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

Abstract

We specialize a recently-proposed determinant formula for the overlap of the zero-momentum Néel 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 Néel 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 Néel 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. The statement that overlaps with non-parity-invariant Bethe states vanish is still valid in the scaling limit to LL which means that the BEC state has zero overlap with non-parity-invariant LL Bethe states.

21 pages

References in corpus (5)

• Thermalization and its mechanism for generic isolated quantum systems
• Relaxation after quantum quenches in the spin-1/2 Heisenberg XXZ chain
• Overlaps of $q$-raised Néel states with XXZ Bethe states and their relation to the Lieb-Liniger Bose gas
• Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions
• A note on the spin-1/2 XXZ chain concerning its relation to the Bose gas

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