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Geometry of Discrete-Time Spin Systems

Artikel i vetenskaplig tidskrift
Författare R. I. McLachlan
Klas Modin
O. Verdier
Publicerad i Journal of Nonlinear Science
Volym 26
Nummer/häfte 5
Sidor 1507-1523
ISSN 0938-8974
Publiceringsår 2016
Publicerad vid Institutionen för matematiska vetenskaper
Institutionen för matematiska vetenskaper, matematik
Sidor 1507-1523
Språk en
Länkar dx.doi.org/10.1007/s00332-016-9311-...
Ämnesord Spin systems, Heisenberg spin chain, Discrete integrable systems, Symplectic integration, Moser-Veselov, Hopf fibration, Collective symplectic integrators, Midpoint method
Ämneskategorier Matematisk analys, Beräkningsmatematik

Sammanfattning

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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