复杂量子网络的几何结构：演化与相变
Complex Quantum Network Geometries: Evolution and Phase Transitions
摘要：
网络是拓扑和几何结构，用于描述不同的系统，如互联网，大脑或时空的量子结构。在这里，我们定义了复杂的量子网络几何，描述了不断增长的单纯2-复合体的基本结构，即由三角形形成的单纯复合体。这些网络是几何网络，其链接的能量根据非平衡动力学增长。几何网络的时间演化是描述定义量子网络状态演化的路径积分的给定路径的经典演化。量子网络状态的特征在于量子占用数，可以分别映射到网络的每个链路的节点、链路和三角形。我们把描述量子网络态演化的几何网络称为量子几何网络。量子几何网络具有复杂网络的小世界性、高聚集系数、高模块性、无标度分布等特性，并且可以区分为分别服从Fermi-Dirac统计和Bose-Einstein统计的Fermi-Dirac网络和Bose-Einstein网络。我们表明，这些网络可以进行结构相变的网络的几何性质发生急剧变化。最后对量子复网络几何、自旋网络和三角剖分之间的关系进行了评述。
英文摘要：
Networks are topological and geometric structures used to describe systems as different as the Internet, the brain or the quantum structure of space-time. Here we define complex quantum network geometries, describing the underlying structure of growing simplicial 2-complexes, i.e. simplicial complexes formed by triangles. These networks are geometric networks with energies of the links that grow according to a non-equilibrium dynamics. The evolution in time of the geometric networks is a classical evolution describing a given path of a path integral defining the evolution of quantum network states. The quantum network states are characterized by quantum occupation numbers that can be mapped respectively to the nodes, links, and triangles incident to each link of the network. We call the geometric networks describing the evolution of quantum network states the quantum geometric networks. The quantum geometric networks have many properties common to complex networks including small-world property, high clustering coefficient, high modularity, scale-free degree distribution.Moreover they can be distinguished between the Fermi-Dirac Network and the Bose-Einstein Network obeying respectively the Fermi-Dirac and Bose-Einstein statistics. We show that these networks can undergo structural phase transitions where the geometrical properties of the networks change drastically. Finally we comment on the relation between Quantum Complex Network Geometries, spin networks and triangulations.
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