论文标题
Ising Hamiltonian具有远距离互动的特征值
Eigenvalues of Ising Hamiltonian with long-range interactions
论文作者
论文摘要
我们在超立方体晶格上获得了多维伊斯林汉密尔顿的分析特征值，并用自旋旋转相互作用常数和一维伊斯林汉密尔顿汉密尔顿（Hamiltonian）的特征值表示（后者是众所周知的）。为此，我们写下了多维的哈密顿特征向量，作为一维伊斯丁汉密尔顿人的特征向量的Kronecker产品。对于周期性的边界条件，可以考虑到与无限数量的相邻自旋的相互作用，可以获得精确的结果。在本文中，我们为平面和立方体的特征值介绍了前五个协调领域的特征值（即与最近的邻居，下一个邻居，下一个next邻居，Next-Next-Next-Next-Next-Next-Next邻居和下一个next-next-next-next-next-next-neighbors的互动）。在自由边界系统的情况下，我们表明，在二维和三维中，只有当我们分别考虑与前两个配位球和前三个配位球的相互作用时，才能获得精确的表达式。
We obtained analytically eigenvalues of a multidimensional Ising Hamiltonian on a hypercube lattice and expressed them in terms of spin-spin interaction constants and the eigenvalues of the one-dimensional Ising Hamiltonian (the latter are well known). To do this we wrote down the multidimensional Hamiltonian eigenvectors as the Kronecker products of the eigenvectors of the one-dimensional Ising Hamiltonian. For periodic boundary conditions, it is possible to obtain exact results taking into account interactions with an unlimited number of neighboring spins. In this paper, we present exact expressions for the eigenvalues for the planar and cubic Ising systems accounting for the first five coordination spheres (that is interactions with the nearest neighbors, the next neighbors, the next-next neighbors, the next-next-next neighbors and the next-next-next-next neighbors). In the case of free-boundary systems, we showed that in the two- and three-dimensions the exact expressions could be obtained only if we account for interactions with spins of first two coordination spheres and first three coordination spheres, respectively.
