Spontaneous magnetization of the 3D quantum Heisenberg ferromagnet
In plain words
A cubic lattice of quantum magnets with parallel-favouring coupling should be magnetized at low temperature, as real iron is. The proof works for classical spins and for the antiparallel quantum case, but not for the quantum ferromagnet.
Precise statement
H = -sum_<xy> S_x . S_y on $Z^3$ with spin $S = 1/2$ (or any S). Prove that for T below some $T_c > 0$ the infinite-volume Gibbs states have spontaneous magnetization, e.g. $\lim_{L} L^{-6} \langle(\sum_x S_x)^2\rangle > 0$ in the box of side L. Answer: a proof.
What would settle it
A proof of long-range order at positive temperature for the spin-$1/2$ quantum Heisenberg ferromagnet on $Z^{3}$.
Status in the literature
Reflection positivity gives order for the classical ferromagnet (Frohlich, Simon and Spencer, 1976) and the quantum antiferromagnet (Dyson, Lieb and Simon, 1978; Kennedy, Lieb and Shastry, 1988) but fails here; the spin-wave free energy is proven at low T (Correggi, Giuliani and Seiringer, Communications in Mathematical Physics 2015, arXiv 1312.7873); a 2017 preprint claim (Suto, arXiv 1710.04441) lists no journal publication.