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The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

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 $ β\to 0 $ in such a way that $ βS $ 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.