MATHPH
In the literature: open
Neel order in the spin-$1/2$ square-lattice Heisenberg antiferromagnet
In plain words
In the ground state of a square grid of spin-$1/2$ magnets with antiparallel coupling, the spins should line up in a checkerboard pattern with finite strength. This is proven for larger spins but not for spin $1/2$.
Precise statement
H = sum_<ij> S_i . S_j on $Z^2$ with $S = 1/2$. Prove long-range order in the ground state: the staggered magnetization $m_s = \lim \sqrt{\langle(1/N \sum_x (-1)^x S_x)^2\rangle}$ is positive (numerically $m_s$ approximately 0.307). Answer: a proof.
What would settle it
A proof of positive staggered magnetization, for example by sharpening reflection-positivity bounds.
Status in the literature
Proven by reflection positivity for $S \ge 1$ in 2D (Neves and Perez, 1986; Kennedy, Lieb and Shastry, 1988) and for $S = 1/2$ in 3D (Kennedy, Lieb and Shastry, 1988).