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

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...

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Bibliographic Details
Published in:arXiv.org 2014-09
Main Authors: Brockmann, Michael, De Nardis, Jacopo, Wouters, Bram, Caux, Jean-Sébastien
Format: Article
Language:English
Subjects:
Bosons
Boundary conditions
Chains
Invariants
Parity
Scaling
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1403.7469