Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A)

Theorem 1.1: for every $d\ge3$ and $S\in\{1/2,1,3/2,\dots\}$ there is $\beta_0(d,S)$ such that for every $\beta\ge\beta_0$ the zero-field nearest-neighbour dynamics has a translation-invariant $\beta$-KMS state with $\omega(S_0^z)\ge S/4$. The proof uses an exchange-cycle (random interchange) representation with boundary pins, stable-polynomial negative dependence and a determinant identity for conditioned walks. Not shown: long-range order in the finite-volume two-point formulation, the critical temperature, or anything in $d\le2$. The companion manuscripts in the family prove Bloch's law and the spherical magnetization law in $d=3$.

Theorem 1.1: for every $d\ge3$ and $S\in\{1/2,1,3/2,\dots\}$ there is $\beta_0(d,S)$ such that for every $\beta\ge\beta_0$ the zero-field nearest-neighbour dynamics has a translation-invariant $\beta$-KMS state with $\omega(S_0^z)\ge S/4$. The proof uses an exchange-cycle (random interchange) representation with boundary pins, stable-polynomial negative dependence and a determinant identity for conditioned walks. Not shown: long-range order in the finite-volume two-point formulation, the critical temperature, or anything in $d\le2$. The companion manuscripts in the family prove Bloch's law and the spherical magnetization law in $d=3$.