# Semi-classical analysis of the inner product of Bethe states

Abstract : We study the inner product of two Bethe states, one of which is taken on-shell, in an inhomogeneous XXX chain in the Sutherland limit, where the number of magnons is comparable with the length L of the chain and the magnon rapidities arrange in a small number of macroscopically large Bethe strings. The leading order in the large L limit is known to be expressed through a contour integral of a dilogarithm. Here we derive the subleading term. Our analysis is based on a new contour-integral representation of the inner product in terms of a Fredholm determinant. We give two derivations of the sub-leading term. Besides a direct derivation by solving a Riemann-Hilbert problem, we give a less rigorous, but more intuitive derivation by field-theoretical methods. For that we represent the Fredholm determinant as an expectation value in a Fock space of chiral fermions and then bosonize. We construct a collective field for the bosonized theory, the short wave-length part of which may be evaluated exactly, while the long wave-length part is amenable to a $1/L$ expansion. Our treatment thus results in a systematic 1/L expansion of structure factors within the Sutherland limit.
Document type :
Journal articles

https://hal-cea.archives-ouvertes.fr/cea-01037998
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Submitted on : Wednesday, July 23, 2014 - 10:16:20 AM
Last modification on : Monday, December 13, 2021 - 9:16:03 AM

### Citation

Eldad Bettelheim, Ivan Kostov. Semi-classical analysis of the inner product of Bethe states. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2014, 47 (24), pp.25401. ⟨10.1088/1751-8113/47/24/245401⟩. ⟨cea-01037998⟩

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