Spontaneous magnetization in the quantum Heisenberg ferromagnet

For every dimension d ≥ 3 and every spin $S\in \{\frac{1}{2},1,\frac{3}{2},\dots \}$, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least $S/4$. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.