Geometry of Discrete-Time Spin Systems
Journal article, 2016

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space . 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 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 Kahler geometry, this provides another geometric proof of symplecticity.

Heisenberg spin chain

Discrete integrable systems


Spin systems

Hopf fibration

Collective symplectic integrators

Symplectic integration

Midpoint method


R. I. McLachlan

Massey University

Klas Modin

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Olivier Verdier

Hogskolen i Bergen

Journal of Nonlinear Science

0938-8974 (ISSN) 1432-1467 (eISSN)

Vol. 26 5 1507-1523

Subject Categories

Computational Mathematics

Mathematical Analysis



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