发表机构
Institute for Condensed Matter Physics, Technical University of Darmstadt（达姆施塔特工业大学凝聚态物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究一维和二维紧束缚模型中幂律分布耦合的局域化性质，通过数值对角化比较拉普拉斯与邻接矩阵，发现拉普拉斯矩阵因守恒律产生系统跨越模式，且二维中耦合指数超0.75时最大弛豫时间模式局域化。
AI 中文摘要
我们通过数值对角化不同系统尺寸和连通性下的拉普拉斯矩阵和邻接矩阵，比较其谱和本征模的局域化性质，研究了具有幂律分布耦合的一维和二维紧束缚模型中的局域化特性。这两种矩阵与不同类型的动力学过程相关。虽然对于足够大的系统尺寸，邻接矩阵的所有本征模都是局域化的，但由于守恒定律，拉普拉斯矩阵总是导致一小部分跨越系统的模式，从而在参与比率的概率分布及其与特征值的关系中出现幂律尾部。在一维中，这些幂律的指数随表征耦合分布的指数连续变化。在二维中，当耦合分布的指数大于0.75时，具有最大弛豫时间的模式从跨越系统变为局域化。我们为所有这些发现提供了现象学解释。
英文摘要
We study the localization properties of 1D and 2D tight-binding models with power-law distributed couplings by comparing the spectrum and the localization properties of the eigenmodes of the Laplacian and the adjacency matrix, using numerical diagonalization of these matrices for different system sizes and connectivities. These two matrices are relevant for different types of dynamical processes. While all eigenmodes of the adjacency matrix are localized for sufficiently large system sizes, the Laplacian matrix always leads to a small proportion of system-spanning modes due to a conservation law, and therefore to power-law tails in the probability distribution of the participation ratio and its relation to the eigenvalues. In one dimension, the exponent of these power laws change continuously with the exponent that characterizes the distribution of couplings. In two dimensions, the modes with the largest relaxation times change from system-spanning to localized when the exponent of the distribution of couplings becomes larger than 0.75. We provide phenomenological explanations for all these findings.
