Neel-XXZ state overlaps: odd particle numbers and Lieb-Liniger scaling limit

We specialize a recently-proposed determinant formula (Brockmann, De Nardis, Wouters and Caux 2014 J. Phys. A: Math. Theor. 47 145003) for the overlap of the zero-momentum Neel state with Bethe states of the spin-1/2 XXZ chain to the case of an odd number of downturned spins, showing that it is still of 'Gaudin-like' form, similar to the case of an even number of down spins. We generalize this result to the overlap of q-raised Neel states with parity-invariant Bethe states lying in a nonzero magnetization sector. The generalized determinant expression can then be used to derive the corresponding determinants and their prefactors in the scaling limit to the Lieb-Liniger (LL) Bose gas. The odd number of down spins directly translates to an odd number of bosons. We furthermore give a proof that the Neel state has no overlap with non-parity-invariant Bethe states. This is based on a determinant expression for overlaps with general Bethe states that was obtained in the context of the XXZ chain with open boundary conditions (Pozsgay 2013 arXiv:1309.4593, Kozlowski and Pozsgay 2012 J. Stat. Mech. P05021, Tsuchiya 1998 J. Math. Phys. 39 5946). The statement that overlaps with non-parity-invariant Bethe states vanish is still valid in the scaling limit to LL which means that the Bose-Einstein condensate state (De Nardis, Wouters, Brockmann and Caux 2014 Phys. Rev. A 89 033601) has zero overlap with non-parity-invariant LL Bethe states.