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Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

Yunfeng Jiang, Yang Zhang

Published Mar 1, 2018• 27 citations
Publisher

Abstract

In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.

DOI
10.3929/ethz-b-000254391

Related papers

Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

Yunfeng Jiang, Yang Zhang

Abstract In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar $$ \mathcal{N}=4 $$ N = 4 super-Yang-Mills theory.

arXiv:1710.04693•Journal of High Energy Physics•Mar 1, 2018•20 citations

Bethe Ansatz

Gleb Eduardovich Arutyunov

The Bethe Ansatz is a powerful method in the theory of quantum integrable models, essential for determining the energy spectrum of dynamical systems - from spin chains in magnetism to models in high-energy physics. This book provides a comprehensive introduction to the Bethe ansatz, from its historical roots to modern developments. First introduced by Hans Bethe in 1931, the method has evolved into a universal framework encompassing algebraic, analytic, thermodynamic, and functional forms. The book explores various Bethe ansatz techniques and their interrelations, covering both coordinate and algebraic versions, with particular attention to nested structures and functional relations involving transfer matrices. Advanced tools such as the separation of variables method are presented in detail. With a wealth of worked examples and precise calculations, this volume serves as an accessible and rigorous reference for graduate students and researchers in mathematical physics and integrable systems.

Cambridge University Press eBooks•Apr 27, 2026•1 citation

The Bethe ansatz

Fedor Levkovich-Maslyuk

Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.

arXiv:1606.02950•Journal of Physics A Mathematical and Theoretical•Jul 20, 2016•70 citations

Completeness of Wronskian Bethe equations for rational $\mathfrak{gl}_{m|n}$ 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 22, 2020•0 citations

Quantum integrable systems

Steve Zelditch

NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX

Regional conference series in mathematics•Dec 12, 2017•18 citations

Algebraic Bethe ansatz

Gleb Eduardovich Arutyunov

This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R -matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.

Cambridge University Press eBooks•Apr 27, 2026•0 citations
In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.
Abstract In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 \mathcal{N}=4 N=4 N = 4 super-Yang-Mills theory.
The Bethe Ansatz is a powerful method in the theory of quantum integrable models, essential for determining the energy spectrum of dynamical systems - from spin chains in magnetism to models in high-energy physics. This book provides a comprehensive introduction to the Bethe ansatz, from its historical roots to modern developments. First introduced by Hans Bethe in 1931, the method has evolved into a universal framework encompassing algebraic, analytic, thermodynamic, and functional forms. The book explores various Bethe ansatz techniques and their interrelations, covering both coordinate and algebraic versions, with particular attention to nested structures and functional relations involving transfer matrices. Advanced tools such as the separation of variables method are presented in detail. With a wealth of worked examples and precise calculations, this volume serves as an accessible and rigorous reference for graduate students and researchers in mathematical physics and integrable systems.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
Completeness of Wronskian Bethe equations for rational glm∣n\mathfrak{gl}_{m|n}glm∣n​ spin chain
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.
NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX
This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R -matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.