图上量子XY和Heisenberg系统的静动力学
Statics and Dynamics of Quantum XY and Heisenberg Systems on Graphs
摘要：
本文研究了任意N阶图G上可区分自旋1/2系统的静力学和动力学性质。特别地，我们考虑量子自旋系统按照两个哈密顿量之一演化：（i）XY哈密顿量H_XY，它包含G中由边连接的每对自旋的XY相互作用;和（ii）海森堡哈密顿量H_Heis，它包含G中由边连接的每对自旋的海森堡相互作用项。我们发现XY（海森堡）哈密顿量在状态空间上的作用等价于序列G_k（k=0，…）的邻接矩阵（组合拉普拉斯）的作用，G（其中G_1=G）导出的图的个数N。这种作用量的等价性表明这两个模型的动力学行为与图G_k上自由粒子跳跃的演化过程是一致的。因此，我们展示了如何在一个更复杂的几何结构上用更简单的动力学来取代原始自旋模型的复杂动力学。一个简单的推论，我们的方法允许我们写一个明确的谱分解的XY模型在磁场的路径上的N个顶点。我们也使用我们的方法，利用谱图理论的结果来解决新的自旋模型：XY模型和海森堡模型在一个磁场上的完整的图形。
英文摘要：
We consider the statics and dynamics of distinguishable spin-1/2 systems on an arbitrary graph G with N vertices. In particular, we consider systems of quantum spins evolving according to one of two hamiltonians: (i) the XY hamiltonian H_XY, which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg hamiltonian H_Heis, which contains a Heisenberg interaction term for every pair of spins connected by an edge in G. We find that the action of the XY (respectively, Heisenberg) hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial laplacian) of a sequence G_k, k=0,..., N of graphs derived from G (with G_1=G). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs G_k. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the XY model in a magnetic field on the path consisting of N vertices. We also use our approach to utilise results from spectral graph theory to solve new spin models: the XY model and heisenberg model in a magnetic field on the complete graph.
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