论文标题
图形和复杂性
Fractons on Graphs and Complexity
论文作者
论文摘要
我们在一般空间图上介绍了两个异国晶格模型。第一个是紧凑的LIFSHITZ标量字段的物质理论，而第二个则是一定的等级-2 $ u（1）$量表理论。两种晶格模型均通过一般图上的离散拉普拉斯运算符定义。我们在这些晶格模型的物理可观察物与图理论量之间的物理可观察物之间提出了一个有趣的对应关系。例如，物质理论的基态退化等于空间图的跨越树的数量，这是图理论中复杂性的常见度量（“ GSD =复杂性”）。离散的全局对称性被识别为图的雅各布群。在量规理论中，分形的超选择扇区与图理论中的除数类是一对一的对应关系。特别是，在空间图上的温和假设下，使用图理论的亚伯 - 雅各比图证明了分形式固定性。
We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory ("GSD = complexity"). The discrete global symmetry is identified as the Jacobian group of the graph. In the gauge theory, superselection sectors of fractons are in one-to-one correspondence with the divisor classes in graph theory. In particular, under mild assumptions on the spatial graph, the fracton immobility is proven using a graph-theoretic Abel-Jacobi map.
