An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

1. Front Matter

   Pages i-xii

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2. The XY Chain

   • Fabio Franchini

   Pages 1-17

3. The Lieb-Liniger Model

   • Fabio Franchini

   Pages 19-45

4. The Heisenberg Chain

   • Fabio Franchini

   Pages 47-70

5. The XXZ Chain

   • Fabio Franchini

   Pages 71-92

6. Algebraic Bethe Ansatz

   • Fabio Franchini

   Pages 93-125

7. Back Matter

   Pages 127-180

   PDF

This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics.


The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model.  Finally, the low energy physics of these integrable models is mapped intothe corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.

• Lieb-Liniger model
• Heisenberg chain
• Yang-Baxter equations
• Toeplitz determinants
• bosonization
• XY chain
• XXZ chain
• Kitaev chain
• Bethe Ansatz
• Yan-Yang equation
• Inverse scattering problem
• Lax Representation
• Braid Limit
• Quantum Groups
• Strong Szegö theorem
• Widom's Theorem
• Fisher-Hartwig Conjecture
• Basor-Tracy Conjecture
• Slavnov's formulas
• Gaudin's formulas

“This lecture note is a good source to learn basic notions and techniques of one-dimensional integrable quantum systems.” (Wencai Liu, zbMATH 1376.82001, 2018)

• Ruđer Bošković Institute, Zagreb, Croatia

  Fabio Franchini

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