Geometry of discrete-time spin systems

TL;DR

This paper explores the geometric properties of the spherical midpoint method for discretizing classical Hamiltonian spin systems, revealing its symplectic nature through novel geometric proofs and introducing a new Riemannian midpoint scheme.

Contribution

It provides two new geometric proofs of symplecticity for the spherical midpoint method and introduces a Riemannian midpoint scheme applicable to non-Euclidean metrics.

Findings

Spherical midpoint method preserves symplecticity.

Extended Hopf fibration links midpoint methods to classical schemes.

New Riemannian midpoint method generalizes geometric discretizations.

Abstract

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 K\"ahler geometry, this provides another geometric proof of symplecticity.