Penrose镶嵌上的经典二聚体
Classical Dimers on Penrose Tilings
摘要：
研究了菱形Penrose平铺上的经典二聚体模型，菱形Penrose平铺的边和顶点可以与二部图的边和顶点等同。我们发现，彭罗斯平铺不承认完美的匹配（无缺陷的二聚体覆盖）。相反，它们的最大匹配在热力学极限中具有$81-50φ\approx 0.098$的单体密度，其中$φ=\left(1+\sqrt{5}\right)/2$是黄金比例。最大匹配将平铺划分为嵌套的封闭区域的分形，这些区域由单体不能穿过的环限定。这些环连接第二近邻的偶价顶点，每个偶价顶点都位于这样的环上。通过对每个单体的电荷与其二分子晶格固定的符号进行比较，我们发现每个有界区域都有一个多余的电荷，以及一组相应的单体，相邻区域具有相反的净电荷。无限平铺是电荷中性的。我们设计了一个简单的算法生成最大匹配，并证明了最大匹配形成一个连接的流形下的局部单体二聚体重排。我们发现，飞镖风筝彭罗斯平铺功能的不平衡的电荷之间的二分子晶格，导致一个最小的单体密度$\left(7-4φ\right)/5\approx 0.106$所有的一个电荷。
英文摘要：
We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of $81-50φ\approx 0.098$ in the thermodynamic limit, with $φ=\left(1+\sqrt{5}\right)/2$ the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings, and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements. We show that dart-kite Penrose tilings feature an imbalance of charge between bipartite sub-lattices, leading to a minimum monomer density of $\left(7-4φ\right)/5\approx 0.106$ all of one charge.
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