准周期Ammann-Beenker镶嵌上二聚体的统计力学
Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings
摘要：
我们研究二维准周期Ammann-Beenker（AB）镶嵌上的经典二聚体。尽管缺乏周期性，我们证明，每一个无限的瓷砖承认'完美匹配'，其中每个顶点是由一个二聚体。我们引入了一个辅助的'AB $^*$ '平铺从AB平铺删除所有8重协调顶点。AB $^*$镶嵌也是二维的、无限的和准周期的。AB $^*$镶嵌有一个连接的组件，它允许完美匹配。我们发现，在所有的完美匹配，二聚体的AB $^*$平铺躺在不相交的一维循环和梯子，分开的“膜”，设置的边缘，二聚体是缺席。结果，AB $^*$平铺的二聚体配分函数分解为沿着这些结构的二聚体配分函数的乘积。我们计算的配分函数和自由能每边的AB $^*$平铺使用解析传递矩阵的方法。回到AB平铺，我们发现，在AB $^*$平铺膜成为“伪膜”，集合的边缘，共同主机最多一个二聚体。这导致了一个显着的离散尺度不变性的匹配问题。结构表明，AB平铺应表现出高度不均匀和缓慢衰减的连接二聚体的相关性。使用蒙特卡罗模拟，我们发现证据支持这一假设的形式连接二聚体相关性符合幂律行为。在一组完美的匹配，我们发现准周期的类似物观察到的交错和柱状阶段的周期性系统。
英文摘要：
We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary 'AB$^*$' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB$^*$ tiling is again two-dimensional, infinite, and quasiperiodic. The AB$^*$ tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB$^*$ tiling lie along disjoint one-dimensional loops and ladders, separated by 'membranes', sets of edges where dimers are absent. As a result, the dimer partition function of the AB$^*$ tiling factorizes into the product of dimer partition functions along these structures. We compute the partition function and free energy per edge on the AB$^*$ tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB$^*$ tiling become 'pseudomembranes', sets of edges which collectively host at most one dimer. This leads to a remarkable discrete scale-invariance in the matching problem. The structure suggests that the AB tiling should exhibit highly inhomogenous and slowly decaying connected dimer correlations. Using Monte Carlo simulations, we find evidence supporting this supposition in the form of connected dimer correlations consistent with power law behaviour. Within the set of perfect matchings we find quasiperiodic analogues to the staggered and columnar phases observed in periodic systems.
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