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:1704.01650

Invitation to the Bethe ansatz

Reiho Sakamoto

Published Apr 5, 2017• 0 citations
arXivHTMLPDF

Abstract

We review the algebraic Bethe ansatz for the Heisenberg model. The exposition includes some of recent advancements with emphasis on a relation with the rigged configurations. We also provide somewhat thorough review of the crystal bases and the rigged configurations. In particular, we provide the inverse scattering transform for the type $D^{(1)}_n$ box-ball systems. We also provide a reformulation of a result of arXiv:0711.4185.

DOI
10.48550/arxiv.1704.01650

Related papers

On the algebraic Bethe Ansatz approach to the correlation functions of the XXZ spin-1/2 Heisenberg chain(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

N Kitanine, Jean Michel Maillet, N. A. Slavnov +1 more

We present a review of the method we have elaborated to compute the correlation functions of the $XXZ$ spin-1/2 Heisenberg chain.This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz bamework, and leads to a mul- tiple integral representation of the dynamical correlation functions.We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral.Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.

数理解析研究所講究録•Apr 1, 2006•5 citations

RIGGED CONFIGURATIONS AND THE BETHE ANSATZ

Anne Schilling

These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of this bijection for the symmetry algebra $D_n^{(1)}$, based on work in collaboration with Okado and Shimozono.

arXiv:math-ph/0210014•International School on Symmetry and Structural Properties of Condensed Matter (SSPCM)•Jul 1, 2003•5 citations

Correlation functions of the XXZ Heisenberg spin chain: Bethe ansatz approach

Jean Michel Maillet

We review recent progress in the computation of correlation functions of the XXZ spin-1/2 chain. We describe both finite and infinite chain results. Long distance asymptotic behavior is discussed. Our method is based on the resolution of the quantum inverse scattering problem in the framework of the algebraic Bethe ansatz.

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006•May 15, 2007•1 citation

Heisenberg model and rigged configurations

Pulak Ranjan Giri, Tetsuo Deguchi

We show a correspondence of all the solutions of the Bethe ansatz equations of the one-dimensional spin-1/2 isotropic Heisenberg model with periodic boundary conditions for N = 12 to the rigged configurations. This mapping is based on the comparison of the set of Takahashi quantum numbers arranged in lexicographical order with the set of riggings of the rigged configurations arranged in co-lexicographical order.

arXiv:1501.07801•Journal of Statistical Mechanics Theory and Experiment•Jul 8, 2015•3 citations

CORRELATION FUNCTIONS OF THE XXZ SPIN-½ HEISENBERG CHAIN: RECENT ADVANCES (REVIEW)

N Kitanine, Jean Michel Maillet, N. A. Slavnov +1 more

We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.

International Journal of Modern Physics A•May 1, 2004•5 citations

The Bethe Ansatz in magnetism: An introduction (invited)

Michael Fowler

After reviewing the basic ideas of the Bethe Ansatz using the simplest possible model—the one-dimensional interacting boson system of Leib and Liniger—we discuss the magnetic systems for which it has been used successfully. These include spin one-half chains, the Hubbard model, and the Kondo problem.

Journal of Applied Physics•Mar 1, 1982•4 citations
We review the algebraic Bethe ansatz for the Heisenberg model. The exposition includes some of recent advancements with emphasis on a relation with the rigged configurations. We also provide somewhat thorough review of the crystal bases and the rigged configurations. In particular, we provide the inverse scattering transform for the type Dn(1)D^{(1)}_nDn(1)​ box-ball systems. We also provide a reformulation of a result of arXiv:0711.4185.
We present a review of the method we have elaborated to compute the correlation functions of the XXZXXZXXZ spin-1/2 Heisenberg chain.This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz bamework, and leads to a mul- tiple integral representation of the dynamical correlation functions.We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral.Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.
These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of this bijection for the symmetry algebra Dn(1)D_n^{(1)}Dn(1)​, based on work in collaboration with Okado and Shimozono.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.