Off-diagonal long-range order implies vanishing charge gap

For a large class of quantum many-body systems with U(1) symmetry, we prove general inequality that relates the (off-diagonal) long-range order charge gap. system bosons or fermions on lattice in continuum, implies ground state off-diagonal inevitably has vanishing gap, and hence is characterized by nonzero susceptibility. spin system, within magnetization plateau cannot have transverse order.