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I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article.

Related MathOverflow question: what-is-an-integrable-system.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

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Do you have anything specific in mind? I think the term integrability is sometimes used in slightly different contexts. – Pieter Naaijkens Sep 22 '11 at 12:25
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The fact that "integrability" can mean so many things sometimes makes the quest to learn about it so challenging! I have found the introductory sections of Etingoff's paper www-math.mit.edu/~etingof/zlecnew.pdf to be a very good mathematical reference for a particular, physically interesting system (Calogero-Moser) which describes particles interacting in one-dimension. – Eric Zaslow Sep 24 '11 at 17:28
up vote 21 down vote accepted

I take "integrable models" to mean "exactly solvable models in statistical physics".

You can take a look at the classic book

Otherwise this new book is quit readable and covers more than just solvable models

Others can probably give you more mathematician-friendly references, but I think it would be good if you could be more specific about what you are looking for.

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Yes, "exactly solvable" is what I mean. Thanks, I will update my question later. – Phira Sep 22 '11 at 14:35
    
Baxter is a dead link. – ja72 Aug 28 '14 at 20:11

This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.

  • Another good recent book: Maciej Dunajski, Solitons, Instantons and Twistors.

  • Some very good reviews on Quantum inverse scattering and Algebraic Bethe Ansatz:

    • Faddeev: How Algebraic Bethe Ansatz works for integrable model

    • Kulish and Sklyanin: Quantum Spectral Transform Method. Recent Developments

    • Takhtajan: Introduction to algebraic Bethe Ansatz

    and the Books:

    • Jimbo and Miwa: Algebraic Analysis of Solvable Lattice Models

    • Korepin et al: Quantum inverse Scattering and Correlation Functions

    • Korepin et al: The One-Dimensional Hubbard Model

    plus the article

    • Martins and Ramos: The Quantum Inverse Scattering Method for Hubbard-like Models

Here are some more

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My references are very good reviews:

Quantum inverse scattering and Algebraic Bethe Ansatz:

Faddeev: How Algebraic Bethe Ansatz works for integrable model

Kulish and Sklyanin: Quantum Spectral Transform Method. Recent Developments

Takhtajan: Introduction to algebraic Bethe ansatz

and the Books:

Jimbo and Miwa: Algebraic Analysis of Solvable Lattice Models

Korepin et al: Quantum inverse Scattering and Correlation Functions

Korepin et al: The One-Dimensional Hubbard Model

plus the article

Martins and Ramos: The Quantum Inverse Scattering Method for Hubbard-like Models

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