Journal of Nonlinear Science

, Volume 26, Issue 5, pp 1507–1523 | Cite as

Geometry of Discrete-Time Spin Systems

  • Robert I. McLachlanEmail author
  • Klas Modin
  • Olivier Verdier


Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space \((S^2)^n\). In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on \(T^*\mathbf {R}^{2n}\) for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.


Spin systems Heisenberg spin chain Discrete integrable systems Symplectic integration Moser–Veselov Hopf fibration Collective symplectic integrators Midpoint method 

Mathematics Subject Classification

37M15 65P10 70H08 70K99 93C55 



The research was supported by the J C Kempe Memorial Fund, the Swedish Foundation for Strategic Research (ICA12-0052), EU Horizon 2020 Marie Sklodowska-Curie Individual Fellowship (661482), the Swedish Foundation for International Cooperation in Research and Higher Education (PT2014-5823), and the Marsden Fund of the Royal Society of New Zealand.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  2. 2.Department of MathematicsChalmers University of TechnologyGothenburgSweden
  3. 3.Department of MathematicsUniversity of GothenburgGothenburgSweden
  4. 4.Department of ComputingMathematics and Physics, Bergen University CollegeBergenNorway

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