How Algebraic Bethe Ansatz works for integrable model

TL;DR

This paper explains the Algebraic Bethe Ansatz method in detail using the spin 1/2 XXX magnetic chain as a primary example, and discusses its application to various integrable models including quantum field theories.

Contribution

It provides a detailed explanation of the Algebraic Bethe Ansatz technique and extends its application to a broad class of integrable models with different parameters.

Findings

Detailed step-by-step solution for the spin 1/2 XXX chain

Extension of the method to models with varying spin, anisotropy, and shifts

Connection to soliton theory and relativistic quantum field models

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.