Abstract
Abstract We study the inner product of Bethe states in the inhomogeneous periodic XXX spin-1/2 chain of length L, which is given by the Slavnov determinant formula. We show that the inner product of an on-shell M -magnon state with a generic M -magnon state is given by the same expression as the inner product of a 2 M -magnon state with a vacuum descendent. The second inner product is proportional to the partition function of the six-vertex model on a rectangular L × 2 M grid, with partial domain-wall boundary conditions.
Highlights
JHEP10(2012)168 state with a vacuum descendent, obtained previously in [7], if one chooses the rapidities w of the Bethe state as w = u ∪ v
We study the inner product of Bethe states in the inhomogeneous periodic XXX spin-1/2 chain of length L, which is given by the Slavnov determinant formula
We show that the inner product of an on-shell M -magnon state with a generic M -magnon state is given by the same expression as the inner product of a 2M -magnon state with a vacuum descendent
Summary
The Slavnov determinant (2.20) can be given a very convenient operator expression [9, 10], whose derivation we review below. We represent the Slavnov kernel Ω(u, v) as the result of the action of two difference operators on the Cauchy kernel. Write the Slavnov determinant as the result of the action of N pairs of difference operators to the Cauchy determinant , Su,v =. The functional Au±[f ] can be expanded as a sum of monomials associated with the partitions of the set u into two disjoint subsets, Au±[f ] =. Under this form, the functional Au±[f ] appeared previously in ref.
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