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At any other filling it is unproven.","posed_since":"1978","precise":"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 $\\rho$.","problem_ref":null,"references":"","settled_by":"A proof of off-diagonal long-range order at some density $\\rho \\ne 1/2$ that does not rely on reflection positivity.","status_note":"The half-filled case follows from reflection positivity (Dyson, Lieb and Simon, Journal of Statistical Physics 1978); a nonzero field breaks reflection positivity.","title":"Condensation of lattice hard-core bosons away from half 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