Condensation of lattice hard-core bosons away from half filling
In plain words
For bosons hopping on a 3D lattice with at most one particle per site, condensation is proven only when exactly half the sites are filled, because the proof uses a mirror symmetry that holds only there. At any other filling it is unproven.
Precise statement
Hard-core bosons on $Z^3$, H = -sum_<xy> (a_x^+ a_y + a_y^+ a_x), equivalently the spin-$1/2$ XY model in a uniform field along z, at density $\rho \text{ in } (0,1), \rho != 1/2$. Prove off-diagonal long-range order, $\operatorname{lim}_{\mid x-y\mid \to \infty} \langle a_x^{+} a_y\rangle > 0$, for $T$ below some $T_c(\rho) > 0$. Answer: a proof for an open interval of $\rho$.
What would settle it
A proof of off-diagonal long-range order at some density $\rho \ne 1/2$ that does not rely on reflection positivity.
Status in the literature
The half-filled case follows from reflection positivity (Dyson, Lieb and Simon, Journal of Statistical Physics 1978); a nonzero field breaks reflection positivity.