Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How Algebraic Bethe Ansatz works for integrable model

L. D. Faddeev

TL;DR

The paper presents a comprehensive, constructive treatment of the Algebraic Bethe Ansatz for integrable lattice models, beginning with the spin-1/2 XXX chain as a concrete exemplar and systematically extending to higher spin and the XXZ model. Through the Lax formalism, R-matrix, and RTT/FCR framework, it derives Bethe Ansatz equations, analyzes the thermodynamic limit, and interprets excitations as quasiparticles with explicit dispersion and S-matrix data. It further demonstrates how continuum field theories like the nonlinear Schrödinger model and the S^2 sigma-model emerge from spin chains in suitable limits, and extends the formalism to inhomogeneous chains and discrete spacetime formulations with applications to Sine–Gordon dynamics. The work highlights the universality of ABA, the interplay with quantum groups, and outlines future directions in higher-rank generalizations, integrable field theories, and rigorous mathematical foundations.

Abstract

I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.

How Algebraic Bethe Ansatz works for integrable model

TL;DR

The paper presents a comprehensive, constructive treatment of the Algebraic Bethe Ansatz for integrable lattice models, beginning with the spin-1/2 XXX chain as a concrete exemplar and systematically extending to higher spin and the XXZ model. Through the Lax formalism, R-matrix, and RTT/FCR framework, it derives Bethe Ansatz equations, analyzes the thermodynamic limit, and interprets excitations as quasiparticles with explicit dispersion and S-matrix data. It further demonstrates how continuum field theories like the nonlinear Schrödinger model and the S^2 sigma-model emerge from spin chains in suitable limits, and extends the formalism to inhomogeneous chains and discrete spacetime formulations with applications to Sine–Gordon dynamics. The work highlights the universality of ABA, the interplay with quantum groups, and outlines future directions in higher-rank generalizations, integrable field theories, and rigorous mathematical foundations.

Abstract

I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin , anisotropy parameter , shift in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.

Paper Structure

This paper contains 14 sections, 421 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General outline of the course.
  3. XXX${}_{1/2}$ model. Description.
  4. XXX${}_{1/2}$ model. Bethe Ansatz equations.
  5. XXX${}_{1/2}$ model. Physical spectrum in the ferromagnetic thermodynamic limit
  6. XXX${}_{1/2}$ model. BAE for an arbitrary configuration
  7. XXX${}_{1/2}$ model. Physical spectrum in the antiferromagnetic case
  8. XXX${}_s$ model
  9. XXX${}_{s}$ spin chain. Applications to the physical systems
  10. XXZ model
  11. Inhomogeneous chains and discrete time shift
  12. Examples of dynamical models in discrete space-time
  13. Conclusions and perspectives
  14. Comments on the literature on BAE

Figures (3)

  • Figure 1: Discrete space time and initial saw
  • Figure 2: Zero curvature equation
  • Figure 3: Elementary plaquette