Semi-classical analysis of the inner product of Bethe states

Eldad Bettelheim, Ivan Kostov

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


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 sub-leading 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.

Original languageAmerican English
Article number245401
JournalJournal of Physics A: Mathematical and Theoretical
Issue number24
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 IOP Publishing Ltd.


  • AdS/CFT
  • Bethe Ansatz
  • Slavnovs formula
  • Sutherland limit
  • Yang-Mills
  • form factors
  • integrability


Dive into the research topics of 'Semi-classical analysis of the inner product of Bethe states'. Together they form a unique fingerprint.

Cite this