Geometry of Discrete-Time Spin Systems
Geometry of Discrete-Time Spin Systems
复制标题
离散时间自旋系统的几何
DOI：
10.1007/s00332-016-9311-z
复制
发表时间：
2015
影响因子：
3
通讯作者：
Olivier Verdier
中科院分区：
文献类型：
--
作者：
R. McLachlan;K. Modin;Olivier Verdier
经典哈密顿自旋系统是辛相空间$$（S^2）^n$（S2）n上的连续动力系统。在本文中，我们调查的基本几何的时间离散计划的经典哈密顿自旋系统称为球形中点方法。事实证明，这种方法显示了一系列有趣的几何特征，产生的见解，并提出了一般战略的几何时间离散化的非正则辛流形上的哈密顿系统。特别是，我们的研究提供了两个新的，完全几何证明，由球面中点方法获得的离散时间自旋系统保持辛性。本研究遵循两条路径。首先，我们引入了一个扩展版本的霍普夫纤维化表明，球形中点方法可以看作是起源于经典的中点方法对$$T^*\mathbf {R}^{2n}$$T <$R2n的集体哈密顿量。辛性是一个直接的几何结果。其次，我们提出了一个新的离散方案的黎曼流形称为黎曼中点方法。我们确定其性质的等距和黎曼淹没，并作为一个特殊的情况下，我们表明，球形中点的方法是这种类型的非欧几里德度量。结合凯勒几何，这提供了辛性的另一个几何证明。
Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $$(S^2)^n$$(S2)n. In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $$T^*\mathbf {R}^{2n}$$T∗R2n 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.