Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.01591 · Journal of High Energy Physics

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

Nikolay Gromov, Nicolò Primi, Paul Ryan

Published Nov 9, 2022• 13 citations
arXivHTMLPDF

Abstract

Abstract Integrable $$ \mathfrak{sl} $$ sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for $$ \mathfrak{sl} $$ sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

DOI
10.1007/jhep11(2022)039

Related papers

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

Ryan, Paul, Nikolay Gromov, Nicolò Primi

Integrable sl(N ) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications.For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions.Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists.We prove that the set of principal operators generates the complete spin chain Yangian.Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators.Finally, we show that the available combinations of multiple insertions includes Sklyanin's SoV B operator.As a result, we are able to derive the B operator for sl(N ) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.B Mapping xΨ A | Ô|Ψ B y to xy| Ô|xy 42 C SoV basis 47 D Existence of basis of Bethe algebra eigenstates 49

•Oct 20, 2022•0 citations

Determinant form of correlators in high rank integrable spin chains via separation of variables

Nikolay Gromov, Fedor Levkovich-Maslyuk, Paul Ryan

Abstract In this paper we take further steps towards developing the separation of variables program for integrable spin chains with $$ \mathfrak{gl}(N) $$ gl N symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determinant form in our approach. Our results also extend our previous works in collaboration with A. Cavaglia and D. Volin to general values of the spin, including the SoV construction in the higher-rank non-compact case for the first time.

arXiv:2011.08229•Journal of High Energy Physics•May 19, 2021•26 citations

Integrable systems, separation of variables and the Yang-Baxter equation

Paul Ryan

This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for $\mathfrak{gl}(n)$ spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.

arXiv:2201.12057•Jan 28, 2022•4 citations

Separation of variables for higher rank integrable models: review of recent progress

Fedor Levkovich-Maslyuk

Abstract Separation of variables (SoVs) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher rank examples, such as SU(N) spin chains with N > 2, has remained elusive for a long time. In this pedagogical review we discuss the major progress achieved recently in understanding SoV for models of this type. In particular, for rational SU(N) spin chains we describe different constructions of the SoV basis, novel compact forms for spin chain eigenstates, the functional SoV approach, and explicit computation of the SoV measure. We also discuss key first applications of these results, namely the new compact determinant representations for many observables such as scalar products and correlators.

arXiv:2503.15398•Journal of Physics A Mathematical and Theoretical•Jun 4, 2025•3 citations

On scalar products in higher rank quantum separation of variables

Jean Michel Maillet, Giuliano Niccoli, Louis Vignoli

Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle ⟨ 𝐡 ̲ ⟩ , there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle ⟨ 𝐡 ̲ | 𝐤 ̲ ⟩ with the vectors |{\underline{\mathbf{k}}}\rangle | 𝐤 ̲ ⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3) 𝒴 ( g l 3 ) lattice model with N N sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}} ⟨ 𝐡 ̲ | 𝐤 ̲ ⟩ ∝ δ 𝐡 ̲ , 𝐤 ̲ ). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2) 𝒴 ( g l 2 )

arXiv:2003.04281•SciPost Physics•Dec 11, 2020•32 citations

Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables

Giuliano Niccoli

The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.

arXiv:1206.2418•Journal of Mathematical Physics•May 1, 2013•81 citations
Abstract Integrable sl\mathfrak{sl}sl sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for sl\mathfrak{sl}sl sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.
Integrable sl(N ) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications.For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions.Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists.We prove that the set of principal operators generates the complete spin chain Yangian.Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators.Finally, we show that the available combinations of multiple insertions includes Sklyanin's SoV B operator.As a result, we are able to derive the B operator for sl(N ) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.B Mapping xΨ A | Ô|Ψ B y to xy| Ô|xy 42 C SoV basis 47 D Existence of basis of Bethe algebra eigenstates 49
Abstract In this paper we take further steps towards developing the separation of variables program for integrable spin chains with gl(N) \mathfrak{gl}(N) gl(N) gl N symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determinant form in our approach. Our results also extend our previous works in collaboration with A. Cavaglia and D. Volin to general values of the spin, including the SoV construction in the higher-rank non-compact case for the first time.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
Abstract Separation of variables (SoVs) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher rank examples, such as SU(N) spin chains with N > 2, has remained elusive for a long time. In this pedagogical review we discuss the major progress achieved recently in understanding SoV for models of this type. In particular, for rational SU(N) spin chains we describe different constructions of the SoV basis, novel compact forms for spin chain eigenstates, the functional SoV approach, and explicit computation of the SoV measure. We also discuss key first applications of these results, namely the new compact determinant representations for many observables such as scalar products and correlators.
Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle ⟨ 𝐡 ̲ ⟩ , there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle ⟨ 𝐡 ̲ | 𝐤 ̲ ⟩ with the vectors |{\underline{\mathbf{k}}}\rangle | 𝐤 ̲ ⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3) 𝒴 ( g l 3 ) lattice model with N N sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}} ⟨ 𝐡 ̲ | 𝐤 ̲ ⟩ ∝ δ 𝐡 ̲ , 𝐤 ̲ ). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2) 𝒴 ( g l 2 )
The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.