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Geometry of Discrete-Time Spin Systems

Journal article
Authors R. I. McLachlan
Klas Modin
O. Verdier
Published in Journal of Nonlinear Science
Volume 26
Issue 5
Pages 1507-1523
ISSN 0938-8974
Publication year 2016
Published at Department of Mathematical Sciences
Department of Mathematical Sciences, Mathematics
Pages 1507-1523
Language en
Links dx.doi.org/10.1007/s00332-016-9311-...
Keywords Spin systems, Heisenberg spin chain, Discrete integrable systems, Symplectic integration, Moser-Veselov, Hopf fibration, Collective symplectic integrators, Midpoint method
Subject categories Mathematical Analysis, Computational Mathematics

Abstract

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.

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