Lectures on Quantum Integrability: Lattices, Symmetries and Physics

Part of the CRM Series in Mathematical Physics book series (CRM)


These lectures notes are intended as a friendly introduction to the many-body physics and methods of integrability. They were presented during the ASIDE12, a school preceding the SIDE12 conference. The aimed audience of them are mathematicians with interests in integrability and physics.



I would like to thank an anonymous referee and Jacek Pawełczyk for substantial help in improving the quality of the presentation. I acknowledge financial support from the National Science Centre under FUGA grant 2015/16/S/ST2/00448 and from Centre de recherches mathématiques in Montréal.


  1. 1.
    V.I. Arnold, Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60 (Springer, Berlin, 1978)Google Scholar
  2. 2.
    R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982)MATHGoogle Scholar
  3. 3.
    J.-S. Caux, J. Mossel, Remarks on the notion of quantum integrability. J. Stat. Mech. Theory Exp. 2011, P02023 (2011)CrossRefGoogle Scholar
  4. 4.
    T. Deguchi, Introduction to solvable lattice models in statistical and mathematical physics, in Classical and Quantum Nonlinear Integrable Systems, ed. by A. Kundu. Mathematical and Computational Physics, chap. 5, pp. 107–146 (Institute of Physics, Bristol, 2003)Google Scholar
  5. 5.
    T. Deguchi, C. Matsui, Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry. Nucl. Phys. B 814(3), 405–438 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Clarendon Press, Oxford, 1959)MATHGoogle Scholar
  7. 7.
    A. Doikou, S. Evangelisti, G. Feverati, N. Karaiskos, Introduction to quantum integrability. Int. J. Modern Phys. A 25(17), 3307–3351 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper, V.E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2005)CrossRefMATHGoogle Scholar
  9. 9.
    L.D. Faddeev, How the algebraic Bethe ansatz works for integrable models, in Symétries Quantiques, ed. by A. Connes, K. Gawedzki, J. Zinn-Justin (North-Holland, Amsterdam, 1998), pp. 149–219Google Scholar
  10. 10.
    F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems. Springer Briefs in Mathematical Physics, vol. 16 (Springer, Cham, 2017)Google Scholar
  11. 11.
    M. Gaudin, La fonction d’onde de Bethe (Masson, Paris, 1983)MATHGoogle Scholar
  12. 12.
    T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004)MATHGoogle Scholar
  13. 13.
    N. Kitanine, J.-M. Maillet, V. Terras, Form factors of the XXZ Heisenberg spin-\(\frac{1} {2}\) finite chain. Nuclear Phys. B 554(3), 647–678 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1993)Google Scholar
  15. 15.
    B. Lake, D.A. Tennant, J.-S. Caux, T. Barthel, U. Schollwöck, S.E. Nagler, C.D. Frost, Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain. Phys. Rev. Lett. 111(13), 137205 (2013)Google Scholar
  16. 16.
    J.-M. Maillet, J. Sánchez de Santos, Drinfeld twists and algebraic Bethe ansatz, in L. D. Faddeev’s Seminar on Mathematical Physics, ed. by M. Semenov-Tian-Shansky. American Mathematical Society Translation Series 2, vol. 201 (American Mathematical Society, Providence, 2000), pp. 137–178Google Scholar
  17. 17.
    J.-M. Maillet, V. Terras, On the quantum inverse scattering problem. Nucl. Phys. B 575(3), 627–644 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    R.I. Nepomechie, A spin chain primer. Int. J. Modern Phys. B 13(24-25), 2973–2985 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    N.A. Slavnov, Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz. Theor. Math. Phys. 79(2), 502–508 (1989)MathSciNetCrossRefGoogle Scholar
  20. 20.
    B. Sutherland, Beautiful Models (World Scientific, Singapore, 2004)CrossRefMATHGoogle Scholar
  21. 21.
    M. Takahashi, Thermodynamical Bethe ansatz and condensed matter, in Conformal Field Theories and Integrable Models, ed. by Z. Horváth, L. Palla. Lecture Notes in Physics, vol. 498 (Springer, Berlin, 1997), pp. 204–250Google Scholar
  22. 22.
    M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, 1999)CrossRefMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Physics, University of WarsawWarszawaPoland

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