论文标题
在Cayley树上的紧密结合模型的完整解决方案：强烈局部与扩展状态
Complete Solution of the Tight Binding Model on a Cayley Tree: Strongly Localised versus Extended States
论文作者
论文摘要
在带有开放边界条件的Cayley树上，最近邻居紧密绑定模型的完整集合和$ m $分支世代相传。我们发现$ n = 1 +3（2^m-1）$总状态在整个Cayley树中仅扩展了$ 3 m +1 $状态。剩下的$ n-（3 m+1）$状态被认为是强烈的定位状态，仅在一部分站点上具有有限的幅度。特别是，对于$ m> 1 $，$ 3 \ times 2^{m-2} $表面状态，它们是Cayley树表面上只有两个位点的每个反对称组合，并且始终在$ e = 0 $的情况下，是乐队中间的。发现Cayley树的基态和前两个激动的状态被发现是扩展的状态，在Cayley Tree的所有地点上都有幅度，所有$ M $。我们在整个本征态和特征值集中使用结果来得出状态的总密度和局部状态密度。
The complete set of Eigenstates and Eigenvalues of the nearest neighbour tight binding model on a Cayley tree with branching number $b=2$ and $M$ branching generations with open boundary conditions is derived. We find that of the $N= 1 +3 (2^M-1)$ total states only $3 M +1$ states are extended throughout the Cayley tree. The remaining $N-(3 M+1)$ states are found to be strongly localised states with finite amplitudes on only a subset of sites. In particular, there are, for $M>1$, $3 \times 2^{M-2}$ surface states which are each antisymmetric combinations of only two sites on the surface of the Cayley tree and have energy eactly at $E=0$, the middle of the band. The ground state and the first two excited states of the Cayley tree are found to be extended states with amplitudes on all sites of the Cayley tree, for all $M$. We use the results on the complete set of Eigenstates and Eigenvalues to derive the total density of states and a local density of states.
