By Fabian H. L. Essler
By Holger Frahm
By Frank Göhmann
By Andreas Klümper
By Vladimir E. Korepin
Publisher: Cambridge University Press
Print Publication Year: 2005
Online Publication Date:August 2009
Online ISBN:9780511534843
Hardback ISBN:9780521802628
Paperback ISBN:9780521143943
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511534843.013
Subjects: Condensed Matter Physics, Nanoscience and Mesoscopic Physics, Theoretical Physics and Mathematical Physics
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Introduction to the quantum inverse scattering method
The quantum inverse scattering method is the modern algebraic theory of exactly solvable quantum systems. It arose [404, 410, 411] as an attempt to carry over the concepts of the inverse scattering method for classical non-linear evolution equations [2, 134] into quantum mechanics. As a result, our understanding of both the theory of integrable partial differential equations and the theory of exactly solvable quantum systems changed, and the algebraic roots of the exact solvability became apparent. These roots originate from the Yang-Baxter equation and its classical counterpart.
Before turning to our actual subject, which is the application of the quantum inverse scattering method to the Hubbard model, we give a brief general introduction. We shall limit our exposition basically to the material which is needed later for the understanding of the algebraic structure of the Hubbard model. The reader who is interested in the general scope of the method and in the history of its development is referred to the excellent books and review articles [131, 270, 276, 277, 407].
Integrability
As a motivation for the definition of the Yang-Baxter algebra in the following subsection we shall first recall the concept of integrability in classical mechanics. Then, by considering the elementary example of the harmonic oscillator, we shall see that this concept does not directly apply to quantum mechanical systems and needs to be extended.
pp. i-iv
pp. v-x
pp. xi-xvi
pp. 1-19
2 - The Hubbard Hamiltonian and its symmetries : Read PDF
pp. 20-49
3 - The Bethe ansatz solution : Read PDF
pp. 50-119
4 - String hypothesis : Read PDF
pp. 120-148
5 - Thermodynamics in the Yang-Yang approach : Read PDF
pp. 149-174
6 - Ground state properties in the thermodynamic limit : Read PDF
pp. 175-208
7 - Excited states at zero temperature : Read PDF
pp. 209-267
8 - Finite size corrections at zero temperature : Read PDF
pp. 268-296
9 - Asymptotics of correlation functions : Read PDF
pp. 297-332
10 - Scaling and continuum limits at half-filling : Read PDF
pp. 333-375
11 - Universal correlations at low density : Read PDF
pp. 376-392
12 - The algebraic approach to the Hubbard model : Read PDF
pp. 393-487
13 - The path integral approach to thermodynamics : Read PDF
pp. 488-562
14 - The Yangian symmetry of the Hubbard model : Read PDF
pp. 563-598
15 - S-matrix and Yangian symmetry in the infinite interval limit : Read PDF
pp. 599-619
16 - Hubbard model in the attractive case : Read PDF
pp. 620-637
17 - Mathematical appendices : Read PDF
pp. 638-642
pp. 643-668
pp. 669-674