任意图上的C类网络模型
Network Models in Class C on Arbitrary Graphs
摘要：
我们考虑量子局域化的网络模型，其中具有双分量波函数的粒子通过节点并沿着任意有向图的边缘传播，在其穿过的每个边缘上受到随机SU（2）旋转。通过每个节点的传播由任意但固定的S-矩阵指定。这样的网络模型的本地化问题在C类的分类Altland和Zirnbauer，并在适当的图形，他们的自旋量子霍尔过渡模型。我们扩展了Gruzberg，Ludwig和Read以及Beamond，Cardy和Chalker的分析，表明在任意图上，平均态密度和平均电导可以根据同一图上经典历史相关随机行走的可观测量来计算。该过程的转移权重与S矩阵的元素显式相关。它们被正确地归一化，但是在度大于4的节点的图上，不一定是非负的（因此可以解释为概率），除非它们中有足够数量的节点恰好消失。我们的方法使用一个超对称的路径积分制定的问题是完全有限的和严格的。
英文摘要：
We consider network models of quantum localisation in which a particle with a two-component wave function propagates through the nodes and along the edges of an arbitrary directed graph, subject to a random SU(2) rotation on each edge it traverses. The propagation through each node is specified by an arbitrary but fixed S-matrix. Such networks model localisation problems in class C of the classification of Altland and Zirnbauer, and, on suitable graphs, they model the spin quantum Hall transition. We extend the analyses of Gruzberg, Ludwig and Read and of Beamond, Cardy and Chalker to show that, on an arbitrary graph, the mean density of states and the mean conductance may be calculated in terms of observables of a classical history-dependent random walk on the same graph. The transition weights for this process are explicitly related to the elements of the S-matrices. They are correctly normalised but, on graphs with nodes of degree greater than 4, not necessarily non-negative (and therefore interpretable as probabilities) unless a sufficient number of them happen to vanish. Our methods use a supersymmetric path integral formulation of the problem which is completely finite and rigorous.
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