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arXiv:1207.2562 · Journal of High Energy Physics

Inner products of Bethe states as partial domain wall partition functions

Ivan K. Kostov, Yutaka Matsuo

Published Oct 1, 2012• 49 citations
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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.

DOI
10.1007/jhep10(2012)168

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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.
A bstract We consider the rational six-vertex model on an L × L lattice with domain wall boundary conditions and restrict N parallel-line rapidities, N ≤ L/ 2, to satisfy length- L XXX $ \mathrm{spin}-\frac{1}{2} $ chain Bethe equations. We show that the partition function is an ( L − 2 N )- parameter extension of Slavnov’s scalar product of a Bethe eigenstate and a generic state, with N magnons each, on a length- L XXX $ \mathrm{spin}-\frac{1}{2} $ chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is Casoratian, and consequently that tree-level $ \mathcal{N}=4 $ SYM structure constants that are known to be determinants, remain determinants at 1-loop level.
In this note, we consider the six-vertex model with domain wall boundary conditions, defined on an M×M lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities λj and μk. For a particular choice of the set of λj we find a new determinant representation for the partition function, which allows evaluation of the bulk free energy in the thermodynamic limit. This provides a new connection between two types of determinant formulas. We also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions.
Abstract We consider the rational six-vertex model on anL×Llattice with domain wall boundary conditions and restrictNparallel-line rapidities,N≤L/2, to satisfy length-LXXX $ \mathrm{spin}-\frac{1}{2} $ chain Bethe equations. We show that the partition function is an (L− 2N)- parameter extension of Slavnov’s scalar product of a Bethe eigenstate and a generic state, withNmagnons each, on a length-LXXX $ \mathrm{spin}-\frac{1}{2} $ chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is Casoratian, and consequently that tree-level $ \mathcal{N}=4 $ SYM structure constants that are known to be determinants, remain determinants at 1-loop level.
We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.
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/L1/L1/L expansion. Our treatment thus results in a systematic 1/L expansion of structure factors within the Sutherland limit.
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/L1/L1/L expansion. Our treatment thus results in a systematic 1/L expansion of structure factors within the Sutherland limit.