Geometry of Discrete-Time Spin Systems

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 * 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.

Collective Symplectic Integrators on $S_2^N \times T^*\mathbb{R}^M$

“…The z-variable is integrated with the midpoint method. As shown in [MMV16], this integrator coincides with the spherical midpoint method for ray-constant vector fields (As we have here, cf. (3))…”

Clebsch canonization of Lie–Poisson systems

“…Our work gives a concrete realization of the general theory of collective integrators developed by [39]; see also [40,41]. The main advantage of collective integrators is that one can construct Lie-Poisson integrators that preserve the coadjoint orbits out of existing symplectic integrators.…”

“…One can show that this is the case with the symplectic Runge-Kutta method if, for example, one can find another momentum map J on T * Q so that the pair of momentum maps M and J constitute a dual pair, as discussed in [39,Theorem 7]. Existing constructions (see, e.g., [39,40,41]) of such momentum maps M and J are rather ad-hoc, and thus are limited to Lie-Poisson equations on relatively simple spaces such as o(p, q, F), sp(2k, F ), gl(n, F), u(p, q) with F = R, C, H, and some semi-direct products.…”

Clebsch Canonization of Lie-Poisson Systems

“…Our work is an extension of a more recent series of works on the so-called collective integrators by McLachlan et al [40,41,42]. Namely, in order to numerically solve a Lie-Poisson equation on g * , one solves the corresponding canonical Hamiltonian system on T * Q using a symplectic integrator and then map the solution to g * by M.…”

“…The main advantage of collective integrators is that one can construct a Lie-Poisson integrator out of existing symplectic integrators. On the other hand, the main disadvantage is that it is not always clear how one can find a suitable cotangent bundle T * Q and momentum map M. Existing works of McLachlan et al [40,41,42] and Ohsawa [57] are limited to Lie-Poisson equations on simple spaces such as so(3) * ∼ = su(2) * and se(3) * , and are based on ad-hoc constructions of such momentum maps M.…”