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arXiv · 1403.7469

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

Michael Brockmann·Jacopo De Nardis·Bram Wouters·Jean-Sébastien Caux
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Abstract

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. We furthermore give a proof that the N\'eel 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.

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Michael Brockmann, Jacopo De Nardis, Bram Wouters, Jean-Sébastien Caux. 2014-03-28. N\'eel-XXZ state overlaps: odd particle numbers and Lieb-Liniger scaling limit. https://doi.org/10.1088/1751-8113/47/34/345003

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