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:1810.10996 · Journal of Mathematical Physics

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

Paul Ryan, Dmytro Volin

Published Mar 1, 2019• 63 citations
arXivHTMLPDF

Abstract

We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.

DOI
10.1063/1.5085387

Related papers

Bethe ansatz without nesting

Gleb Eduardovich Arutyunov, Hrachya M. Babujian, Minghao Gao

A bstract We develop a non-nested Bethe ansatz description of rational 𝔤𝔩 ℓ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔤𝔩 3 and 𝔤𝔩 4 spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- ℓ equation ℛ ℓ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q 1 ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔤𝔩 ℓ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔤𝔩 3 spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔤𝔩 3 oper.

arXiv:2607.11617•Journal of High Energy Physics•Oct 8, 2026•0 citations

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

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

Nikolay Gromov, Nicolò Primi, Paul Ryan

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.

arXiv:2202.01591•Journal of High Energy Physics•Nov 9, 2022•13 citations

Completeness of Wronskian Bethe equations for rational gl(m|n) spin chain

Dmitry Chernyak, Sébastien Leurent, Dmytro Volin

We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities $\theta_\ell$, $\ell=1,\ldots,L$ satisfy $\theta_\ell+\hbar\neq\theta_{\ell'}$ for $\ell<\ell'$. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $\theta_\ell$. In particular, we consider the regime $\theta_{\ell+1}/\theta_{\ell}\gg 1$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

•Apr 6, 2020•4 citations

New construction of eigenstates and separation of variables for SU(N) quantum spin chains

Nikolay Gromov, Fedor Levkovich-Maslyuk, Grigory Sizov

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.

arXiv:1610.08032•Journal of High Energy Physics•Sep 1, 2017•108 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
We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.
A bstract We develop a non-nested Bethe ansatz description of rational 𝔤𝔩 ℓ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔤𝔩 3 and 𝔤𝔩 4 spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- ℓ equation ℛ ℓ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q 1 ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔤𝔩 ℓ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔤𝔩 3 spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔤𝔩 3 oper.
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 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.
We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities θℓ\theta_\ellθℓ​, ℓ=1,…,L\ell=1,\ldots,Lℓ=1,…,L satisfy θℓ+ℏ≠θℓ′\theta_\ell+\hbar\neq\theta_{\ell'}θℓ​+ℏ=θℓ′​ for ℓ<ℓ′\ell<\ell'ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of θℓ\theta_\ellθℓ​. In particular, we consider the regime θℓ+1/θℓ≫1\theta_{\ell+1}/\theta_{\ell}\gg 1θℓ+1​/θℓ​≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.
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.