnet to spin wave theory in the large spin and high-temperature limit.

Findings

Free energy converges to spin wave theory prediction

Magnons behave as non-interacting bosons in the limit

Explicit error bounds on the free energy approximation

Abstract

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if $ S \to \infty $ and the inverse temperature $ \beta \to 0 $ in such a way that $ \beta S $ stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.