Bloch's $T^{3/2}$ law for the three-dimensional quantum Heisenberg ferromagnet

Theorem 1.1: for every fixed spin $S$ and every nonnegative symmetric finite-range coupling $J$ on $\mathbb Z^3$ whose support generates $\mathbb Z^3$, $\lim_{\beta\to\infty}\beta^{3/2}(S-m_{S,J}(\beta))=\zeta(3/2)/(8\pi^{3/2}\sqrt{\det D_{S,J}})$, with $D_{S,J}=\frac S2\sum_z J(z)zz^{\mathsf T}$; nearest-neighbour gives $\zeta(3/2)/(8\pi^{3/2}S^{3/2})$. Companions: the first lattice correction $3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2})$ for nearest-neighbour couplings, deficit equal to the ideal-magnon density up to $o(\beta^{-5/2})$; and a spherical magnetization law for the symmetric zero-field magnetization on periodic cubes. Not shown: Dyson's $T^4$ interaction term, dimensions other than three, or uniformity in $S$.

Theorem 1.1: for every fixed spin $S$ and every nonnegative symmetric finite-range coupling $J$ on $\mathbb Z^3$ whose support generates $\mathbb Z^3$, $\lim_{\beta\to\infty}\beta^{3/2}(S-m_{S,J}(\beta))=\zeta(3/2)/(8\pi^{3/2}\sqrt{\det D_{S,J}})$, with $D_{S,J}=\frac S2\sum_z J(z)zz^{\mathsf T}$; nearest-neighbour gives $\zeta(3/2)/(8\pi^{3/2}S^{3/2})$. Companions: the first lattice correction $3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2})$ for nearest-neighbour couplings, deficit equal to the ideal-magnon density up to $o(\beta^{-5/2})$; and a spherical magnetization law for the symmetric zero-field magnetization on periodic cubes. Not shown: Dyson's $T^4$ interaction term, dimensions other than three, or uniformity in $S$.