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How Algebraic Bethe Ansatz works for integrable model

Lyudvig Dmitrievich Faddeev

Published May 26, 1996• 236 citations
OpenAlex

Abstract

In my Les–Houches lectures of 1982 I described the inverse scattering method of solving the integrable field–theoretical models in 1+1 dimensional space–time. Both classical case, stemming from the famous paper by Gardner, Green, Kruskal and Miura of 1967 on KdV equation, and its quantum counterpart, developed mostly by

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How Algebraic Bethe Ansatz works for integrable model

Lyudvig Dmitrievich Faddeev

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.

arXiv:hep-th/9605187•May 26, 1996•509 citations

Final Conclusion

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

We have explained the theory of exactly solvable models on both the quantum and classical levels. We hope that the reader appreciates the beauty and perfection of the Bethe Ansatz. The theory of completely integrable nonlinear partial differential equations plays an equally important role in the book. It is deeply related to the Bethe Ansatz. We hope that our book can be used for further development of the theory of exactly solvable models (we think that the interested reader can find problems to work out in our book). Our book does not close the theory of the Bethe Ansatz; it opens new possibilities for further development. We have explained how to construct eigenfunctions of the Hamiltonian and how to describe the ground state and its excitations. We have also evaluated dispersion curves and scattering matrices for the excitations. The thermodynamics of exactly solvable models is explained in this book in all the details, including the evaluation of temperature correlation functions (even if they depend on real time). The problem of the evaluation of correlation functions (for Bethe Ansatz solvable models) is solved in our book, using the example of the one-dimensional Bose gas (quantum nonlinear Schrödinger equation). It was done using the ideas of the inverse scattering method (the Lax representation, the Riemann-Hilbert problem, the Gel'fand-Levitan-Marchenko equation, etc.). We would like also to emphasize that all other Bethe Ansatz solvable models are constructed in a similar way.

Cambridge University Press eBooks•Aug 19, 1993•0 citations

HOW THE ALGEBRAIC BETHE ANSATZ WORKS FOR INTEGRABLE MODELS

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Integrable Kondo impurity in one-dimensional q-deformed $t-J$ models

Xiangyu Ge, M. D. Gould, Jon Links +1 more

Integrable Kondo impurities in two cases of the one-dimensional q-deformed $t-J$ models are studied by means of the boundary ${\bf Z}_2$-graded quantum inverse scattering method. The boundary $K$ matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, these models are solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.

arXiv:cond-mat/9908191•Aug 13, 1999•2 citations

Algebraic Bethe ansatz for U(1) invariant integrable models: The method and general results

C.S. Melo, Marcio Jose Martins

arXiv:0806.2404•Nuclear Physics B•Jul 31, 2008•13 citations

Integrable Kondo impurity in one-dimensionalq-deformedt-Jmodels

Xiangyu Ge, M. D. Gould, Jon Links +1 more

Integrable Kondo impurities in two cases of one-dimensional q -deformed t - J models are studied by means of the boundary Z 2 -graded quantum inverse scattering method. The boundary K matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, these models are solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.

Journal of Physics A Mathematical and General•Oct 8, 2001•6 citations
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 sss, 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.
We have explained the theory of exactly solvable models on both the quantum and classical levels. We hope that the reader appreciates the beauty and perfection of the Bethe Ansatz. The theory of completely integrable nonlinear partial differential equations plays an equally important role in the book. It is deeply related to the Bethe Ansatz. We hope that our book can be used for further development of the theory of exactly solvable models (we think that the interested reader can find problems to work out in our book). Our book does not close the theory of the Bethe Ansatz; it opens new possibilities for further development. We have explained how to construct eigenfunctions of the Hamiltonian and how to describe the ground state and its excitations. We have also evaluated dispersion curves and scattering matrices for the excitations. The thermodynamics of exactly solvable models is explained in this book in all the details, including the evaluation of temperature correlation functions (even if they depend on real time). The problem of the evaluation of correlation functions (for Bethe Ansatz solvable models) is solved in our book, using the example of the one-dimensional Bose gas (quantum nonlinear Schrödinger equation). It was done using the ideas of the inverse scattering method (the Lax representation, the Riemann-Hilbert problem, the Gel'fand-Levitan-Marchenko equation, etc.). We would like also to emphasize that all other Bethe Ansatz solvable models are constructed in a similar way.
Integrable Kondo impurity in one-dimensional q-deformed t−Jt-Jt−J models
Integrable Kondo impurities in two cases of the one-dimensional q-deformed t−Jt-Jt−J models are studied by means of the boundary Z2{\bf Z}_2Z2​-graded quantum inverse scattering method. The boundary KKK matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, these models are solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.