Quantum Spin Chains and Integrable Many-Body Systems of Classical Mechanics

  • A. ZabrodinEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 163)


This note is a review of the recently revealed intriguing connection between integrable quantum spin chains and integrable many-body systems of classical mechanics. The essence of this connection lies in the fact that the spectral problem for quantum Hamiltonians of the former models is closely related to a sort of inverse spectral problem for Lax matrices of the latter ones. For simplicity, we focus on the most transparent and familiar case of spin chains on \(N\) sites constructed by means of the GL(2)-invariant \(R\)-matrix. They are related to the classical Ruijsenaars-Schneider system of \(N\) particles, which is known to be an integrable deformation of the Calogero-Moser system. As an explicit example the case \(N=2\) is considered in detail.


Spin Chain Inverse Spectral Problem Twisted Boundary Condition Bethe Root Gaudin Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Discussions with A. Alexandrov, A. Gorsky, V. Kazakov, S. Khoroshkin, I. Krichever, S. Leurent, M. Olshanetsky, A. Orlov, T. Takebe, Z. Tsuboi, and A. Zotov are gratefully acknowledged. Some of these results were reported at the International School and Workshop “Nonlinear Mathematical Physics and Natural Hazards” (November 28–December 2 2013, Sofia, Bulgaria). The author thanks the organizers and especially professors B. Aneva and V. Gerdzhikov for the invitation and support. This work was supported in part by RFBR grant 12-01-00525, by joint RFBR grants 12-02-91052-CNRS, 14-01-90405-Ukr and grant NSh-1500.2014.2 for support of leading scientific schools.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Biochemical PhysicsMoscowRussia
  2. 2.ITEPMoscowRussia
  3. 3.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussia

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