有限自旋网络的多重分形几何
Multi-fractal Geometry of Finite Networks of Spins
摘要：
量子自旋网络通过将信息编码为自旋自由度来克服传统基于电荷的电子学的挑战。虽然与基于电荷的对应物相比，自旋网络有利于以最小的损失传输信息，但由于其复杂的缩放特性，自旋网络中信息传播的数学形式化具有挑战性。在本文中，我们提出了一个几何方法-特定于有限网络-解开自旋链和环的信息理论现象，通过抽象它们作为加权图，其中的顶点对应于自旋激发态和边缘代表节点对之间的信息理论距离。量子自旋网络动力学的加权图表示呈现出复杂的自相似结构（其中子图在不同的空间尺度上在一定程度上重复）。为了量化这种复杂的行为，我们开发了一种新的盒计数启发算法，该算法评估量子自旋网络的单分形与多重分形特性。除了特定于有限网络之外，多重分形还通过“工程”或“偏置”网络以进行选择性传输来进一步复合，因为选择性使网络更加异构。为了证明有限尺寸系统的临界性，我们定义了一个热力学框架来描述信息传播，并证明了一些自旋链和环表现出信息相变现象，类似于有限介质中安德森局域化中的金属到绝缘体相变。
英文摘要：
Quantum spin networks overcome the challenges of traditional charge-based electronics by encoding the information into spin degrees of freedom. Although beneficial for transmitting information with minimal losses when compared to their charge-based counterparts, the mathematical formalization of the information propagation in a spin(tronic) network is challenging due to its complicated scaling properties. In this paper, we propose a geometric approach---specific to finite networks---for unraveling the information-theoretic phenomena of spin chains and rings by abstracting them as weighted graphs, where the vertices correspond to the spin excitation states and the edges represent the information theoretic distance between pair of nodes. The weighted graph representation of the quantum spin network dynamics exhibits a complex self-similar structure (where subgraphs repeat to some extent over various space scales). To quantify this complex behavior, we develop a new box counting inspired algorithm which assesses the mono-fractal versus multi-fractal properties of quantum spin networks. Besides specific to finite networks, multi-fractality is further compounded by "engineering" or "biasing" the network for selective transfer, as selectivity makes the network more heterogeneous. To demonstrate criticality in finite size systems, we define a thermodynamics inspired framework for describing information propagation and show evidence that some spin chains and rings exhibit an informational phase transition phenomenon, akin to the metal-to-insulator phase transition in Anderson localization in finite media.}
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