Overlaps of q-raised Néel states with XXZ Bethe states and their relation to the Lieb–Liniger Bose gas

Abstract

We present a ‘Gaudin-like’ determinant expression for overlaps of q-raised Néel states with Bethe states of the spin-1/2 XXZ chain in the non-zero-magnetization sector. The former is constructed by applying global Uq(sl2) spin raising operators to the Néel state, the ground state of the antiferromagnetic Ising chain. The formulas presented are derived from recently-obtained results for the overlap of the Néel state with XXZ Bethe states (Brockmann et al, 2014 J. Phys. A: Math. Theor. 47 145003, Pozsgay, 2013 arXiv:1309.4593, Kozlowski and Pozsgay, 2012 J. Stat. Mech. P05021, Tsuchiya, 1998 J. Math. Phys. 39 5946). The determinants as well as their prefactors can be evaluated in the scaling limit of the XXZ spin chain to the Lieb–Liniger Bose gas. Within this limit a q-raised Néel state that contains finitely many down spins corresponds to the ground state of finitely many free bosons. This allows for a rigorous proof of the overlap formula of De Nardis et al (2014 Phys. Rev. A 89 033601) for Lieb–Liniger Bethe states and a Bose–Einstein condensate (BEC) state with an arbitrary even number of particles.