N\'eel-XXZ state overlaps: odd particle numbers and Lieb-Liniger scaling limit

Understand

We specialize a recently-proposed determinant formula for the overlap of the zero-momentum N\'eel 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 N\'eel 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.