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 $S=\tfrac12,1,\tfrac32,\ldots$. 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.
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