The first lattice correction to Bloch's law

We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin $S=\frac{1}{2},1,\frac{3}{2},\dots$. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to $o({\beta }^{-5/2})$. In addition to the leading Bloch term, this gives the correction $3\zeta (5/2)(\beta S{)}^{-5/2}/(128{\pi }^{3/2})$. The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.