2 VARIATIONS ON SLAVNOV’S SCALAR PRODUCT

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.

Related papers

Variations on Slavnov’s scalar product

O. Foda, M. Wheeler

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.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.

arXiv:1207.6871Journal of High Energy PhysicsOct 1, 201210 citations

Inner products of Bethe states as partial domain wall partition functions

Ivan K. Kostov, Yutaka Matsuo

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.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.

arXiv:1207.2562Journal of High Energy PhysicsOct 1, 201249 citations

The open XXZ spin chain in the SoV framework: scalar product of separate states

N Kitanine, Jean Michel Maillet, Giuliano Niccoli +1 more

Abstract In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor . 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K -matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.Abstract In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor . 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K -matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.

arXiv:1807.05197Journal of Physics A Mathematical and TheoreticalOct 10, 201824 citations

On the determinant representations of Gaudin models’ scalar products and form factors

Alexandre Faribault, Dirk Schuricht

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.

arXiv:1207.2352Journal of Physics A Mathematical and TheoreticalNov 19, 201240 citations

More on Slavnov Products of Spin Chains and KP Hierarchy Tau Functions

Thiago Araujo

Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally takes the form of determinants. Building on recent results on the structure of rational and trigonometric models, we show that, provided certain conditions are satisfied, the Slavnov product of a given model can be interpreted as a tau function of the KP hierarchy, thus extending known results in a more general setting. Moreover, we show that Slavnov products can be expanded in terms of other tau functions. We also prove that their homogeneous limit can be systematically expressed as a Wronskian of functions related to the eigenvalues of the transfer matrices. Finally, we compute the Baker-Akhiezer functions associated with these Slavnov products and show that, apart from a universal multiplicative factor, they admit a closed determinantal representation.Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally takes the form of determinants. Building on recent results on the structure of rational and trigonometric models, we show that, provided certain conditions are satisfied, the Slavnov product of a given model can be interpreted as a tau function of the KP hierarchy, thus extending known results in a more general setting. Moreover, we show that Slavnov products can be expanded in terms of other tau functions. We also prove that their homogeneous limit can be systematically expressed as a Wronskian of functions related to the eigenvalues of the transfer matrices. Finally, we compute the Baker-Akhiezer functions associated with these Slavnov products and show that, apart from a universal multiplicative factor, they admit a closed determinantal representation.

arXiv:2505.21166May 27, 20250 citations

General scalar products in the arbitrary six-vertex model

G A P Ribeiro

In this work we use the algebraic Bethe ansatz to derive the general scalar product in the asymmetric six-vertex model for generic Boltzmann weights. We performed this calculation using only the unitarity property, the Yang–Baxter algebra and the Yang–Baxter equation. We have derived a recurrence relation for the scalar product. The solution of this relation was written in terms of the domain wall partition functions. In turn, these partition functions were also obtained for generic Boltzmann weights, which provided us with an explicit expression for the general scalar product.In this work we use the algebraic Bethe ansatz to derive the general scalar product in the asymmetric six-vertex model for generic Boltzmann weights. We performed this calculation using only the unitarity property, the Yang–Baxter algebra and the Yang–Baxter equation. We have derived a recurrence relation for the scalar product. The solution of this relation was written in terms of the domain wall partition functions. In turn, these partition functions were also obtained for generic Boltzmann weights, which provided us with an explicit expression for the general scalar product.

arXiv:1109.3686Journal of Statistical Mechanics Theory and ExperimentNov 18, 20111 citation

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}

\mathrm{spin}-\frac{1}{2}

\mathcal{N}=4

SYM structure constants that are known to be determinants, remain determinants at 1-loop level.